NONEQUILIBRIUM STATISTICAL
SPECIAL
SYSTEMS
SPECIALSECTION:
SECTION:
Persistence in nonequilibrium systems
Satya N. Majumdar
Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India
This is a brief review of recent theoretical efforts to
understand persistence in nonequilibrium systems.
THE problem of persistence in spatially extended nonequilibrium systems has recently generated a lot of interest
both theoretically1–7 and experimentally8–10. Persistence is
simply the probability that the local value of the fluctuating
nonequilibrium field does not change sign up to time t. It
has been studied in various systems, including several
models undergoing phase separation1–4,11–15, the simple
diffusion equation with random initial conditions5,6, several
reaction diffusion systems in both pure16 and disordered17
environments, fluctuating interfaces18–20, Lotka–Volterra
models of population dynamics21, and granular media22.
The precise definition of persistence is as follows. Let
φ(x, t) be a nonequilibrium field fluctuating in space and time
according to some dynamics. For example, it could represent
the coarsening spin field in the Ising model after being
quenched to low temperature from an initial high
temperature. It could also be simply a diffusing field starting
from random initial configuration or the height of a
fluctuating interface. Persistence is simply the probability
P0(t) that at a fixed point in space, the quantity sgn[φ(x, t) –
⟨φ(x, t)⟩] does not change up to time t. In all the examples
mentioned above this probability decays as a power law
P0(t) ~ t–θ at late times, where the persistence exponent θ is
usually nontrivial.
In this article, we review some recent theoretical efforts in
calculating this nontrivial exponent in various models and
also mention some recent experiments that measured this
exponent. The plan of the paper is as follows. We first
discuss the persistence in very simple single variable
systems. This makes the ground for later study of
persistence in more complex many-body systems. Next, we
consider many-body systems such as the Ising model and
discuss where the complexity is coming from. We follow it
up with the calculation of this exponent for a simpler manybody system namely diffusion equation and see that even
in this simple case, the exponent θ is nontrivial. Next, we
show that all these examples can be viewed within the
general framework of the ‘zero crossing’ problem of a
Gaussian stationary process (GSP). We review the new
results obtained for this general Gaussian problem in
various special cases. Finally, we mention the emerging new
directions towards different generalizations of persistence.
We start with a very simple system namely the onedimensional Brownian walker. Let φ(t) represent the position
of a 1-D Brownian walker at time t. This is a single-body
system in the sense that the field φhas no x dependence but
only t dependence. The position of the walker evolves as,
dφ
= η(t ),
dt
(1)
where η(t) is a white noise with zero mean and delta
correlated, ⟨φ(t)φ(t′)⟩ = δ(t – t′). Then persistence P0(t) is
simply the probability that φ(t) does not change sign up to
time t, i.e. the walker does not cross the origin up to time t.
This problem can be very easily solved exactly by writing
down the corresponding Fokker–Planck equation with an
absorbing boundary condition at the origin23. The
persistence decays as P0(t) ~ t–1/2 and hence θ = 1/2. The
important point to note here is that the exact calculation is
possible here due to the Markovian nature of the process in
eq. (1). Note that φ evolves according to a first order
equation in time, i.e. to know φ(t), we just need the value of
φ(t – ∆t) but not on the previous history. This is precisely
the definition of a Markov process.
In order to make contact with the general framework to be
developed in this article, we now solve the same process by
a different method. We note from eq. (1) that η(t) is a
Gaussian noise and eq. (1) is linear in φ. Hence, φ is also a
Gaussian process with zero mean and a two-time correlator,
⟨φ(t)φ(t′)⟩ = min(t, t') obtained by integrating eq. (1). We
recall that a Gaussian process can be completely
characterized by just the two-time correlator. Any higher
order correlator can be simply calculated by using Wick’s
theorem. Since min(t, t′) depends on both time t and t′ and
not just on their difference |t – t′|, clearly φ is a Gaussian
non-stationary process. From the technical point of view,
stationary processes are often preferable to non-stationary
processes. Fortunately there turns out to be a simple
transformation by which one can convert this nonstationary process into a stationary one. It turns out that
this transformation is more general and will work even for
more complicated examples to follow. Therefore we illustrate
it in detail for the Brownian walker problem in the following
paragraph.
The transformation works as follows. Consider first the
normalized process, X (t ) = φ(t ) / ⟨φ2 (t )⟩Then,
X(t) is also
.
e-mail: satya@theory.tifr.res.in
370
t ′) / (tt′ ) .
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
a Gaussian process with zero mean and its two-time
correlator is given by, ⟨X(t)X(t′)⟩ = min(t,
Now we
define a new ‘time’ variable, T = log(t). Then, in this new
time variable T, the two-time correlator becomes,
⟨X(T )X(T ′)⟩ = exp(– |T – T ′|/2) and hence is stationary in T.
1
3
Thus, the persistence
problem reduces
to calculating the
2
2
probability P0(T ) of no zero crossing of X(T ), a GSP
characterized by its two-time correlator, ⟨X(T )X(T ′)⟩ =
exp(– |T – T ′|/2).
3
One could,
of course, ask the same question for an
2
arbitrary GSP with a given correlator ⟨X(T )X(T ′)⟩ = f (|T –
T ′|) [in case of Brownian motion, f (T ) = exp(– T / 2)]. This
general zero crossing problem of a GSP has been studied by
mathematicians for a long time24. Few results are known
exactly. For example, it is known that if f (T ) < 1/T for large
T, then P0(T ) ~ exp(–µT ) for large T. Exact result is known
only
for
Markov
GSP
which
are characterized by purely exponential correlator,
f (T ) = exp(–λT ). In that case, P0(T ) = (2/π) sin–1
[exp(–λT )] (ref. 24). Our example of Brownian motion
corresponds to the case when λ= 1/2 and therefore the
persistence P0(T ) ~ exp(– T/2) for large T. Reverting to the
original time using T = log(t), we recover the result, P0(t) ~ t–
1/2
. Thus the inverse of the decay rate in T becomes the
power law exponent in t by virtue of this ‘log–time’
transformation. Note that when the correlator f (T ) is
different from pure exponential, the process is nonMarkovian and in that case no general answer is known.
Having described the simplest one-body Markov process, we now consider another one-body process which
however is non-Markovian. Let φ(t) (still independent of x)
now represent the position of a particle undergoing random
acceleration,
d2φ
dt 2
= η( t),
(2)
where η(t) is a white noise as before. What is the
probability P0(t) that the particle does not cross zero up to
time t? This problem was first proposed in the review article
by Wang and Uhlenbeck25 way back in 1945 and it got
solved only very recently in 1992, first by Sinai26, followed
by Burkhardt27 by a different method. The answer is,
P0(t) ~ t–1/4 for large t and the persistence exponent is
θ = 1/4. Thus even for this apparently simple looking
problem, the calculation of θ is nontrivial. This nontriviality
can be traced back to the fact that this process is nonMarkovian. Note that eq. (2) is a second order equation and
to know φ(t + ∆t), we need to know its values at two
previous points φ(t) and φ(t – ∆t). Thus, it depends on two
previous steps as opposed to just the previous step as in
eq. (1). Hence it is a non-Markovian process.
We notice that eq. (2) is still linear and hence φ(t) is still a
Gaussian process with a non-stationary correlator.
However, using the same T = log(t) transformation as
defined in the previous paragraph, we can convert this to
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
the zero crossing problem in time T of a GSP with
correlator, f (T ) = exp(–T/2) – exp(– 3T/2). Note that
this is different from pure exponential and hence is nonMarkovian. We also notice another important point: It is not
correct to just consider the asymptotic form of
f (T ) ~ exp(–T/2) and conclude that the exponent is
therefore 1/2. The fact that the exponent is exactly 1/4,
reflects that the ‘no zero crossing’ probability P0(T )
depends very crucially on the full functional form of f (T )
and not just on its asymptotic form. This example thus
illustrates the history dependence of the exponent θ which
makes its calculation nontrivial.
Having discussed the single particle system, let us now
turn to many body systems where the field φ(x, t) now has x
dependence also. The first example that was studied is
when φ(x, t) represents the spin field of one-dimensional
Ising model undergoing zero temperature coarsening
dynamics, starting from a random high temperature
configuration. Let us consider for simplicity a discrete
lattice where φ(i, t) = ± 1 representing Ising spins. One
starts from a random initial configuration of these spins.
The zero temperature dynamics proceeds as follows: at
every step, a spin is chosen at random and its value is
updated to that of one of its neighbours chosen at random
and then time is incremented by ∆t and one keeps repeating
this process. Then persistence is simply the probability that
a given spin (say at site i ) does not flip up to time t. Even in
one dimension, the calculation of P0(t) is quite nontrivial.
Derrida et al.2 solved this problem exactly and found
P0(t) ~ t–θ
for
large
t
with
θ = 3/8.
They also generalized this to q-state Potts model in
1-D and found an exact formula, θ(q) = – + [cos–1
{(2 – q)/ q}]2 for all q.
This calculation however cannot be easily extended to
d = 2 which is more relevant from an experimental point of
2
view. Early numerical results indicated that the18 exponent
θ
2
π
~ 0.22 (ref.
2 3) for d = 2 Ising model evolving with zero
temperature spin flip dynamics. It was therefore important to
have a theory in d = 2 which, if not exact, at least could give
approximate results. We will discuss later about our efforts
towards such an approximate theory of Ising model in
higher dimensions. But before that let us try to understand
the main difficulties that one encounters in general in manybody systems.
In a many-body system, if one sits at a particular point x
in space and monitors the local field φ(x, t) there as a
function of t, how would this ‘effective’ stochastic process
(as a function of time only) look like? If one knows enough
properties of this single site process as a function of time,
then the next step is to ask what is the probability that this
stochastic process viewed from x as a function of t, does
not change sign up to time t. So, the general strategy
involves two steps: first, one has to solve the underlying
many-body dynamics to find out what the ‘effective’ single
site process looks like and second, given this single site
process, what is its no zero crossing probability.
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SPECIAL SECTION:
Before discussing the higher dimensional Ising model
where both of these steps are quite hard, let us discuss a
simple example (which however is quite abundant in nature),
namely the diffusion equation. This is a many-body system
but at least the first step of the two-step strategy can be
carried out exactly and quite simply. The second step
cannot
be
carried
out
exactly
even
for
this simple example, but one can obtain very good
approximate results.
Let φ(x, t) (which depends on both x and t) denote field
that is evolving via the simple diffusion equation,
∂φ
= ∇ 2 φ.
∂t
(3)
This equation is deterministic and the only randomness is in
the initial condition φ(x, 0) which can be chosen as a
Gaussian random variable with zero mean. For example,
φ(x, t) could simply represent the density fluctuation,
φ(x, t) = ρ(x, t) – ⟨ρ⟩ of a diffusing gas. The persistence, as
usual, is simply the probability that φ(x, t) at some x does
not change sign up to time t. This classical diffusion
equation is so simple that it came as a surprise to find that
even in this case, the persistence P0(t) ~ t–θ numerically with
nontrivial θ ≈ 0.1207, 0.1875, 0.2380 in d = 1, 2 and 3,
respectively.
In the light of our previous discussion, it is however easy
to see why one would expect a nontrivial answer even in
this simple case. Since the diffusion equation (3) is linear,
the field φ(x, t) at a fixed point x as a function of t is clearly a
Gaussian process with zero mean and is simply given by
ρ ρ
the solution of eq. (3), φ(x, t) = ∫ d dx′ G( x − x ′, t)φ(x′, 0),
ρ
where G( x , t) = (4πt)–d/2 exp[– x2/4t] is the Green's function
in d. Note that by solving eq. (3), we have already reduced
the many-body diffusion problem to an ‘effective’ singlesite Gaussian process in time t at fixed x. This therefore
completes the first step of the two-step strategy mentioned
earlier exactly. Now we turn to the second step, namely the
‘no zero crossing’ probability of this single-site Gaussian
process. The two-time correlator of this can be easily
computed from above and turns out to be non-stationary as
in the examples specified by eqs (1) and (2). However by
using the T = log(t) transformation as before, the normalized
field reduces to a GSP in time T with correlator,
⟨X(T1)X(T2)⟩ = [sech(T/2)]d/2, where T = |T1–T2|. Thus, once
again, we are back to the general problem of the zero
crossing of a GSP, this time with a correlator
f(T) = [sech(T/2)]d/2 which is very different from pure exponential form and hence is non Markovian. The persistence,
P0(T ) will still decay as P0(T ) ~ exp(– θT ) ~ t–θ for large T
(since f (T ) decays faster than 1/T for large T ) but clearly
with a nontrivial exponent.
Since persistence in all the examples that we have
discussed so far (except the Ising model) reduces to the
zero crossing probability of a GSP with correlator f (T )
[where f (T ) of course varies from problem to problem], let
us now discuss some general properties of such a process.
372
It turns out that a lot of information can already be inferred
by
examining
the
short-time
properties
of
the correlator f (T ). In case of Brownian motion, we
found f(T) = exp(–T/2) ~ 1 – T/2 + O(T2) for small T. For
the random acceleration problem, f (T ) = exp(– T/2) –
exp(– 3T/2) ~ 1 – 3T 2/8 + O(T 3) for small T and for
the diffusion problem, f (T ) = [sech(T/2)]d/2 ~ 1 –
T2
3
α
+ O(T ) as T → 0. In general f (T ) = 1 – aT + . . . for small
T, where 0 < α ≤ 2 (ref. 24). It turns out that processes for
which α= 2 are ‘smooth’ in the sense that the density of
zero crossings ρ is finite, i.e. the number of zero crossings
of the process in a given time T scales linearly with T.
Indeed there exists an exact formula due to Rice28, ρ =
when α= 2. However, for α< 2, f″ (0) does not
exist and this formula breaks down. It turns out that the
density is infinite for α< 2 and once the process crosses
zero, it immediately crosses many times and then makes a
long excursion before crossing the zero again. In other
words, the zero’s are not uniformly distributed over a given
interval and in general the set of zeros has a fractal
structure.29
Let us first consider ‘smooth’ processes with α= 2 such
as random acceleration or the diffusion problem. It turns out
that for such processes, one can make very good progress
in calculating the persistence exponent θ.
The first approach consists of using an ‘independent
interval approximation’ (IIA)5. Consider the ‘effective’
single-site process φ(T ) as a function of the ‘log–time’
T = log(t). As a first step, one introduces the ‘clipped’
variable σ= sgn(φ), which changes sign at the zeros of φ(T ).
Given that φ(T ) is a Gaussian process, it is easy to compute
the correlator, A(T ) = ⟨σ(0)σ(T )⟩ = sin–1 [ f (T )], π2where
f (T ) is the correlator of φ(T ). Since the ‘clipped’ process
σ(T ) can take values ± 1 only, one can express A(T ) as,
A(T ) =
∞
∑ ( −1) n Pn (T ),
(4)
n= 0
where Pn(T ) is the probability that the interval T contains n
zeros of φ(T ). So far, there is no approximation. The strategy
next is to use the following approximation,
Pn (T ) = ⟨T ⟩ −1 ∫ dT1 ∫ 2 dT 2 Λ
T
T
0
T1
∫T
T
n −1
dT n ×
Q (Tn ) P (T2 − T1 ) Λ P(Tn − Tn −1 ) Q (T − Tn ), (5)
where P(T ) is the distribution of intervals between two
successive zeros and Q(T ) is the probability that an interval
of size T to the right or left of a zero contains no further
zeros. Clearly, P(T ) = – Q′(T ). ⟨T ⟩ = 1/ρ is the mean
interval size. We have made the IIA by writing the joint
distribution of n successive zero-crossing intervals as the
product of the distribution of single intervals. The rest is
straightforward5. By taking the Laplace transform of the
~
above equations, one finally obtains, P (s) = [2 – F
(s)]/F (s), where
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
3
2
1
2
NONEQUILIBRIUM STATISTICAL SYSTEMS
1
~
F (s ) = 1 +
s[1 − s A( s )],
2ρ
(6)
~
where the Laplace transform A (s) of A(T ) can be easily
computed knowing f (T ). The expectation that the
persistence, P0(T ) and hence the interval distribution, P(T )
~
~ exp(– θT ) for large T, suggests a simple pole in the P (s)
at s = – θ. The exponent θ is therefore given by the first
zero on the negative s axis of the function,
F (s ) = 1 +
1
s
2ρ
2s ∞


−1
 1 − ∫0 d T exp( − sT ) sin [ f (T )]  .
π


(7)
For the diffusion equation, f(T) = [sech(T/2)]d/2 and
ρ=
We then get the IIA estimates of
θ = 0.1203, 0.1862 and 0.2358 in d = 1, 2 and 3 respectively,
which should be compared with the simulation
values,
0.1207 ± 0.0005,
0.1875 ± 0.0010
and
0.2380 ± 0.0015. For the
random acceleration problem, f(T) = exp(–T/2) – exp(– 3T/2) and ρ = /2π and
we get,
d/8θπiia2=. 0.2647 which can be compared with its exact
value, θ = 1/4.
Though the IIA approach produces excellent results
when compared to numerical simulations, it cannot however
be systematically improved. For this purpose, we turn to the
3
1 6
‘series expansion’
approach
which can be improved
3
2
2
systematically order by order. The idea is to consider the
generating function,
P( p , t ) =
by Bendat30. We have computed the third moment as well6.
For example, for 2-D diffusion equation, we get the series,
θ( p = 1 − ε) =
P0 (T ) =
2∫
where Pn(t) is the probability of n zero crossings in time t of
the ‘effective’ single-site process. For p = 0, P(0, t) is the
usual persistence, decaying as t–θ(0) as usual. Note that we
have used θ(0) instead of the usual notation θ, because it
turns out 6 that for general p, P(p, t) ~ t–θ (p) for large t, where
θ(p) depends continuously on p for ‘smooth’ Gaussian
processes. This has been checked numerically as well as
within IIA approach6. Note that for p = 1, P(1, t) = 1
implying θ(1) = 0. For smooth Gaussian processes, one can
then derive an exact series expansion of θ(p) near p = 1.
Writing p n = exp(n log p) and expanding the exponential, we
then obtain an expansion in terms of moments of n, the
number of zero crossings,
∞
(log p ) r
r
(9)
⟨n ⟩ c ,
r
!
r =1
r
where ⟨n ⟩ c are the cumulants of the moments. Using p = 1 –
ε, we express the right hand side as a series in powers of ε.
Fortunately, the computation of the moments of n is
relatively straightforward, though tedious for higher
moments. We have already mentioned the result of Rice for
the first moment. The second moment ⟨n 2⟩ was computed
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
φ >0
D φ(τ) exp[ − S ]
∫ D φ(τ) exp[ − S ]
=
Z1
,
Z0
(11)
(8)
n= 0
log P ( p , t ) = ∑
(10)
Keeping terms up to second order and putting ε = 1 (in the
same spirit as ε expansion in critical phenomena) gives,
θ(0) = (π+ 4)/4π2 = 0.180899. . ., just 3.5% below the
simulation value, θsim = 0.1875 ± 0.001. Thus, this gives us a
systematic series expansion approach for calculating the
persistence exponent for any smooth Gaussian process.
Note that both the above approaches (IIA and series
expansion) are valid only for ‘smooth’ Gaussian processes
(α= 2) with finite density ρ of zero crossings. What about
the nonsmooth processes where 0 < α< 2, where such
approaches fail? Even the Markov process, for which
f (T ) = exp(– λT ) is a non-smooth process with α= 1.
However, for the Markov case, one knows that the
persistence exponent θ = λ exactly. One expects therefore
that for Gaussian processes which may be non-smooth but
‘close’ to a Markov process, it may be possible to compute
θ by perturbing around the Markov result.
In order to achieve this, we note that the persistence
P0(T ) in stationary time T, can be written formally4 as the
ratio of two path integrals,
∞
∑ p n Pn ( t),
1
1  2
 1
3
ε+ 2 −
 ε + O (ε ) .
2π
4π 
π
where Z1 denotes the total weight of all paths which never
crossed zero, i.e. paths restricted to either positive or
negative (which accounts for the factor 2) side of φ= 0 and
Z0 denotes the weight of all paths completely unrestricted.
1 T T
Here S =
φ(τ1 ) G (τ1 − τ2 ) φ(τ2 ) d τ1 dτ 2
2 ∫0 ∫0
is the ‘action’ with G(τ1–τ2) being the inverse matrix of the
Gaussian correlator f (τ1–τ2). Since P0(T ) is expected to
decay as exp(– θT ) for large T, we get,
1
log P0 (T ).
T →∞ T
θ = − lim
(12)
If we now interpret the time T as inverse temperature β, then
θ = E1 – E0, where E1 and E0 are respectively the ground
states of two ‘quantum’ problems, one with a ‘hard’ wall at
the origin and the other without the wall.
For concreteness, first consider the Markov process,
f (T ) = exp(– λ|T |). In this case, it is easy to see that S is the
action of a harmonic oscillator with frequency λ. The
ground state energy, E0 = λ/2 for an unrestricted oscillator
with frequency λ. Whereas, for an oscillator with a ‘hard’
wall at the origin, it is well known that E1 = 3λ/2. This then
reproduces the Markovian result, θ = E1 – E0 = λ. For
processes close to Markov process, such that f (T ) = exp(–
λT ) + ε f1(T ), where ε is small, it is then straightforward to
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SPECIAL SECTION:
λT ) + ε f1(T ), where ε is small, it is then straightforward to
carry out a perturbation expansion around the harmonic
oscillator action in orders of ε (ref. 4). The exponent θ, to
order ε, can be expressed as,


2λ ∞
θ = λ1 − ε
f1 (T )[1 − exp( − 2λT )] −3 / 2 dT  .
∫


π 0


(13)
At this point, we go back momentarily to the zero
temperature Glauber dynamics of Ising model. Note that the
spin at a site in the Ising model takes values either 1 or – 1
at any given time. Therefore, one really cannot consider the
single-site process s(t) as a Gaussian process. However,
one can make a useful approximation in order to make
contact with the Gaussian processes discussed so far. This
is achieved by the so-called Gaussian closure
approximation, first used by Mazenko31 in the context
of phase ordering kinetics. The idea is to write,
s(t) = sgn[φ(t)], where φ(t) now is assumed to be Gaussian.
This is clearly an approximation. However, for phase
ordering kinetics with nonconserved order parameter, this
approximation has been quite accurate31. Note that within
this approximation, the persistence or no flipping probability of the Ising spin s(t) is same as the no zero crossing
probability of the underlying Gaussian process φ(t).
Assuming φ(t) to be a Gaussian process, one can compute
its two-point non-stationary correlator self-consistently.
Then, using the same ‘log–time’ transformation (with
T = log(t)) mentioned earlier, one can evaluate the
corresponding stationary correlator f (T ). We are thus back
to the general problem of zero crossing of a GSP even for
the Ising case, though only approximately.
In 1-D, the correlator f (T ) of the underlying process can
be computed exactly, f (T ) =
2 /(1 + exp( 2 | T |(ref.
)) 4)
and in higher dimensions, it can be obtained numerically as
the solution of a closed differential equation. By expanding
around, T = 0, we find that in all dimensions, α= 1 and
hence they represent non-smooth processes with infinite
density of zero crossings. Hence, we cannot use IIA or
series expansion result for θ. Also due to the lack of a small
parameter, we cannot think of this process as ‘close’ to a
Markov process and hence cannot use the perturbation
result. However, since θ = E1 – E0 quite generally and since
α= 1, we can use a variational approximation to estimate E1
and E0. We use as trial Hamiltonian that of a harmonic
oscillator whose frequency λ is our tunable variational
parameter4. We just mention the results here, the details can
be found in Majumdar and Sire, and Sire et al.4. For example,
in d = 1, we find θ ≈ 0.35 compared to the exact result
θ = 3/8. In d = 2 and 3, we find θ ≈ 0.195 and 0.156. The
exponent in 2-D has recently been measured experimentally9
in a liquid crystal system which has an effective Glauber
dynamics and is in good agreement with our variational
prediction.
So far we have been discussing about the persistence of
374
a single spin in the Ising model11. This can be immediately
generalized to the persistence of ‘global’ order parameter in
the Ising model. For example, what is the probability that the
total magnetization (sum of all the spins) does not change
sign up to time t in the Ising model? It turns out that when
quenched to zero temperature, this probability also decays
as a power law
with an exponent θg that is different
from the single spin persistence exponent θ. For example, in
1-D, θg = 1/4 exactly11 as opposed to θ = 3/8 (ref. 2). A
natural interpolation between the local and global
persistence can be established via introducing the idea of
‘block’ persistence. The ‘block’ persistence is the
probability p l(t) that a block of size l does not flip up to time
t. As l increases from 0 to ∞, the exponent crosses over
from its ‘local’ value θ to its ‘global’ value θg.
When quenched to the critical temperature Tc of the Ising
model, the local persistence decays exponentially with time
due to the flips induced by thermal fluctuations but the
‘global’
persistence
still
decays
algebraically,
~ t −θ c , where the exponent θc is a new non-equilibrium
critical exponent11. It has been computed in mean field
theory, in the n →∞ limit of the O(n) model, to first order in
ε = 4 – d expansion11. Recently this epsilon expansion has
been carried out to order ε 2 (ref. 12).
Recently, the persistence of a single spin has also been
generalized to persistence of ‘patterns’ in the zero
temperature dynamics of 1-D Ising or more generally q-state
Potts model. For example, the survival probability of a given
‘domain’ was found to decay algebraically in time as
t −θ d where the q-dependent exponent θd(2) ≈ 0.126 (ref.
(ref.~ 14),
14) for q = 2 (Ising case), different from θ = 3/8 and θ0 = 1/4.
Also, the probability that a ‘domain’ wall has not
encountered any other domain wall up to time t was found
to decay as
with yet another~ new
t −θ1 exponent θ1(q),
where θ1(2) = 1/2 and θ1(3) ≈ 0.72 (ref. 15). Thus, it seems
that there is a whole hierarchy of nontrivial exponents
associated with the decay of persistence of different
patterns in phase ordering systems.
Another direction of generalization has been to
investigate the ‘residence time’ distribution, whose limiting
behaviour determines the persistence exponent32. Consider
the effective single-site stochastic process φ(t) discussed in
this paper. Let r(t) denote the fraction of time the process
φ(t) is positive (or negative) within time window [0, t]. The
distribution f (r, t) of the random variable r is the residence
time distribution. In the limits r → 0 or r → 1, this
distribution is proportional to usual persistence. However,
the full function f (r, t) obviously gives more detailed
information about the process than its limiting behaviours.
This quantity has been studied extensively for diffusion
equation32,33, Ising model34, Le’vy processes35, interface
models20 and generalized Gaussian Markov processes36.
The various persistence probabilities in pure systems
have recently been generalized to systems with disorder17.
For example, what is the probability that a random walker in
a random environment (such as in Sinai model) does not
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
cross the origin? Analytical predictions for the persistence
in disordered environment have been made recently based
on an asymptotically exact renormalization group
approach17.
Another important application of some of these
persistence ideas, experimentally somewhat more relevant
perhaps, is in the area of interface fluctuations18,19. The
persistence in Gaussian interfaces such as the Edwards–
Wilkinson model, the problem can again be mapped to a
general GSP but with a non-Markovian correlator18. In this
case, several upper and lower bounds have been obtained
analytically18. For nonlinear interfaces of KPZ types, one
has to mostly resort to numerical means19. The study of
history dependence via persistence has provided some
deeper insights into the problems of interface
fluctuations19,20.
On the experimental side, the persistence exponent has
been measured in systems with breath figures8, soap
bubbles10 and twisted nematic liquid crystal exhibiting
planar Glauber dynamics9. It has also been noted recently37
that persistence exponent for diffusion equation may
possibly be measured in dense spin-polarized noble gases
(Helium-3 and Xenon-129) using NMR spectroscopy and
imaging38. In these systems, the polarization acts like a
diffusing field. With some modifications these systems may
possibly also be used to measure the persistence of
‘patterns’ discussed in this paper.
In conclusion, persistence is an interesting and
challenging problem with many applications in the area of
nonequilibrium statistical physics. Some aspects of the
problem have been understood recently as reviewed here.
But there still exist many questions and emerging new
directions open to more theoretical and experimental efforts.
1. Derrida, B., Bray, A. J. and Godrèche, C., J. Phys. A, 1994, 27,
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L., Ben-Naim, E. and Redner, S., Phys. Rev. E, 1994, 50, 2474.
2. Derrida, B., Hakim, V. and Pasquier, V., Phys. Rev. Lett., 1995,
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and Yekutieli, I., Physica, 1995, D214, 396.
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9. Yurke, B., Pargellis, A. N., Majumdar, S. N. and Sire, C., Phys.
Rev. E, 1997, 56, R40.
10. Tam, W. Y., Zeitak, R., Szeto, K. Y. and Stavans, J., Phys. Rev.
Lett., 1997, 78, 1588.
11. Majumdar, S. N., Bray, A. J., Cornell, S. J. and Sire, C., Phys.
Rev. Lett., 1996, 77, 3704.
12. Oerding, K., Cornell, S. J. and Bray, A. J., Phys. Rev. E, 1997,
56, R25; Oerding, K. and van Wijland, F ., J. Phys. A, 1998, 31,
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13. Cueille, S. and Sire, C., J. Phys. A, 1997, 30, L791; Euro. Phys.
J. B, 1999, 7, 111.
14. Krapivsky, P. L. and Ben-Naim, E., Phys. Rev. E, 1998, 56,
3788.
15. Majumdar, S. N. and Cornell, S. J., Phys. Rev. E, 1998, 57, 3757.
16. Cardy, J., J. Phys. A, 1995, 28, L19; Ben-Naim, E., Phys. Rev.
E, 1996, 53, 1566; Howard, M., J. Phys. A, 1996, 29, 3437;
Monthus, C., Phys. Rev. E, 1996, 54, 4844.
17. Fisher, D. S., Le Doussal, P. and Monthus, C., Phys. Rev. Lett.,
1998, 80, 3539; cond-mat/9811300; Le Doussal, P. and
Monthus, C., cond-mat/9901306, (to appear).
18. Krug, J., Kallabis, H., Majumdar, S. N., Cornell, S. J., Bray, A. J.
and Sire, C., Phys. Rev. E, 1997, 56, 2702.
19. Kallabis, H. and Krug, J., cond-mat/9809241, (to appear).
20. Toroczkai, Z., Newman, T. J. and Das Sarma, S., cond-mat/
9810359, (to appear).
21. Frachebourg, L., Krapivsky, P. L. and Ben-Naim, E., Phys. Rev.
Lett., 1996, 77, 2125; Phys. Rev. E, 1996, 54, 6186.
22. Swift, M. R. and Bray, A. J., cond-mat/9811422, (to appear).
23. Feller, W., Introduction to Probability Theory and its
Applications, Wiley, New York, 3rd edn, 1968, vol. 1.
24. Slepian, D., Bell Syst. Tech. J., 1962, 41, 463.
25. Wang, M. C. and Uhlenbeck, G. E., Rev. Mod. Phys., 1945, 17,
323.
26. Sinai, Y. G., Theor. Math. Phys., 1992, 90, 219.
27. Burkhardt, T . W., J. Phys. A, 1993, 26, L1157.
28. Rice, S. O., Bell Syst. Tech. J., 1944, 23, 282; 1945, 24, 46.
29. Kac, M., SIAM Rev., 1962, 4, 1; Blake, I. F. and Lindsey, W. C.,
IEEE Trans. Inf. Theory, 1973, 19, 295.
30. Bendat J. S., Principles and Applications of Random Noise
Theory, Wiley, New York, 1958.
31. Mazenko, G. F., Phys. Rev. Lett., 1989, 63, 1605.
32. Dornic, I. and Godrèche, C., J. Phys. A, 1998, 31, 5413.
33. Newman, T. J. and Toroczkai, Z., Phys. Rev. E, 1998, 58,
R2685.
34. Drouffe , J-M. and Godrèche, C., cond-mat/9808153, (to
appear).
35. Baldassarri, A., Bouchaud, J. P., Dornic, I. and Godrèche, C.,
cond-mat/9805212, (to appear).
36. Dhar, A. and Majumdar, S. N., Phys. Rev. E., condmat/9902004, (to appear).
37. Walsworth , R. L. (private communication).
38. Tseng, C. H., Peled, S., Nascimben, L., Oteiza, E., Walsworth,
R. L. and Jolesz, F. A., J. Magn. Reson., Ser. B, (to appear).
ACKNOWLEDGEMENTS. I thank my collaborators C. Sire, A. J.
Bray, S. J. Cornell, J. Krug, H. Kallabis, B. Yurke, A. Pargellis and A.
Dhar. I also thank D. Dhar for many valuable suggestions and
discussions. I am grateful to M. Barma, B. Derrida and C. Godrèche
for useful discussions and to CNRS, Universite’ Paul Sabatier for
hospitality where the whole series of work began.
375
SPECIAL SECTION:
Kinetics of phase ordering
Sanjay Puri
School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India
We briefly review our current understanding of phase
ordering kinetics, viz. the far-from-equilibrium evolution
of a homogeneous two-phase mixture which has been
rendered thermodynamically unstable by a rapid change in
parameters. In particular, we emphasize the large number
of unsolved problems in this area.
THERE now exists a good understanding of phase
transitions in physical systems. It has long been known that
a particular physical system (e.g. water) can exist in more
than one phase (e.g. vapour, liquid or solid) depending
upon values of external parameters (e.g. pressure,
temperature). In statistical physics, one formulates
microscopic or macroscopic models for such systems,
which have different states as free-energy minima in
different regions of the phase diagram. Calculations with
these models are not necessarily straightforward but there
are few conceptual hurdles left in understanding the static
aspects of phase transitions.
Recent attention has turned to the dynamics of phase
transitions, and this article provides an overview of a
particularly important problem in this area. We will consider
the evolution of a homogeneous two-phase mixture which
has been rendered thermodynamically unstable by a sudden
change in parameters. The evolution of the homogeneous
system towards its new equilibrium state is referred to as
‘phase ordering dynamics’ and has been the subject of
intense
experimental,
numerical
and
theoretical
investigation1. In this article, we focus upon the successes
and outstanding challenges of research in this area.
This article is organized as follows. There are two
prototypical problems of phase ordering dynamics. The
ordering of a ferromagnet or evolution with a nonconserved
order parameter is discussed first. The phase separation of a
binary mixture or evolution with a conserved order
parameter is discussed next. Then, we briefly discuss future
directions for studies of phase ordering dynamics.
H = −J
∑ Si S j
⟨ i, j⟩
− h∑ Si ,
S i = ± 1,
(1)
i
where S i is the z-component of the spin at site i. We
consider the simplest case of a two-state spin so that
S i = ± 1 in dimensionless units. In eq. (1), J(> 0) is the
strength of the exchange interaction which makes it
energetically preferable for neighbouring spins to align
parallel; and h is a magnetic field pointing along the zdirection. The notation Σ refers to a sum over nearest⟨ i, j ⟩
neighbour pairs.
The Ising model in eq. (1) has been a paradigm for
understanding phase transitions in a ferromagnet. With
minimal effort (using mean-field theory2), we can obtain
qualitative features of the phase diagram. Figure 1 shows
the spontaneous magnetization M as a function of
temperature T for a ferromagnet in zero magnetic field
(h = 0). Of course, various behaviours around the critical
point (T = Tc , h c = 0) are not captured correctly in meanfield theory, but that will not concern us here.
We are interested in the following dynamical problem. A
disordered ferromagnet at temperature TI > Tc is rapidly
cooled to a temperature TF < Tc . Clearly, the ferromagnet
would now be in equilibrium in a spontaneously-magnetized
state, with spins pointing either ‘up’ or ‘down’. This is our
first prototypical phase ordering problem, i.e. the far-fromequilibrium evolution of the disordered initial condition to
the ordered final state. The appropriate order parameter
Case with nonconserved order parameter
Consider a ferromagnet, which is an assembly of atoms with
residual spin angular momentum. Due to an exchange
interaction, spins at neighbouring sites tend to align parallel
to each other. This is encapsulated in the simple Ising
Hamiltonian defined on a lattice:
e-mail: puri@jnuniv.ernet.in
376
Figure 1. Spontaneous magnetization M as a function of
temperature T for a ferromagnet in zero field (h = 0). We consider
temperature quenches from TI > Tc to TF < Tc at time t = 0.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
which labels the state of the system is the spontaneous
magnetization, which takes a value 0 in the disordered state;
and ± M0 in the ordered state. There is no constraint on the
evolution of the order parameter with time – hence, this
evolution is referred to as the case with nonconserved order
parameter.
To obtain a dynamical model for this evolution, we again
consider the Ising model. Unfortunately, this model has no
intrinsic dynamics, as is seen by constructing the
appropriate Poisson bracket. This problem is circumvented
by introducing a stochastic kinetics, which is presumed to
result from thermal fluctuations of a heat-bath at
temperature T. The simplest nonconserved kinetics is the
so-called Glauber spin-flip kinetics, where a randomlychosen spin is flipped as S i → – S i. The change in
configuration is accepted with a probability which must
satisfy the detailed balance condition3. This condition
ensures that the system evolves towards statistical
equilibrium. The Ising model, in conjunction with Glauber
spin-flip kinetics, constitutes a reasonable microscopic
model for phase ordering dynamics with a nonconserved
order parameter.
We can obtain an equivalent model at the macroscopic
ρ
level, in terms of a continuum magnetization field ψ( r , t ).
This field is obtained by coarse-graining the microscopic
ρ
spins, with r and t being space and time variables. The
appropriate dynamical equation for nonconserved ordering
is the time-dependent Ginzburg–Landau (TDGL) equation or
Model A (ref. 4):
ρ
ρ
∂ψ( r , t)
δF [ψ( r , t )]
ρ
= −L
+ σ( r , t ),
ρ
∂t
δψ( r , t )
(2)
where L is an Onsager coefficient. The free-energy
ρ
functional F [ψ(r , t)] is usually taken to be of the form:
ρ
ρ a
ρ
b ρ
F [ψ( r , t )] = ∫ d r  ψ(r , t ) 2 + ψ( r , t ) 4
4
2
ρ
ρ
K
ρ

− h ψ( r , t ) + [ ∇ψ( r , t )] 2  ,
2

(3)
where a ∝ (T – Tc ), b, h and K are phenomenological
ρ
constants. The Gaussian white noise σ(r , t) in eq. (2) must
be chosen to satisfy the fluctuation–dissipation relation.
Eq. (2) can be interpreted in either of two ways. Firstly, it
can be thought of as a generalized Newton’s equation in the
overdamped limit with ψ as ‘coordinates’; F [ψ] as the
‘potential’; and L–1 as the ‘friction constant’. Secondly, at a
more formal level, it can be obtained from a master equation
formulation for the Ising model with spin-flip kinetics3.
Though all our statements are in the context of an
ordering ferromagnet, it should be stressed that the above
modelling is applicable to a wide range of physical systems,
with appropriate changes in nomenclature and/or
generalizations of the Hamiltonian or free-energy functional.
Furthermore, in most phase ordering problems, thermal
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
noise is asymptotically irrelevant and the ordering dynamics
is governed by a zero-temperature fixed point. At an
intuitive level, this can be understood as follows.
Essentially, thermal fluctuations affect only the interfaces
between domains and have no effect on the bulk domain
structure. However, the interfacial structure is asymptotically irrelevant in comparison with the growing domain
length scale. Thus, thermal noise does not alter the
asymptotic behaviour of phase ordering systems.
Equations (2) and (3) are the starting points for most
analytical studies of this problem, which have been
reviewed by Bray1. Essentially, theoretical approaches
assume that there are well-defined domains of ‘up’ and
‘down’ phases, with sharp interfaces between them. The
dynamical evolution of these interfaces is governed by the
local curvature. This interface-dynamics formulation was
used by Ohta et al.5 (OJK) to obtain (a) the growth law for
the characteristic domain size L(t) ~ t1/2; and (b) the
dynamical-scaling form for the time-dependent correlation
function
ρ
ρ ρ
ρ
 r 
G ( r , t) = ⟨ψ( R, t )ψ( R + r , t) ⟩ ≡ g 
 .
 L (t ) 
(4)
In eq. (4), the angular brackets denote an average over
initial conditions and thermal fluctuations; and g(x) is a
time-independent master function, which characterizes the
domain morphology of the evolving system. In physical
terms, the dynamical-scaling property reflects the fact that
the coarsening morphology is self-similar in time. Thus, the
only change with time is in the scale of the morphology.
The experimentally relevant quantity inρ ordering systems is
the structure factor
which is
S (the
k , t ),
Fourier transform
ρ
of G (r , t). Scattering experiments on phase ordering
systems give an amplitude proportional to the structure
factor.
An important extension of the OJK result is due to Oono
and Puri6, who incorporated the nonuniversal effects of
nonzero interfacial thickness into the analytical form for the
correlation function. This extension was of considerable
experimental and numerical relevance because the nonzero
interfacial thickness has a severe impact on the tail of the
structure factor.
ρ
The vector version of the TDGL equation, with ψ(r , t)
ρ ρ
replaced by an n-component vector ψ(r , t), is also of great
experimental relevance. For example, the n = 2 case
(dynamical XY model) is relevant in the ordering of
superconductors, superfluids and liquid crystals. The n = 3
case (dynamical Heisenberg model) is also of relevance in
the ordering of liquid crystals; and even in the evolution
dynamics of the early universe!! Bray and Puri7 and
(independently) Toyoki8 have used a defect-dynamics
approach to solve the ordering problem for the ncomponent TDGL equation in d-dimensional space for
arbitrary n and d (with n ≤ d). They have demonstrated that
the characteristic length scale L(t) ~ t1/2 in this case also.
377
Σ
SPECIAL SECTION:
i
Furthermore, they also obtained an explicit scaling form for
the correlation function. The analytical results of Bray and
Puri7 and Toyoki8 have stimulated much experimental and
numerical work.
We should stress that the case with n > d is unusual in
that there are no topological defects, and it is not possible
to characterize the evolution of the system in terms of the
annealing of ‘defects’. To date, there are no general
analytical results available for the case of the n-component
TDGL equation with n > d.
Thus, it would be fair to say that we have a good
understanding of phase ordering dynamics in the
nonconserved case. However, we should stress that the
defect-dynamics approach discussed above is essentially
mean-field like and valid only when d → ∞. There are
important corrections in the finite-dimensional case, which
we do not yet clearly understand. Nevertheless, most
researchers in this area would agree that the nonconserved
problem is ‘well understood’. Perhaps this optimistic
evaluation is a result of comparison with the rather bleak
picture which emerges when we consider the dynamics of
phase separation.
Case with conserved order parameter
Consider next a binary mixture of atoms A and B, with
similar atoms attracting and dissimilar atoms repelling each
other. This is a fairly ubiquitous situation in metallurgy and
materials science. Again, we assume that the system is
defined on a lattice, whose sites are occupied by either Aor B-atoms; and there are NA(NB) atoms of A(B). We further
assume that there are only nearest-neighbour interactions,
as in the case of a ferromagnet. Then, we can formulate a
Hamiltonian for the binary mixture as follows:
H = εAA
∑ niA n Aj + εBB ∑ niB n Bj
⟨ i, j ⟩
+ ε AB
∑
⟨ i, j ⟩
⟨i , j ⟩
( n iA n Bj
+
n iB n Aj ),
(5)
where we have introduced occupation-number variables
n αi = 1 or 0, depending on whether or not a site i is occupied
by an α-atom. The quantities εAA and εBB (both less than 0)
refer to the attractive energy of an A–A and B–B pair,
respectively. The quantity εAB (> 0) refers to the repulsive
energy of an A–B pair.
It is straightforward to transform the Hamiltonian in eq.
(5) into the Ising model by introducing spin variables
S i = + 1 or – 1 if site i is occupied by A or B, respectively.
Clearly, n Ai = (1 + S i)/2 and n Bi = (1 – S i)/2, which transforms
the Hamiltonian in eq. (5) into the Ising Hamiltonian of eq.
(1) (ref. 2). Again, it is simple to perform a mean-field
calculation with this Hamiltonian but one has to work in a
‘fixed-magnetization’ ensemble, as
S i = NA – NB is fixed
by the composition of the mixture. The phase diagram for a
378
symmetric binary mixture is shown in Figure 2. There is a
high-temperature disordered phase, where A and B are
homogeneously mixed; and a low-temperature ordered
phase, where A and B prefer to phase separate.
The corresponding dynamical problem considers a
homogeneous binary mixture in the one-phase region,
which is rendered thermodynamically unstable by a rapid
quench below the coexistence curve. This is our second
prototypical phase ordering problem and pictures of the
resultant evolution will be shown later. The appropriate
order parameter in this case is the local density difference
ρ
ρ
ρ
ρ
ψ( r , t) =ρA ( r , t) – ρB ( r , t) , where ρα ( r , t) denotes the
ρ
density of species α at point r at time t. In contrast to the
ordering dynamics of the ferromagnet, the order parameter
evolution in this case must satisfy a local conservation
constraint as phase separation proceeds by the diffusion of
A- and B-atoms. Hence, this evolution is said to be
characterized by a conserved order parameter. Typically, the
coarsening dynamics proceeds via the slow process of
evaporation of A-rich (or B-rich) domains; diffusion of this
material through domains rich in the other component B (or
A); and condensation on A-rich (or B-rich) domains
elsewhere.
A reasonable microscopic model for phase separation
associates a suitable stochastic kinetic process with the
Ising model, as before. The simplest conserved kinetics is
the Kawasaki spin-exchange process, which interchanges
spins at neighbouring sites. This simple model has been the
basis of Monte Carlo (MC) simulations of phase separation
dynamics.
We can also formulate a phenomenological model for the
ρ
dynamical evolution of the order parameter ψ( r , t ), which
is the local density difference of the two species. The freeenergy functional in eq. (3) is still reasonable as it
corresponds to a coarse-grained description of the Ising
model. Currents are set up in the phase-separating system
ρ
due to gradients in the chemical potential µ( r , t) as
follows:
ρ ρ
ρ
ρ
ρ ρ
J ( r , t ) = − M (ψ)∇ µ(r , t ) + η( r , t)
ρ
ρ  δF [ψ( r , t )]  ρ ρ
= − M (ψ)∇ 
ρ
 + η( r , t) .
 δψ( r , t) 
(6)
In eq. (6), M(ψ) is the mobility and we have introduced a
ρ ρ
vector Gaussian white noise η( r , t) , which models thermal
fluctuations in the current. The evolution of the order
parameter is obtained from the continuity equation as
(7)
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
∂ψ( r , t)
∂t
ρ
= −∇ ⋅ J ( r , t)
ρ
ρ 
ρ  δF [ψ( r , t )]  

 ρ ρ ρ
= −∇ ⋅  M (ψ) ∇ 
ρ
  − ∇ ⋅ η( r , t) .

 δψ( r , t)  SYSTEMS
NONEQUILIBRIUM
STATISTICAL
Equation (7) is well known in the literature as the Cahn–
Hilliard–Cook (CHC) equation (or Model B (ref. 4)) and
describes phase separation in binary mixtures when
hydrodynamic effects are not relevant, e.g. binary alloys.
For binary fluid mixtures, hydrodynamic effects play a
crucial role and eq. (7) has to be coupled with the Navier–
Stokes equation for the fluid velocity field – the extended
model is described by the Kawasaki equations and is
referred to as Model H (ref. 4).
There have been numerous experimental studies of phase
separation in binary alloys and binary fluid mixtures1,9,10.
These studies found that there is a growing characteristic
domain length scale L(t), which asymptotically exhibits a
power-law behaviour in time, i.e. L(t) ~ tφ, where φ is the
growth exponent. The value of the exponent is φ= 1/3 for
binary alloys9; and φ= 1 for binary fluids10. These
experiments have also investigated the scaling form for the
structure factor and its various features, which we do not
enumerate here.
There have also been many numerical studies of phase
separation in binary mixtures (alloys or fluids) and these are
in agreement with the experimental results quoted above.
The most extensive simulations to date are due to Shinozaki
and Oono11, who used the coarse-grained Cell Dynamical
System models developed earlier by Oono and Puri12.
However, it is not the purpose of this article to review the
numerous experimental and numerical results in this field.
Rather, we would like to focus on the current analytical
understanding of the phase separation problem. It is
relatively easy to analytically obtain the growth exponents
for phase-separating binary alloys and fluids1. However,
there has been only limited progress in developing a good
theory for the scaling form of the correlation function or
Figure 2. Phase diagram of a binary mixture AB in the (c, T )plane, where c refers to the concentration of (say) A. We consider
quenches from the one-phase (disordered) region to the two-phase
(ordered) region. There are two different possibilities for the
evolution of the system. If A and B are present in approximately
equal proportions (cf. quench labelled as ‘u’), the system is
spontaneously unstable and segregates via ‘spinodal decomposition’.
If one of the components is present in a much larger fraction (cf.
quench labelled as ‘m’), the mixture separates via ‘nucleation and
growth’ of critical droplets.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
structure factor. Typically, approaches based on interface
dynamics have not worked well in the conserved case
because of the long-range correlations between movements
of interfaces. Sadly enough, the most complete analytical
work in this context is still the classic work of Lifshitz and
Slyozov13, who considered the limiting case of phase
separation in binary alloys when one of the components is
present in a vanishingly small fraction.
Therefore, unlike the nonconserved case, there remain
many challenging analytical problems in understanding
domain growth in the conserved case!! We understand
various limiting behaviours of the scaled structure factor for
the conserved problem, but there is still no comprehensive
theory which gives the entire structure factor.
Future directions
We have discussed in some detail the two prototypical
problems of phase ordering dynamics, viz. the
nonconserved and conserved cases. These two simple
problems provide the launch-pad for a range of further
studies, as we will briefly elucidate here. Recall that the
earlier discussions were in the context of pure and isotropic
systems. Of course, real experimental systems are neither
pure nor isotropic. Recent research in this area has
attempted to incorporate and study various experimentally
relevant effects in phase ordering systems. In what follows,
we discuss some of these recent directions.
Phase ordering systems typically contain disorder, either
quenched or annealed. Quenched (or immobile) disorder is
in the form of large impurities, which act as pinning centres
for domain interfaces. Thus, the coarsening of domains is
driven by a curvature-reduction mechanism only for a
transient period. This is followed by a crossover to a regime
in which domains can grow only by thermally-activated
hopping over disorder traps. The presence of quenched
disorder drastically changes the nature of the asymptotic
domain growth law, but does not appreciably alter the
domain morphology of the evolving system14. Domain
growth with quenched disorder has received considerable
attention in the literature and there are still many issues to
be clarified in this context.
Another class of important problems concerns the role of
annealed (or mobile) disorder. Let us consider two
particularly important classes of annealed disorder, viz.
surfactants and vacancies. Surfactants are amphiphilic
molecules which reduce the surface tension between two
immiscible fluids (e.g. oil and water) and promote mixing.
Consider a phase-separating binary fluid with a small
concentration of surfactants. (This would constitute a
ternary or three-component mixture.) Typically, the
surfactants migrate rapidly to interfacial regions –
diminishing the surface tension and, thereby, the drive to
segregate. Thus, a binary fluid with surfactants can exhibit a
range of fascinating meso-scale structures15. Vacancies in
binary alloys also play a similar role as surfactants16.
However, we should stress that phase separation in binary
379
SPECIAL SECTION:
should stress that phase separation in binary alloys is
mediated by vacancies rather than direct A–B interchanges.
Therefore, vacancies are necessary for the phase separation
of binary alloys and are not just a complicating feature in
the phase diagram!!
At a coarse-grained level, phase ordering in ternary
mixtures is described in terms of coupled dynamical equations for two conserved order parameters, referred to as
Model D (ref. 4). We could also consider the problem of
ordering of a ferromagnet with vacancies and this would be
described by coupled dynamical equations for one nonconserved order parameter (i.e. spontaneous magnetization)
and one conserved order parameter (i.e. atom-vacancy
density-difference field). These equations are referred to as
Model C in the classification of Hohenberg and Halperin4.
Next, we consider the role of anisotropies in phase
ordering systems. Many interesting physical situations give
rise to anisotropic phase ordering dynamics. Thus,
anisotropy can result from external fields, e.g. originating
from a surface with a preferential attraction for one of the
components of a phase-separating binary mixture. We have
studied the fascinating dynamical interplay of two timedependent phenomena in this problem, viz. dynamics of
surface wetting by the preferred component; and dynamics
of phase separation in the bulk17. Figure 3 shows an example
of a numerical simulation of ‘surface-directed phase
separation’.
Anisotropies in phase ordering systems can result from
internal fields also. In our earlier discussion, we have
referred to phase separation in binary alloys as a realization
of the CHC equation. This is a gross over-simplification of
the actual situation. In real binary alloys, strain fields are
invariably set up at interfaces between A-rich and B-rich
domains due to lattice parameter mismatches. These longranged strain fields strongly affect the intermediate and late
stages of phase separation in binary alloys, inducing
Figure 3. Dynamics of phase separation of a binary mixture AB in the presence of a surface with a preferential attraction for one of the
components (say, A). These evolution pictures were obtained from the numerical solution of a 2-dimensional version of the appropriate
dynamical equations (Puri and Frisch17 ). The surface is located at Z = 0; and the system size was LX × LZ ≡ 400 × 300. We show snapshots for
dimensionless times 30, 90, 900 and 9000. The A-rich and B-rich regions are denoted by black and white, respectively. There are three distinct
regions: (i) the bulk region (Z large), where one has usual bulk phase separation; (ii) the surface region (Z ~
– 0), where there is a growing wetting
layer; and (iii) the region between the bulk and surface regions, which is of maximum interest to us in the present context.
380
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
various anisotropies18.
The examples quoted above merely constitute the tip of
the iceberg. There are a large number of experimentally
relevant effects in phase ordering systems (e.g. gravity,
viscoelasticity, shear flows, chemical reactions, etc.), which
are presently being investigated by various groups
worldwide. As yet, we do not even have a comprehensive
experimental or numerical understanding of the asymptotic
behaviour of phase ordering dynamics in most of
the above situations – leave alone an analytical understanding. The field of phase ordering dynamics continues
to be a fascinating realization of far-from-equilibrium
statistical mechanics, and promises to be so for many years
to come.
1. For reviews, see Binder, K., in Materials Science and
Technology: Phase Transformations of Materials (eds Cahn, R.
W., Haasen, P. and Kramer, E. J.) VCH, Weinheim, 1991, vol.
5,
p.
405;
Bray,
A. J., Adv. Phys., 1994, 43, 357.
2. Plischke, M. and Bergersen, B., Equilibrium Statistical Physics,
Prentice–Hall, New Jersey, 1989.
3. Kawasaki, K., in Phase Transitions and Critical Phenomena (eds
Domb, C. and Green, M. S.), Academic Press, New York, 1972,
vol. 2, p. 443 and references therein.
4. Hohenberg, P. C. and Halperin, B. I., Rev. Mod. Phys., 1977, 49,
435.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
5. Ohta, T., Jasnow, D. and Kawasaki, K., Phys. Rev. Lett., 1982,
49, 1223.
6. Oono, Y. and Puri, S., Mod. Phys. Lett. B, 1988, 2, 861.
7. Bray, A. J. and Puri, S., Phys. Rev. Lett., 1991, 67, 2670.
8. Toyoki, H., Phys. Rev. B, 1992, 45, 1965.
9. For example, see Gaulin, B. D., Spooner, S. and Morii, Y., Phys.
Rev. Lett., 1987, 59, 668; Malik, A., Sandy, A. R., Lurio, L. B.,
Stephenson, G. B., Mochrie, S. G. J., McNulty, I. and Sutton, M.,
Phys. Rev. Lett., 1998, 81, 5832.
10. For example, see Wong, N. C. and Knobler, C. M., Phys. Rev. A,
1981, 24, 3205; Tanaka, H., J. Chem. Phys., 1996, 105, 10099.
11. Shinozaki, A. and Oono, Y., Phys. Rev. E, 1993, 48, 2622.
12. Oono, Y. and Puri, S., Phys. Rev. Lett., 1987, 58, 836; Phys.
Rev. A, 1988, 38, 434; Puri, S. and Oono, Y., Phys. Rev. A,
1988, 38, 1542.
13. Lifshitz, I. M. and Slyozov, V. V., J. Phys. Chem. Solids, 1961,
19, 35.
14. Puri, S., Chowdhury, D. and Parekh, N., J. Phys. A, 1991, 24,
L1087; Gyure, M. F., Harrington, S. T., Strilka, R. and Stanley,
H. E., Phys. Rev. E, 1995, 52, 4632.
15. For reviews, see Chowdhury, D., J. Phys. Condens. Matter, 1994,
6, 2435; Kawakatsu, T., Kawasaki, K., Furusaka, M.,
Okabayashi, H. and Kanaya, T., J. Phys. Condens. Matter, 1994,
6, 6385.
16. Puri, S., Phys. Rev. E, 1997, 55, 1752; Puri, S. and Sharma, R.,
Phys. Rev. E, 1998, 57, 1873.
17. For a review of experimental results, see Krausch, G., Mater. Sci.
Eng. Rep., 1995, R14, 1; For a review of modelling and
numerical simulations, see Puri, S. and Frisch, H. L., J. Phys.
Condens. Matter, 1997, 9, 2109.
18. Onuki, A. and Nishimori, H., Phys. Rev. B, 1991, 43, 13649;
Sagui, C., Somoza, A. M. and Desai, R. C., Phys. Rev. E, 1994,
50, 4865.
381
SPECIAL SECTION:
Arrested states of solids
Madan Rao*† and Surajit Sengupta**,§
*Raman Research Institute, C.V. Raman Avenue, Sadashivanagar, Bangalore 560 080, India
**Material Science Division, Indira Gandhi Center for Atomic Research, Kalpakkam 603 102, India
§
Current address: Institut für Physik, Johannes Gutenberg Universität Mainz, 55099 Mainz, Germany
Solids produced as a result of a fast quench across a
freezing or a structural transition get stuck in long-lived
metastable configurations of distinct morphology,
sensitively dependent on the processing history.
Martensites are particularly well-studied examples of
nonequilibrium solid–solid transformations. Since there
are some excellent reviews on the subject, we shall, in this
brief article, mainly present our viewpoint.
Nonequilibrium structures in solids
What determines the final microstructure of a solid under
changes of temperature or pressure? This is a complex
issue, since a rigid solid finds it difficult to flow along its
free energy landscape to settle into a unique equilibrium
configuration. Solids often get stuck in long-lived metastable or jammed states because the energy barriers that
need to be surmounted in order to get unstuck are much
larger than kBT.
Such nonequilibrium solid structures may be obtained
either by quenching from the liquid phase across a freezing
transition (see Caroli et al.1 for a comprehensive review), or
by cooling from the solid phase across a structural
transition. Unlike the former, nonequilibrium structures
resulting from structural transformations do not seem to
have attracted much attention amongst physicists, apart
from Barsch et al.2 and Gooding et al.3, possibly because
the microstructures and mechanical properties obtained
appear nongeneric and sensitively dependent on details of
processing history.
Metallurgical studies have however classified some of
the more generic nonequilibrium microstructures obtained in
solid (parent/austenite)–solid (product/ferrite) transformations depending on the kind of shape change and the
mobility of atoms. To cite a few:
• Martensites are the result of solid state transformations
involving shear and no atomic transport. Martensites
occur in a wide variety of alloys, polymeric solids and
ceramics, and exhibit very distinct plate-like structures
built from twinned variants of the product.
• Bainites are similar to martensites, but in addition
possess a small concentration of impurities (e.g. carbon
in iron) which diffuse and preferentially dissolve in the
†
On leave of absence from Intitute of Mathematical Sciences, CIT
Campus, Taramani, Chennai 600 113, India.
*For correspondence. (e-mail: madan@rri.ernet.in)
382
parent phase.
• Widmanstätten ferrites result from structural transformations involving shape changes and are accompanied by short-range atomic diffusion.
• Pearlites are an eutectic mixture of bcc Fe and the
carbide consisting of alternating stripes.
• Amorphous alloys, a result of a fast quench, typically
possess some short range ordering of atoms.
• Polycrystalline materials of the product phase are a
result of a slower quench across a structural transition
and display macroscopic regions of ordered configurations of atoms separated by grain boundaries.
That the morphology of a solid depends on the detailed
dynamics across a solid–solid transformation, has been
recognized by metallurgists who routinely use time–
temperature–transformation (TTT) diagrams to determine
heat treatment schedules. The TTT diagram is a family of
curves parametrized by a fraction δ of transformed product.
Each curve is a plot of the time required to obtain δ versus
temperature of the quench (Figure 1). The TTT curves for
an alloy of fixed composition may be viewed as a ‘kinetic
phase diagram’. For example, starting from a hot alloy at
Figure
1. TTT
curves 11
for
steel
AISI
1090
(0.84%C + 0.60% Mn). A: austenite (fcc), F: ferrite (bcc), C: carbide
(Fe3 C). Curves correspond to 0, 50 and 100% transformation. Below
a temperature Ms, the metastable martensite (M) is formed – the
transformation curves for martensites are horizontal.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
t = 0 equilibrated above the transition temperature (upper
left corner) one could, depending on the quench rate
(obtained from the slope of a line T(t)), avoid the nose of the
curve and go directly into the martensitic region or obtain a
mixture of ferrite and carbide when cooled slowly.
It appears from these studies that several qualitative
features of the kinetics and morphology of microstructures
are common to a wide variety of materials. This would
suggest that there must be a set of general principles
underlying
such
nonequilibrium
solid–solid
transformations. Since most of the microstructures exhibit
features at length scales ranging from 100 Å to 100 µm, it
seems reasonable to describe the phenomenon at the
mesoscopic scale, wherein the solid is treated as a
continuum. Such a coarse-grained description would ignore
atomic details and instead involve effective continuum
theories based on symmetry principles, conservation laws
and broken symmetry.
Let us state the general problem in its simplest context.
Consider a solid state phase diagram exhibiting two
different equilibrium crystalline phases separated by a first
order boundary (Figure 2). An adiabatically slow quench
from Tin → Tfin across the phase boundary in which the
cooling rate is so small that at any instant the solid is in
equilibrium corresponding to the instantaneous temperature
would clearly result in an equilibrium final product at Tfin. On
the other hand, an instantaneous quench would result in a
metastable product bearing some specific relation to the
parent phase. The task is to develop a nonequilibrium
theory of solid state transformations which would relate the
nature of the final arrested state and the dynamics leading
to it to the type of structural change, the quench rate and
the mobility of atoms.
In this article, we concentrate on the dynamical and
structural features of a class of solid–solid transformations
called Martensites. Because of its commercial importance,
martensitic transformations are a well-studied field in
metallurgy and materials science. Several classic review
articles and books discuss various aspects of martensites in
great detail4. The growing literature on the subject is a clear
indication that the dynamics of solid state transformations
is still not well understood. We would like to take this
opportunity to present, for discussion and criticism, our
point of view on this very complex area of nonequilibrium
physics 5,6.
We next review the phenomenology of martensites and
highlight generic features that need to be explained by a
nonequilibrium theory of solid state transformations.
nucleates a grain of the ferrite which grows isotropically,
leading to a polycrystalline bcc solid. A faster quench from
Tin > Tc to Tfin < Ms < Tc (where Ms: martensite start
temperature) instead produces a rapidly transformed
metastable phase called the martensite, preempting the
formation of the equilibrium ferrite. It is believed that
martensites form by a process of heterogeneous nucleation.
On nucleation, martensite ‘plates’ grow radially with a
constant front velocity ~ 105 cm/s, comparable to the speed
of sound. Since the transformation is not accompanied by
the diffusion of atoms, either in the parent or the product, it
is called a diffusionless transformation. Electron microscopy
reveals that each plate consists of an alternating array of
twinned or slipped bcc regions of size ≈ 100 Å. Such
martensites are called acicular martensites.
The plates grow to a size of approximately 1 µm before
they collide with other plates and stop. Most often the
nucleation of plates is athermal; the amount of martensite
nucleated at any temperature is independent of time. This
implies that there is always some retained fcc, partitioned by
martensite plates. Optical micrographs reveal that the
jammed plates lie along specific directions known as habit
planes. Martensites, characterized by such a configuration
of jammed plates, are long lived since the elastic energy
barriers for reorganization are much larger than kBT.
A theoretical analysis of the dynamics of the martensitic
transformation in Fe–C is complicated by the fact that the
deformation is 3-dimensional (Bain strain) with 3 twin
variants of the bcc phase. Alloys like In–Tl, In–Pb and Mn–
Fe however, offer the simplest examples of martensitic
transformations having only two twin variants. In–Tl alloys
undergo a tetragonal to orthorhombic transformation when
cooled below 72°C (ref. 2). The orthorhombic phase can be
obtained from the tetragonal phase by a two-dimensional
deformation, essentially a square to rhombus transition.
Experiments indicate that all along the kinetic pathway, the
local configurations can be obtained from a twodimensional deformation of the tetragonal cell. This would
imply that the movement of atoms is strongly anisotropic
and confined to the ab-plane. Thus as far as the physics of
this transformation is concerned, the ab-planes are in
perfect registry (no variation of the strain along the c-axis).
In the next two sections we shall discuss our work on the
dynamics of the square to a rhombus transformation in
2-dimensions using a molecular dynamics simulation and a
coarse-grained mode coupling theory.
Phenomenology of martensites
One of the most studied alloys undergoing martensitic
transformations is iron–carbon4. As the temperature is
reduced, Fe with less than 0.02%C undergoes an equilibrium
structural transition (Figure 2) from fcc (austenite) to bcc
(ferrite) at Tc = 910°C. An adiabatic cooling across Tc
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
Figure 2. Phase diagram of Fe–C (weight per cent of C < 0.02%).
Ms is the martensite start temperature.
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SPECIAL SECTION:
Molecular dynamics simulation of solid–solid
transformations
Our aim in this will be to study the simplest molecular
dynamics (MD) simulation of the square to rhombus
transformation. We would like to use the simulation results
to construct the complete set of coarse grained variables
needed in a continuum description of the dynamics of solid
state tranformations. We carry out the MD simulation in the
constant NVT ensemble using a Nosé–Hoover thermostat
(N = 12000)7.
Our MD simulation is to be thought of as a ‘coarsegrained’ MD simulation, where the effective potential is a
result of a complicated many-body interaction. One part of
the interaction is a purely repulsive two-body potential
V2(rij) = v 2/r1i 2j , where rij is the distance between particles i and j. The two-body interaction favours a triangular
lattice ground state. In addition, triplets of particles interact
via a short-range three-body potential V3(ri, rj, rk ) = v 3w(rij,
rjk, rik) [sin2 (4θijk) + sin2 (4θjki) + sin2 (4θkij)], where w(r) is a
smooth short-range function and θijk is the bond angle at j
between particles (ijk). Since V3 is minimized when θijk = 0 or
π/2, the three-body term favours a square lattice ground
state. Thus at sufficiently low temperatures, we can induce
a square to triangular lattice transformation by tuning v 3.
The phase diagram in the T – v 3 plane is shown in Figure 3.
We define elastic variables, coarse-grained over a spatial
block of size ξ and a time interval τ, from the instantaneous
positions u of the particles. These include the deformation
tensor ∂u i/∂xk , the full nonlinear strain εij, and the vacancy
field
φ= ρ – ρ (ρ = coarse-grained
local
density, ρ = average density). We have kept track of the
time dependence of these coarse-grained fields during the
MD simulation.
Consider two ‘quench’ scenarios – a high and low
temperature quench (upper and lower arrows in Figure 3
respectively) across the phase boundary. In both cases the
solid is initially at equilibrium in the square phase.
The high temperature quench across the phase boundary, induces a homogeneous nucleation (i.e. strain
inhomogeneities created by thermal fluctuations are
sufficient to induce critical nucleation) and growth of a
triangular region. The product nucleus grows isotropically
with the size R ~ t1/2. A plot of the vacancy/interstitial fields
shows that, at these temperatures they diffuse fast to their
equilibrium value (vacancy diffusion obeys an Arrenhius
form Dv = D0 exp (– A/kBT ), where A is an activation energy,
and so is larger at higher temperatures). The final
morphology
is
a
polycrystalline
triangular
solid.
The low temperature quench on the other hand, needs
defects (either vacancies or dislocations) to seed nucleation
in an appreciable time. This heterogeneous nucleation
initiates an embryo of triangular phase, which grows
anisotropically along specific directions (Figure 4). Two
aspects are immediately apparent, the growing nucleus is
twinned and the front velocities are high. Indeed, the
velocity of the front is a constant and roughly half the
velocity of longitudinal sound. A plot of the
vacancy/interstitial field shows a high concentration at the
parent–product interface. The vacancy field now diffuses
very slowly and so appears to get stuck to the interface
over the time scale of the simulation. If we force the
vacancies and interstitials to annihilate each other, then the
anisotropic twinned nucleus changes in the course of time
to an isotropic untwinned one!
Therefore the lessons from the MD simulation are: (i)
There are two scenarios of nucleation of a product in a
parent depending on the temperature of quench. The
product grows via homogeneous nucleation at high T, and
via heterogeneous nucleation at low T. (ii) The complete set
of slow variables necessary to describe the nucleation of
solid–solid transformations should include the strain tensor
and defects (vacancies and dislocations) which are
generated at the parent–product interface at the onset of
nucleation. (iii) The relaxation times of these defects dictate
the final morphology. At high temperatures the defects relax
fast and the grains grow isotropically with a diffusive front.
The final morphology is a polycrystalline triangular solid.
At low temperatures, the interfacial defects (vacancies)
created by the nucleating grain relax slowly and get stuck at
the parent–product interface. The grains grow
anisotropically along specific directions. The critical
nucleus is twinned and the front grows ballistically (with a
velocity comparable to the sound speed). The final
morphology is a twinned martensite.
Mode coupling theory of solid–solid
transformations
v3
Figure 3. T–v 3 phase diagram from the MD simulations showing
the freezing and structural transitions. The upper and lower arrows
correspond to the high and low temperature quenches, respectively.
384
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
Figure 4. MD snapshot of (a) the nucleating grain at some intermediate time initiated by the low temperature quench across the square–
triangle transition. The dark (white) region is the triangular (square) phase, respectively. Notice that the nucleus is twinned and highly
anisotropic. (b) The vacancy (white)/interstitial (black) density profile at the same time as a. Notice that the vacancies and interstitials are
well separated and cluster around the parent–product interface.
Armed with the lessons from the MD simulation, let us now
construct a continuum elastic theory of solid-state
nucleation. The analysis follows in part the theories of
Krumhansl et al.2, but has important points of departure.
The procedure is to define a coarse grained free energy
functional in terms of all the relevant ‘slow’ variables. From
the simulation results, we found that every configuration is
described in terms of the local (nonsingular) strain field εij,
the vacancy field φ, and singular defect fields like the
dislocation density b ij. These variables are essentially
related to the phase and amplitudes of the density wave
describing the solid {ρG }.
It is clear from the simulation that the strain tensor,
defined with respect to the ideal parent, gets to be of O(1) in
the interfacial region between the parent and the product.
Thus we need to use the full nonlinear strain tensor
εij = (∂iu j + ∂ju i + ∂iu k ∂ju k )/2. Further, since the strain is
inhomogeneous during the nucleation process, the free
energy functional should have derivatives of the strain
tensor ∂k εij (this has unfortunately been termed ‘nonlocal
strain’ by some authors).
In general, the form of the free energy functional can be
very complicated, but in the context of the square-torhombus transformation, the free energy density may be
approximated by a simple form,
f = c(∇ε)2 + ε2 – aε4 + ε6 + χvφ2 + χd b 2 + k d bε,
(1)
where ε is the nonzero component of the strain
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
corresponding to the transformation between a square and
a rhombus, φ is the vacancy field and b is the (scalar)
dislocation density. The tuning parameter a induces a
transition from a square (described by ε = 0) to a rhombus
(ε = ± e0).
Starting with ε = 0 corresponding to the equilibrium
square parent phase at a temperature T > Tc , we quench
across the structural transition. The initial configuration of ε
is now metastable at this lower temperature, and would
decay towards the true equilibrium configuration by
nucleating a small ‘droplet’ of the product. As we saw in the
last section, as soon as a droplet of the product appears
embedded in the parent matrix, atomic mismatch at the
parent–product interface gives rise to interfacial defects like
vacancies and dislocations.
Let us confine ourselves to solids for which the energy
cost of producing dislocations is prohibitively large. This
would imply that the interfacial defects consist of only
vacancies and interstitials. The dynamics of nucleation now
written in terms of ε, g (the conserved momentum density)
and vacancy φare complicated5. For the present purpose, all
we need to realize is that φ couples to the strain and is
diffusive with a diffusion coefficient Dv depending on
temperature.
As in the MD simulation, we find that the morphology
and growth of the droplet of the product depends critically
on the diffusion of these vacancies. If the temperature of
quench is high, φdiffuses to zero before the critical nucleus
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SPECIAL SECTION:
size is attained and the nucleus eventually grows into an
equilibrium (or polycrystalline) triangular solid. In this case,
the nucleus grows isotropically with R ~ t1/2. However, a
quench to lower temperatures results in a low vacancy
diffusion coefficient. In the limit Dv → 0, the φ-field remains
frozen at the moving parent–product interface. In this case,
a constrained variational calculation of the morphology of
the nucleus shows that it is energetically favourable to form
a twinned martensite rather than a uniform triangular
structure. The growth of the twinned nucleus is not
isotropic, but along habit planes. Lastly, the growth along
the
longer
direction
is ballistic with a velocity proportional to (χv)1/2 (of the order
of the sound velocity). All these results are consistent with
the results of the previous section and with martensite
phenomenology. Let us try and understand in more
physical terms, why the growing nucleus might want to
form twins.
As soon as a droplet of the triangular phase of dimension
L is nucleated, it creates vacancies at the parent–product
interface. The free energy of such an inclusion is
F = Fbulk + Fpp + Fφ. The first term is simply the bulk free
energy gain equal to ∆FL2, where ∆F is the free energy
difference between the square and triangular phases. The
next two terms are interfacial terms. Fpp is the elastic
contribution to the parent–product interface coming from
the gradient terms in the free energy density eq. (1), and is
equal to 4σppL, where σpp is the surface tension at the
parent–product interface. Fφ is the contribution from the
interfacial vacancy field glued to the parent–product
interface and is proportional to φ2 ~ L2 (since the atomic
mismatch should scale with the amount of parent–product
interface). This last contribution dominates at large L
setting a prohibitive price to the growth of the triangular
nucleus. The solid gets around this by nucleating a twin
with a strain opposite to the one initially nucleated, thereby
reducing φ. Indeed for an equal size twin, φ→ 0 on the
average, and leads to a much lower interfacial energy Fφ ~ L.
However, the solid now pays the price of having created an
additional twin interface whose energy cost is Ftw = σtwL.
Considering now an (in general) anisotropic inclusion of
length L, width W consisting of N twins, the free energy
calculation goes as
F = ∆FLW + σpp(L + W) + σtw NW + β(L/N)2Ν,
(2)
where the last term is the vacancy contribution. Minimization with respect to N gives L/N ~ W1/2, a relation that is
well known for 2-dimensional martensites like In–Tl.
Our next task is to solve the coupled dynamical equations
with appropriate initial conditions numerically, to obtain the
full morphology phase diagram as a function of the type of
structural change, the parameters entering the free energy
functional and kinetic parameters like Dv.
It should be mentioned that our theory takes off from the
386
theories of Krumhansl et al.2, in that we write the elastic
energy in terms of the nonlinear strain tensor and its
derivatives. In addition, we have shown that the process of
creating a solid nucleus in a parent generates interfacial
defects which evolve in time. The importance of defects has
been stressed by a few metallurgists8. We note also that the
parent–product interface is studded with an array of
vacancies with a separation equal to the twin size. This
implies that the strain decays exponentially from the
interface over a distance of order L/N. This has been called
‘fringing field’2. They obtain this by imposing boundary
conditions on the parent–product interface, whereas here it
appears dynamically.
Patterning in solid–solid tranformations:
Growth and arrest
So far we have discussed the nucleation and growth of
single grains. This description is clearly valid at very early
times, for as time progresses the grains grow to a size of
approximately 1 µm and start colliding, whereupon in most
alloys they stop. Optical micrographs of acicular
martensites reveal that the jammed plates lie along habit
planes that criss-cross and partition the surrounding fcc
(parent) matrix.
Can we quantify the patterning seen in martensite
aggregates over a scale of a millimeter? A useful measure is
the size distribution of the martensite grains embedded in a
given volume of the parent. The appropriate (but difficult!)
calculation at this stage would be the analogue of a Becker–
Döring theory for nucleation in solids. In the absence of
such a theory, we shall take a phenomenological approach.
Clearly the size distribution P(l, t) depends on the spatiotemporal distribution I of nucleation sites and the growth
velocity v. We have analysed the problem explicitly in a
simple 2-dimensional context. Since the nucleating
martensitic grains are highly anisotropic and grow along
certain directions with a uniform velocity, a good
approximation is to treat the grains as lines or rays. These
rays (lines) emanate from nucleation sites along certain
directions, and grow with a constant velocity v. The rays
stop on meeting other rays and eventually after a time T, the
2-dimensional space is fragmented by N colliding rays. The
size distribution of rays, expressed in terms of a scaling
variable y = y(I, v), has two geometrical limits – the Γ -fixed
point (at y = 0) and the L-fixed point (at y = ∞). The Γ -fixed
point corresponds to the limit where the rays nucleate
simultaneously with a uniform spatial distribution. The
stationary distribution P(l) is a Gamma distribution with an
exponentially decaying tail. The L-fixed point, corresponds
to the limit where the rays are nucleated sequentially in time
(and uniformly in space) and grow with infinite velocity. By
mapping on to a multifragmentation problem, Ben Naim and
Krapivsky9 were able to derive the exact asymptotic form for
the moments of P(l) at the L-fixed point. The distribution
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
function P(l) has a multiscaling form, characterized by its
moments ⟨lq⟩ ~ N –µ(q), where µ(q) = (q + 2 – (q 2 + 4)1/2)/2.
At intermediate values of the scaling variable y, there is a
smooth crossover from the Γ -fixed point to the L-fixed
point with a kinematical crossover function and crossover
exponents.
The emergence of scale invariant microstructures in
martensites as arising out of a competition between the
nucleation rate and growth is a novel feature well worth
experimental investigation. There have been similar
suggestions in the literature, but as far as we know there
have been no direct visualization studies of the microstructure of acicular martensites using optical micrographs.
Recent acoustic emission experiments10 on the thermoelastic
reversible martensite Cu–Zn–Al, may be argued to provide
indirect support of the above claim5, but the theory of
acoustic emission in martensites is not understood well
enough to make such an assertion with any confidence.
Open questions
We hope this short review makes clear how far we are in our
understanding of the dynamics of solid–solid
transformations. A deeper understanding of the field will
only come about with systematic experiments on carefully
selected systems. For instance, a crucial feature of our
nonequilibrium theory of martensitic transformations is the
existence of a dynamical interfacial defect field. In
conventional Fe based alloys, the martensitic front grows
incredibly fast, making it difficult to test this using in situ
transmission electron microscopy. Colloidal solutions of
polysterene spheres (polyballs) however, are excellent
systems for studying materials properties. Polyballs
exhibiting fcc → bcc structural transitions have been seen
to undergo twinned martensitic transformations. The length
and time scales associated with colloids are large, making it
comfortable to study these systems using light scattering
and optical microscopy.
In this article we have focused on a small part of the
dynamics of solid state transformations, namely the
dynamics and morphology of martensites. Even so our
presentation here is far from complete and there are crucial
unresolved questions that we need to address.
Let us list the issues as they appear following a
nucleation event.
The physics of heterogeneous nucleation in solids is
very poorly understood. For instance, it appears from our
simulations that the morphology of the growing nucleus
depends on the nature of the defects seeding the nucleation
process (e.g. vacancies, dislocations and grain
boundaries). In addition, several martensitic transformations
are associated with correlated nucleation events and
autocatalysis. Though these features are not central to the
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
issue of martensites, such a study would lead to a better
understanding of the origins of athermal, isothermal and
burst nucleation. This in conjunction with a ‘Becker–Döring
theory’ for multiple martensite grains would be a first step
towards the computation of TTT curves.
We still do not understand the details of the dynamics of
twinning and how subsequent twins are added to the
growing nucleus. Moreover, the structure and dynamics of
the parent–product interface and of the defects embedded
in it have not been clearly analysed.
It would be desirable to have a more complete theory
which displays a morphology phase diagram (for a single
nucleus) given the type of structural transition and the
kinetic, thermal and elastic parameters.
Certain new directions immediately suggest themselves.
For instance, the role of carbon in interstitial alloys like Fe–
C leading to the formation of bainites; the coupling of the
strain to an external stress and the shape memory effect;
tweed phases and pre-martensitic phenomena (role of
quenched impurities).
The study of the dynamics of solid–solid transformations
and the resulting long-lived morphologies lies at the
intersection of metallurgy, materials science and
nonequilibrium statistical mechanics. The diversity of
phenomena makes this an extremely challenging area of
nonequilibrium physics.
1. Caroli, B., Caroli, C. and Roulet, B., in Solids Far from
Equilibrium (ed. Godreche, C.), Cambridge University Press,
1992.
2. Barsch, G. R. and Krumhansl, J. A., Phys. Rev. Lett., 1974, 37,
9328; Barsch, G. R., Horovitz, B. and Krumhansl, J. A., Phys.
Rev. Lett., 1987, 59, 1251.
3. Bales, G. S. and Gooding, R. J., Phys. Rev. Lett., 1991, 67, 3412;
Reed, A. C. E. and Gooding, R. J., Phys. Rev., 1994, B50, 3588;
van
Zyl,
B.
P.
and
Gooding,
R.
J.,
http://xxx.lanl.gov/archive/cond-mat/9602109.
4. Roitburd, A., in Solid State Physics (eds Seitz and Turnbull),
Academic Press, NY, 1958; Nishiyama, Z., Martensitic Transformation, Academic Press, NY, 1978; Kachaturyan, A. G.,
Theory of Structural Transformations in Solids, Wiley, NY,
1983; Martensite (eds Olson, G. B. and Owen, W. S.), ASM
International, The Materials Information Society, 1992.
5. Rao, M. and Sengupta, S., Phys. Rev. Lett., 1997, 78, 2168; Rao,
M. and Sengupta, S., lanl e-print: cond-mat/9709022; Rao, M.
and Sengupta, S., IMSc preprint, 1998.
6. Rao, M., Sengupta, S. and Sahu, H. K., Phys. Rev. Lett., 1995,
75, 2164; Rao, M. and Sengupta, S., Phys. Rev. Lett., 1996, 76,
3235; Rao, M. and Sengupta, S., Physica A, 1996, 224, 403.
7. Sengupta, S. and Rao, M., to be published.
8. Olson, G. B. and Cohen, M., Acta Metall., 1979, 27, 1907;
Olson, G. B., Acta Metall., 1981, 29, 1475; Christian, J. W.,
Metall. Trans. A, 1982, 13, 509.
9. Ben Naim, E. and Krapivsky, P., Phys. Rev. Lett., 1996, 76,
3235.
10. Vives, E. et al., Phys. Rev. Lett., 1994, 72, 1694.
11. Metals Handbook, ASM, Ohio, 9th edition, vol. 4, 1981.
ACKNOWLEDGEMENT. We thank Yashodhan Hatwalne for a
critical reading of the manuscript.
387
SPECIAL SECTION:
Sandpile models of self-organized criticality
S. S. Manna
Laboratoire de Physique et Mecanique des Milieux Heterogenes, École Supérieure de Physique et de Chimie Industrielles,
10 rue Vauquelin, 75231 Paris Cedex 05, France and
Satyendra Nath Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Calcutta 700 091, India
Self-organized criticality is the emergence of long-ranged
spatio-temporal correlations in non-equilibrium steady
states of slowly driven systems without fine tuning of any
control parameter. Sandpiles were proposed as prototypical
examples of self-organized criticality. However, only some
of the laboratory experiments looking for the evidence of
criticality in sandpiles have reported a positive outcome.
On the other hand, a large number of theoretical models
have been constructed that do show the existence of such a
critical state. We discuss here some of the theoretical
models as well as some experiments.
THE concept of self-organized criticality (SOC) was
introduced by Bak, Tang and Wiesenfeld (BTW) in 1987
(ref. 1). It says that there is a certain class of systems in
nature whose members become critical under their own
dynamical evolutions. An external agency drives the system
by injecting some mass (in other examples, it could be the
slope, energy or even local voids) into it. This starts a
transport process within the system: Whenever the mass at
some local region becomes too large, it is distributed to the
neighbourhood by using some local relaxation rules.
Globally, mass is transported by many such successive
local relaxation events. In the language of sandpiles, these
together constitute a burst of activity called an avalanche. If
we start with an initial uncritical state, initially most of the
avalanches are small, but the range of sizes of avalanches
grows with time. After a long time, the system arrives at a
critical state, in which the avalanches extend over all length
and time scales. Customarily, critical states have measure
zero in the phase space. However, with self-organizing
dynamics, the system finds these states in polynomial
times, irrespective of the initial state2–4.
BTW used the example of a sandpile to illustrate their
ideas about SOC. If a sandpile is formed on a horizontal
circular base with any arbitrary initial distribution of sand
grains, a sandpile of fixed conical shape (steady state) is
formed by slowly adding sand grains one after another
(external drive). The surface of the sandpile in the steady
state, on the average, makes a constant angle known as the
angle of repose, with the horizontal plane. Addition of each
sand grain results in some activity on the surface of the pile:
an avalanche of sand mass follows, which propagates on
the surface of the sandpile. Avalanches are of many
different sizes and BTW argued that they would have a
e-mail: manna@boson.bose.res.in
388
power law distribution in the steady state.
There are also some other naturally occurring phenomena
which are considered to be examples of SOC. Slow creeping
of tectonic plates against each other results in intermittent
burst of stress release during earthquakes. The energy
released is known to follow power law distributions as
described by the well-known Gutenberg–Richter Law5. The
phenomenon of earthquakes is being studied using SOC
models6. River networks have been found to have fractal
properties. Water flow causes erosion in river beds, which
in turn changes the flow distribution in the network. It has
been argued that the evolution of river pattern is a selforganized dynamical process7. Propagation of forest fires8
and biological evolution processes9 have also been
suggested to be examples of SOC.
Laboratory experiments on sandpiles, however, have not
always found evidence of criticality in sandpiles. In the first
experiment, the granular material was kept in a semicircular
drum which was slowly rotated about the horizontal axis,
thus slowly tilting the free surface of the pile. Grains fell
vertically downward and were allowed to pass through the
plates of a capacitor. Power spectrum analysis of the time
series for the fluctuating capacitance however showed a
broad peak, contrary to the expectation of a power law
decay, from the SOC theory10.
In a second experiment, sand was slowly dropped on to a
horizontal circular disc, to form a conical pile in the steady
state. On further addition of sand, avalanches were created
on the surface of the pile, and the outflow statistics was
recorded. The size of the avalanche was measured by the
amount of sand mass that dropped out of the system. It was
observed that the avalanche size distribution obeys a
scaling behaviour for small piles. For large piles, however,
scaling did not work very well. It was suggested that SOC
behaviour is seen only for small sizes, and very large
systems would not show SOC11.
Another experiment used a pile of rice between two
vertical glass plates separated by a small gap. Rice grains
were slowly dropped on to the pile. Due to the anisotropy
of grains, various packing configurations were observed. In
the steady state, avalanches of moving rice grains refreshed
the surface repeatedly. SOC behaviour was observed for
grains of large aspect ratio, but not for the less elongated
grains12.
Theoretically, however, a large number of models have
been proposed and studied. Most of these models study
the system using cellular automata where discrete or
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
continuous variables are used for the heights of sand
columns. Among them, the Abelian Sandpile Model (ASM)
is the most popular1,13. Other models of SOC have been
studied but will not be discussed here. These include the
Zhang model which has modified rules for sandpile
evolution14, a model for Abelian distributed processors and
other stochastic rule models15, the Eulerian Walkers model16
and the Takayasu aggregation model17.
In the ASM, we associate a non-negative integer variable
h representing the height of the ‘sand column’ with every
lattice site on a d-dimensional lattice (in general on any
connected graph). One often starts with an arbitrary initial
distribution of heights. Grains are added one at a time at
randomly selected sites
The
O :sand
h O →column
h O + 1.
at any arbitrary site i becomes unstable when h i exceeds a
previously selected threshold value h c for the stability.
Without loss of generality, one usually chooses h c = 2d –1.
An unstable sand column always topples. In a toppling, the
height is reduced as: h i → h i –2d and all the 2d
neighbouring sites {j} gain a unit sand grain each:
h j → h j + 1. This toppling may make some of the
neighbouring sites unstable. Consequently, these sites will
topple again, possibly making further neighbours unstable.
In this way a cascade of topplings propagates, which finally
terminates when all sites in the system become stable
(Figure 1). One waits until this avalanche stops before
adding the next grain. This is equivalent to assuming that
the rate of adding sand is much slower than the natural rate
of relaxation of the system. The wide separation of the ‘time
scale of drive’ and ‘time scale of relaxation’ is common in
many models of SOC. For instance, in earthquakes, the drive
is the slow tectonic movement of continental plates, which
occurs over a time scale of centuries, while the actual stress
relaxation occurs in quakes, whose duration is only a few
seconds. This separation of time scales is usually
considered to be a defining characteristic of SOC. However,
Dhar has argued that the wide separation of time scales
should not be considered as a necessary condition for SOC
in general4. Finally, the system must have an outlet, through
which the grains go out of the system, which is absolutely
necessary to attain a steady state. Most popularly, the
outlet is chosen as the (d –1)-dimensional surface of a ddimensional hypercubic system.
The beauty of the ASM is that the final stable height
configuration of the system is independent of the sequence
in which sand grains are added to the system to reach this
stable configuration13. On a stable configuration C, if two
grains are added, first at i and then at j, the resulting stable
configuration C′ is exactly the same in case the grains were
added first at j and then at i. In other sandpile models,
where the stability of a sand column depends on the local
slope or the local Laplacian, the dynamics is not Abelian,
since toppling of one unstable site may convert another
unstable site to a stable site (Figure 2). Many such rules
have been studied in the literature18,19.
An avalanche is a cascade of topplings of a number of
sites created on the addition of a sand grain. The strength
of an avalanche in general, is a measure of the effect of the
external perturbation created due to the addition of the sand
grain. Quantitatively, the strength of an avalanche is
estimated in four different ways: (i) size (s): the total number
topplings in the avalanche, (ii) area (a): the number of
distinct sites which toppled, (iii) life-time (t): the duration of
the avalanche, and (iv) radius (r): the maximum distance of a
toppled site from the origin. These four different quantities
are not independent and are related to each other by scaling
laws. Between any two measures x, y ∈{s, a, t, r} one can
define a mutual dependence as: ⟨y⟩ ~ xγ . These exponents
are related to one another, e.g. γts = γtr γrs. For the ASM, it
can be proved that the avalanche clusters cannot have any
holes. It has been shown that γrs = 2 in two-dimensions. It
has also been proved that γr t = 5/4 (ref. 21). A better way to
estimate the γtx exponents is to average over the
intermediate values of the size, area and radius at every
intermediate time step during the growth of the avalanche.
Quite generally, the finite size scaling form for
the probability distribution function for any measure
x ∈{s, a, t, r} is taken to be:
xy
Figure 2. Example to show that a directed slope model is nonAbelian. Two slopes are measured from any site (i, j) as h(i, j) –
h(i, j + 1) and h(i, j) – h(i + 1, j + 1). If either of them is greater than
Figure
1. Avalanche
of the from
Abelian
Sandpile
1, two grains
are transferred
(i, j)
and areModel,
given generated
one each on
to
a(i, j3+× 1)
3 and
square
lattice.
graina is
dropped
a stable
(i + 1,
j + 1). AOnsand
dropping
grain
on the on
initial
configuration
site. two
Thedifferent
avalanche
created
has size
configuration, at
we the
see central
that finally
height
configuration
sresult
= 6, area
a=
6, different
life-time sequences
t =24 and the
r =20 . .
due to
two
of radius
topplings
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
389
SPECIAL SECTION:
P(x) ~ x–τ x f x (x/Lσx).
The exponent σx determines the variation of the cut-off of
the quantity x with the system size L. Alternatively,
sometimes it is helpful to consider
the cumulative
σx
probability distribution F (x) = ∫xL P(x) dx which varies as
x1–τ x . However, in the case of τx = 1, the variation should be
in the form F (x) = C – log(x). Between any two measures,
scaling relations like γxy = (τx –1)/(τy –1) exist. Recently, the
scaling assumptions for the avalanche sizes have been
questioned. It has been argued that there actually exists a
multifractal distribution instead22.
Numerical estimation for the exponents has yielded
scattered values. For example estimates of the exponent
τs range from 1.20 (ref. 18) to 1.27 (ref. 23) and 1.29
(ref. 24).
We will now look into the structure of avalanches in more
detail. A site i can topple more than once in the same
avalanche. The set of its neighbouring sites {j}, can be
divided into two subsets. Except at the origin O, where a
grain is added from the outside, for a toppling, the site i
must receive some grains from some of the neighbouring
sites {j1} to exceed the threshold h c . These sites must have
toppled before the site i. When the site i topples, it loses 2d
grains to the neighbours, by giving back the grains it has
received from {j1}, and also donating grains to the other
neighbours {j2}. Some of these neighbours may topple later,
which returns grains to the site i and its height h i is raised.
The following possibilities may arise: (i) some sites of {j2}
may not topple at all; then the site i will never re-topple and
is a singly toppled site on the surface of the avalanche. (ii)
all sites in {j2} topple, but no site in {j1} topples again; then
i will be a singly toppled site, surrounded by singly toppled
sites. (iii) all sites in {j2} topple, and some sites of {j1} retopple; then i will remain a singly toppled site, adjacent to
the doubly toppled sites. (iv) all sites in {j2} topple, and all
sites of {j1} re-topple; then the site i must be a doubly
toppled site. This implies that the set of at least doubly
toppled sites must be surrounded by the set of singly
toppled sites. Arguing in a similar way will reveal that sites
which toppled at least n times, must be a subset and also
are surrounded by the set of sites which toppled at least
(n –1) times. Finally, there will be a central region in the
avalanche, where all sites have toppled a maximum of m
times. The origin of the avalanche O where the sand grain
was dropped, must be a site in this maximum toppled zone.
Also, the origin must be at the boundary of this mth zone,
since otherwise it should have toppled (m + 1) times25.
Using this idea, we see that the boundary sites on any
arbitrary system can topple at most once in any arbitrary
number of avalanches. Similar restrictions are true for inner
sites also. A (2n + 1) × (2n + 1) square lattice can be divided
into (n + 1) subsets which are concentric squares. Sites on
the mth such square from the boundary can topple at most
m times, whereas the central site cannot topple more than n
390
times in any avalanche.
Avalanches can also be decomposed in a different
way, using Waves of Toppling. Suppose, on a stable
configuration C a sand grain is added at the site O. The site
is toppled once, but is not allowed to topple for the second
time, till all other sites become stable. This is called the first
wave. It may happen that after the first wave, the site O is
stable; in that case the avalanche has terminated. If the O is
still unstable, it is toppled for the second time and all other
sites are allowed to become stable again; this is called the
second wave, and so on. It was shown, that in a sample
where all waves occur with equal weights, the probability of
occurrence of a wave of area a is D(a) ~ 1/a (ref. 26).
It is known that the stable height configurations in ASM
are of two types: Recurrent configurations appear only in
the steady state with uniform probabilities, whereas
Transient configurations occur in the steady state with zero
probability. Since long-range correlations appear only in the
steady states, it implies that the recurrent configurations are
correlated. This correlation is manifested by the fact that
certain clusters of connected sites with some specific
distributions of heights never appear in any recurrent
configuration. Such clusters are called the forbidden subconfigurations. It is easy to show that two zero heights at
the neighbouring sites: (0–0) or, an unit height with two
zero
heights
at
its
two
sides:
(0–1–0) never occur in the steady state. There are also many
more forbidden sub-configurations of bigger sizes.
An L × L lattice is a graph, which has all the sites and all
the nearest neighbour edges (bonds). A Spanning tree is a
connected sub-graph having all sites but no loops.
Therefore, between any pair of sites there exists an unique
path through a sequence of bonds. There can be many
possible Spanning trees on a lattice. These trees have
interesting statistics in a sample where they are equally
likely. Suppose we randomly select such a tree and then
randomly select one of the unoccupied bonds and occupy
it, it forms a loop of length l . It has been shown that
these loops have the length distribution D(l ) ~ l –8/5.
Similarly, if a bond of a Spanning tree is randomly selected
and deleted, then it divides into two fragments. The sizes of
the two fragments generated follow a probability
distribution D(a) ~ a –11/8 (ref. 27). It was also shown that
every recurrent configuration of the ASM on an arbitrary
lattice has a one-to-one correspondence to a random
Spanning tree graph on the same lattice. Therefore, there are
exactly the same number of distinct Spanning trees as the
number of recurrent ASM configurations on any arbitrary
lattice21. Given a stable height configuration, there exists an
unique prescription to obtain the equivalent Spanning tree.
This is called the Burning method21. A fire front, initially at
every site outside the boundary, gradually penetrates
(burns) into the system using a deterministic rule. The paths
of the fire front constitute the Spanning tree. A fully burnt
system is recurrent, otherwise it is transient (Figure 3).
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
Suppose, addition of a grain at the site O of a stable
recurrent configuration C, leads to another stable configuration C′. Is it possible to get back the configuration
C knowing C′ and the position of O ? This is done by
Inverse toppling28. Since C′ is recurrent, a corresponding
Spanning tree ST(C′) exists. Now, one grain at O is taken
out from C′ and the configuration C″ = C′ – δO j is obtained.
This means on ST(C′), one bond is deleted at O and it is
divided into two fragments. Therefore one cannot burn the
configuration C″completely since the resulting tree has a
hole consisting of at least the sites of the smaller fragment.
This implies that C″ has a forbidden sub-configuration (F1)
a
b
c
Figure 3. a, An example of the height distribution in a recurrent configuration C ′ on a 24 × 24 square lattice. This configuration is
obtained by dropping a grain a some previous configuration C at the encircled site; b, The spanning tree representation of the configuration
C ′; c, A new configuration C ″ is obtained by taking out one grain at the encircled site from the configuration C ′. A spanning tree cannot be
obtained for C ″. The bonds of the spanning tree corresonding to the forbidden sub-configuration in C ″ are shown by the thin lines.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
391
SPECIAL SECTION:
of equal size and C″ is not recurrent. On (F1), one runs the
inverse toppling process: 4 grains are added to each site i,
and one grain each is taken out from all its neighbours {j}.
The cluster of f1 sites in F1 is called the first inverse
avalanche. The lattice is burnt again. If it still has a
forbidden sub-configuration (F2), another inverse toppling
process is executed, and is called the second inverse
avalanche. The size of the avalanche is: s = f1 + f2 + f3 + . . .,
and f1 is related to the maximum toppled zone of the
avalanche. From the statistics of random spanning trees27 it
is clear that f1 should have the same statistics of the two
fragments of the tree generated on deleting one bond.
Therefore, the maximum toppled zone also has a power law
distribution of the size, D(a) ~ a –11/8.
Sandpile models with stochastic evolution rules have
also been studied. The simplest of these is a two-state
sandpile model. A stable configuration of this system
consists of sites, either vacant or occupied by at most one
grain. If there are two or more grains at a site at the same
time we say there is a collision. In this case, all grains at
that site are moved. Each grain chooses a randomly selected
site from the neighbours and is moved to that site. The
avalanche size is the total number of collisions in an
avalanche. From the numerical simulations, the distribution
of avalanche sizes is found to follow a power law,
characterized by an exponent τs ≈ 1.27 (ref. 29). This twostate model has a nontrivial dynamics even in onedimension30. Recently, it has been shown that instead of
moving all grains, if only two grains are moved randomly
leaving others at the site, the dynamics is Abelian31.
Some other stochastic models also have nontrivial critical
behaviour in one dimension. To model the dynamics of rice
piles, Christensen et al.32 studied the following slope model.
On a one-dimensional lattice of length L, non-negative
integer variable h i represents the height of the sand column
at the site i. The local slope zi = h i – h i + 1 is defined,
maintaining zero height on the right boundary. Grains are
added only at the left boundary i = 1. Addition of one grain
h i → h i + 1 implies an increase in the slope zi → zi + 1. If at
any
site,
the
local
slope
exceeeds
a
pre-assigned threshold value zci , one grain is transferred
from the column at i to the column at (i + 1). This
implies a change in the local slope as: zi → zi – 2 and
zi ± 1 → zi ± 1+ 1. The thresholds of the instability zci are
dynamical variables and are randomly chosen between 1
and 2 in each toppling. Numerically, the avalanche sizes are
found to follow a power law distribution with an exponent
τs ≈ 1.55 and the cutoff exponent was found to be σs ≈ 2.25.
This model is referred as the Oslo model.
Addition of one grain at a time, and allowing the system
to relax to its stable state, implies a zero rate of driving of
the system. What happens when the driving rate is finite?
Corral and Paczuski studied the Oslo model in the situation
of nonzero flow rate. Grains were added at a rate r, i.e. at
every (1/r) time updates, one grain is dropped at the left
boundary i = 1. They observed a dynamical transition
392
separating intermittent and continuous flows33.
Many different versions of the sandpile model have been
studied. However the precise classification of various
models in different universality classes in terms of their
critical exponents is not yet available and still attracts much
attention18,19. Exact values of the critical exponents of the
most widely studied ASM are still not known in twodimensions. Some effort has also been made towards the
analytical calculation of avalanche size exponents34–36.
Numerical studies for these exponents are found to give
scattered values. On the other hand, the two-state sandpile
model is believed to be better behaved and there is good
agreement of numerical values of its exponents by different
investigators. However, whether the ASM and the twostate model belong to the same universality class or not is
still an unsettled question37.
If a real sandpile is to be modelled in terms of any of
these sandpile models or their modifications, it must be a
slope model, rather than a height model. However, not much
work has been done to study the slope models of
sandpiles18,19. Another old question is whether the conservation of the grain number in the toppling rules is a
necessary condition to obtain a critical state. It has been
shown already that too much non-conservation leads to
avalanches of characteristic sizes36. However, if grains are
taken out of the system slowly, the system is found to be
critical in some situations. A non-conservative version of
the ASM with directional bias shows a mean field type
critical behaviour39. Therefore, the detailed role of the
conservation of the grain numbers during the topplings is
still an open question.
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ACKNOWLEDGEMENTS. We thank D. Dhar for a critical reading
of the manuscript and for useful comments.
393
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Nonequilibrium growth problems
Sutapa Mukherji* and Somendra M. Bhattacharjee†§
*Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur 721 302, India
†
Institute of Physics, Bhubaneswar 751 005, India
We discuss the features of nonequilibrium growth
problems, their scaling description and their differences
from equilibrium problems. The emphasis is on the
Kardar–Parisi–Zhang equation and the renormalization
group point of view. Some of the recent developments along
these lines are mentioned.
HOW to characterize the degree of roughness of a surface
as it grows and how the roughness varies in time
have evolved into an important topic due to diverse
interests in physics, biology, chemistry and in technological
applications. One crucial aspect of these nonequilibrium
growth processes is the scale invariance of surface
fluctuations similar to the scale invariance observed in
equilibrium critical point phenomena. Although different
kinds of growths may be governed by distinct natural
processes, they share a common feature that the surface,
crudely speaking, looks similar under any magnification and
at various times. This nonequilibrium generalization of
scaling involving space and time (called ‘dynamic scaling’)
makes this subject of growth problems important in
statistical mechanics.
Growth problems are both of near-equilibrium and
nonequilibrium varieties. Therefore, they provide us with a
fertile ground to study the differences and the
extra features that might emerge in a nonequilibrium
situation1,2. Take for example the case of crystal growth. In
equilibrium, entropic contributions generally lead to a rough
or fluctuating surface, an effect called thermal roughening,
but, for crystals, because of the lattice periodicity a
roughening transition from a smooth to rough surface
occurs at some temperature. The nature of the growth of a
crystal close to equilibrium expectedly depends on whether
the surface is smooth or rough. One can also think of a
crystal growth process which is far away from equilibrium
by subjecting it to an external drive, for instance by random
deposition of particles on the surface. The roughening that
occurs in the nonequilibrium case is called kinetic
roughening. Is the nature of the surface any different in
kinetic roughening? Crystals are definitely not the only
example of growth processes; some other examples of such
nonequilibrium growths would be the growth of bacterial
colonies in a petri dish, sedimentation of colloids in a drop,
the formation of clouds in the upper atmosphere, and so on.
Note the large variation of length scales of these problems.
§
For correspondence. (e-mail: somen@polymer.iopb.res.in)
394
In many such examples, it is difficult if not impossible to
think of an equilibrium counterpart.
Scale invariance in interface fluctuations implies that
fluctuations look statistically the same when viewed at
different length scales. A quantitative measure of the height
fluctuation (height measured from an arbitrary base) is
provided by the correlation function
C(x, t) = ⟨[h(x + x0, t + t0) – h(x0, t0)]2⟩,
(1)
where x and t denote the d-dimensional coordinate on the
substrate and time, respectively. The averaging in eq. (1) is
over all x0, and, by definition, C(x, t) is independent of the
choice of the arbitrary base. In simple language, scale
invariance then means that when the system is, say,
amplified by a scaling x → bx and t → b zt, the height
fluctuations reveal the same features as the original, up to
an overall scale factor. Quantitatively, there exists a
generalized scaling
C(x, t) = b –2χC(bx, b zt),
(2)
where b is a scale factor, and χ and z are known as the
roughening and dynamic exponents which are also
universal. As a direct consequence of eq. (2), a scaling form
for C(x, t) can be obtained by choosing b = 1/x
C(x, t) = x2χ Ĉ (t/xz),
(3)
a form that also explains the origin of the name ‘dynamic
exponent’ for z. The power law behaviour (as opposed to
say exponential decay) of the correlation function implies
absence of any scale, neither in space nor in time. All the
underlying length scales required to define the problem
dropped out of the leading behaviour in eq. (3). Such a scale
invariance is one of the most important features of
equilibrium phase transitions and is observed when a
parameter, say the temperature, approaches its critical value.
However, here there is no special tuning parameter; the
scale invariance appears from the interplay of competing
processes which in the simplest case can be the surface
tension and noise present due to inherent randomness in
the growth. There can be, of course, more complex events
like a phase transition between surfaces with different
roughness but scale invariance (not only of the correlation
function but of any physical quantity) is generically
preserved in all these surfaces.
It is worth emphasizing the enormous simplification that
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
occurs in the scaling description. It is only a very few
quantities that define the asymptotic behaviour of the
system. Consequently, the idea of studying the universal
aspects of growth processes is to classify and characterize
the various universality classes as determined by the
exponents, e.g. χ and z, the scaling function and if
necessary certain other important universal quantities.
At this point, it might be helpful to compare the
equilibrium and nonequilibrium cases again. In equilibrium,
thanks to thermal energy (or ‘random kicks’ from a heat
reservoir), all configurations of a system are accessible and
do occur, but no net flow of probability between any two
states is expected (called ‘detailed balance’). Consequently,
the
knowledge
of
the
states
(and the energies) of a system allows one to obtain
the thermodynamic free energy by summing over the
Boltzmann factors exp(– E/kBT ), where E is the energy of
the state, T the temperature and kB is the Boltzmann
constant. In a nonequilibrium situation, either or both of the
above two conditions may be violated, and the framework
of predicting the properties of a system from free energy is
not necessarily available. A dynamic formulation is needed.
By assigning a time-dependent probability for the system to
be in a configuration at a particular time, one may study the
time evolution of the probability. The equilibrium problem
can be viewed from a dynamical point also. This description
must give back the Boltzmann distribution in the infinite
time steady state limit. This is the Fokker–Planck approach.
The probabilistic description comes from the ensemble
picture where identical copies of the same system exchange
energy with the bath independently. An alternative
approach which finds easy generalization to the nonequilibrium cases is the Langevin approach where one
describes the time evolution of the degrees of freedom, in
our example h(x, t), taking care of the random exchange of
energy by a noise. The dynamics we would consider is
dissipative so that the system in absence of any noise
would tend to a steady state. However, for it to reach the
equilibrium Boltzmann distribution in the presence of noise,
it is clear that the noise must satisfy certain conditions
(Einstein relation) connecting it to the system parameters.
The nonequilibrium case does not have any thermodynamic
free energy as a guiding light and therefore, there is no
requirement to reach the Boltzmann distribution. In the
Langevin approach, the noise term can be completely
independent. In the equilibrium case, the Langevin equation
will be determined by the Hamiltonian or the free energy of
the system, but for nonequilibrium cases there might be
terms which cannot be obtained from Hamiltonians. Since
for t → ∞, the probability distribution for equilibrium cases
attains the Boltzmann distribution, the roughness exponent
χ is determined even in dynamics by the stationary state
while the details of the dynamics is encoded in the dynamic
exponent z. In other words, the two exponents χ and z are
independent quantities. In the nonequilibrium case, there is
no compulsion to reach any predetermined stationary state
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
and therefore the surface roughness is related to the
growth, i.e. χ and z need not be independent. We see below
that there is in fact a specific relation connecting these two
exponents.
The existence of scale invariance and universal exponents implies that as far as the exponents are concerned, the
theory should be insensitive to the microscopic details, or,
in other words, one may integrate out all the small length
scale features. The universal exponents come out as an
output of this process of coarse graining of say the
Langevin equation, followed by a length rescaling that
brings the system back to its original form. The new system
will however have different values of the parameters and
one can study the flow of these parameters in the long
length and time scale limit. This is the basic idea behind the
renormalization group (RG). In this approach, the
importance of an interaction or a term is judged not by its
numerical value but by its relevance. One may start with any
physically possible process in dynamics and see how it
appears as the length scale or resolution changes. For large
length scales, one is left only with the relevant terms that
grow with length, and marginal terms that do not change;
the irrelevant terms or interactions that decay with length
scale automatically drop out from the theory. The exponents
are determined at the fixed points of the flows in the
parameter space. These fixed points, which remain invariant
under renormalization, characterize the macroscopic or
asymptotic behaviour of the system. Clearly from this
viewpoint of RG, one can explain why the microscopic
details can be ignored and how the idea of ‘universality’
emerges. All systems whose dynamical behaviour would
flow to the same fixed point under RG transformation will
have identical scaling behaviour. The various universality
classes can then be associated with the various fixed points
of RG transformations, and phase transitions or criticality
with unstable fixed points or special flows in the parameter
space. An RG approach therefore seems rather natural and
well suited for studying any scale invariant phenomena in
general, and growth problems in particular. Quite
expectedly, the modern approach to growth problems is
based on these views of RG.
For a quantitative discussion, we consider two simple
equations that, from the historical point of view, played a
crucial role in the development of the subject in the last two
decades.
A simple Langevin equation describing the dynamics of a
surface is the Edwards–Wilkinson (EW) equation3
∂h
= ν∇ 2 h + η( x, t),
∂t
(4)
where x represents in general the coordinate on the ddimensional substrate. ν is the coefficient of the diffusion
term trying to smoothen the surface and η is the Langevin
noise which tries to roughen the surface3. One may add a
constant current c to the right hand side, but by going over
to a moving frame of reference (h → h + ct) one recovers eq.
395
SPECIAL SECTION:
(4). The noise here is chosen to have zero mean and shortrange correlation as ⟨η(x, t)η(x′,t′)⟩ = 2Dδ(x –x′) δ(t – t′).
One of the important assumptions in this equation is that
the surface is single-valued and there are no overhangs.
One
can
solve
eq.
(4)
exactly
just
by
taking the Fourier transform and obtain the exponents
χ = (2 –d)/2 and z = 2. That the dynamic exponent z is 2
follows from the simple fact that the equation involves a
first derivative in time but a second derivative in space. The
surface is logarithmically rough at d = 2. For d > 2,
fluctuations in the height are bounded and such a surface is more or less flat, better called ‘asymptotically
flat’. From the growth equation one can also derive
the stationary probability distribution for the height
h(x) which takes the form of a Boltzmann factor
P(h(x)) ∝ exp[– (ν/D)∫(∇h)2d dx] resembling an equilibrium
system at a temperature given by D = kBT. This is the
Einstein relation that noise should satisfy to recover
equilibrium probability distribution. Conversely, given a
hamiltonian of the form ∫(∇h)2d dx, the equilibrium dynamics
will be given by eq. (4) with D determined by the
temperature. Nevertheless, if we do not ascribe any thermal
meaning to D, eq. (4) is good enough to describe a
nonequilibrium dynamic process as well. Such a
nonequilibrium growth will have many similarities with
equilibrium processes, differing only in the origin of the
noise, e.g. the expected symmetry h → – h, with ⟨h⟩ = 0 in
equilibrium will be preserved in the nonequilibrium
case also. The growing
Ĉ surface with a correlation C(x, t) =
| x |(2 – d)/2 (t/| x |2) will be similar in both cases for d > 2.
A genuine nonequilibrium process will involve breaking
the up-down symmetry which in equilibrium follows from
detailed balance. It should therefore be represented by a
term involving even powers of h. We already saw that a
constant current (zeroth power) does not add anything new.
Since the origin in space or time or the position of the basal
plane should not matter, the first possible term is (∇h)2. By
looking at the geometry of a rough surface, it is easy to see
that such a term implies a lateral growth that would happen
if a deposited particle sticks to the first particle it touches
on the surface. One gets the Kardar–Parisi–Zhang (KPZ)
equation4
∂h
λ
= ν∇ 2 h + (∇ h ) 2 + η(x , t ) .
∂t
2
(5)
As a consequence of its mapping to the noisy Burger’s
equation, to the statistical mechanics of directed polymer in
a random medium and other equilibrium and nonequilibrium
systems, the KPZ equation has become a model of quite
widespread interest in statistical mechanics. Though we
focus on growth problems in this paper, the KPZ equation
is also applicable in erosion processes.
Taking cue from the development in understanding the
growth phenomenon through the KPZ equation, a vast
class of simulational and analytical models have evolved to
396
explain different experimentally observed growth processes.
Diverse technical tools ranging from simulations with
various dynamical rules to different versions of RG
techniques, mode coupling theory, transfer matrix techniques, and scaling arguments have been employed to
understand kinetic roughening. In this review we attempt to
provide an overview of this phenomenon of roughening of
a growing surface. It is almost beyond the scope of this
review to describe in detail various models and their
experimental relevance. Rather, we focus our attention on a
few examples which may broadly represent a few different
routes along which research has continued.
The plan of this article is as follows. In the next section
we focus on the KPZ equation and its RG description. We
also point out the connection of the KPZ equation to some
other problems of physics. Next, a more generalized growth
mechanism involving nonlocal interactions is presented.
Finally, the progress in understanding the roughening and
super roughening transitions which appear in a very
distinct class of models involving lattice pinning potential is
presented.
KPZ equation and more
Let us first look at the origin of the various terms in eqs (4)
and (5). In both the equations, the noise term represents
random deposition, the fluctuation around the steady value.
As already mentioned, a steady current can be removed
from the equation by going over to a moving frame. The
term involving second derivative of h can represent either
of the two processes. It could be a surface tension
controlled diffusion process, in which a particle comes to
the surface and then does a random walk on the surface to
settle at the minimum height position, thereby smoothening
the surface. An alternative interpretation would be that
there is desorption from the surface and the process is
proportional to the chemical potential gradient. The
chemical potential of the particles on the surface cannot
depend on h or gradient of h because it is independent of
the arbitrary base or its tilt. The chemical potential then is
related to the second derivative of h. This also has a
geometric meaning that ∇2h is related to the local curvature.
The larger the curvature, the higher is the chance to desorb
because of a lesser number of neighbours. In the KPZ
equation, the nonlinear term represents lateral growth. The
diffusion-like term can then be thought of either (a) as an
alternative that a particle coming to the surface instead of
sticking to the first particle it touches, deposits on the
surface and then diffuses, or (b) as a random deposition
process with desorption. In either case, the noise term tends
to roughen the surface, the diffusion term, of whatever
origin, smoothens it while the nonlinear term leads to a
laterally growing surface. Even if the smoothening linear
term is not present, RG or the scaling argument indicates
that such a term is generated on a large length scale.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
The KPZ equation has a special symmetry not present in
the EW case. This is the tilt symmetry (often called Galilean
invariance – a misnomer, though, in this context). If we tilt
the surface by a small angle, then with a reparametrization
h′ = h +ε· x and x = x + λεt′ and t = t′, the equation remains
invariant for small ε. This transformation depends only on λ,
the coefficient of the nonlinear term and fails for λ= 0. Since
this tilt symmetry is to be maintained no matter at what
lengthscale we look at, λmust be an RG invariant.
Let us now perform a length rescaling analysis. Under a
change of scale as x → bx, t → b zt and h → b χh, KPZ
equation transforms as
b χ–z
∂h
λ
= νb χ–2∇2h + b 2χ–2(∇h)2 + b –d/2–z/2η,
∂t
2
(6)
where the noise correlation has been used to obtain the
scaling of the noise term. Therefore under this scale
transformation different parameters scale as ν → b z–2ν,
D → b z–d–2χD and λ→ b χ+z–2λ. For λ= 0, the equation
remains invariant provided z = 2 and χ = (2 – d)/2. These
are just the exponents one expects from the EW model.
(Such surfaces with anisotropic scaling in different
directions like x and h are called self-affine.) Though we
cannot predict the exponents from eq. (6) when λ≠ 0, it
does tell us that a small nonlinearity added to the EW
equation scales with a scaling dimension χ + z – 2. This term
is always relevant in one-dimension, because it scales like
b 1/2. This type of scaling argument also shows that no other
integral powers of derivatives of h need be considered in
eq. (5) as they are all irrelevant, except (∂h/∂x)3 at d = 1,
which, however, detailed analysis shows to be marginally
irrelevant. Based on this analysis, we reach an important
conclusion that the nonequilibrium behaviour in onedimension, and in fact for any dimension below two, would
be distinctly different from the equilibrium behaviour. For
dimensions greater than two, EW or equilibrium surfaces, as
already mentioned, are asymptotically flat with χ = 0, z = 2,
and so, a small nonlinearity is irrelevant because it will
decay with b. In other words, the growth in higher
dimensions for small λ would be very similar to equilibrium
problems because the EW model is stable with respect to a
small perturbation with nonlinearity. The simple scaling
argument does not tell us if the nature of the surface
changes for large λfor d > 2, but an RG analysis shows that
it does change. That the nonequilibrium growth is always
different in lower dimensions and in higher dimensions
(greater than two), and that there will be a dynamic phase
transition from an equilibrium-like to a genuine
nonequilibrium behaviour, explains the source of excitement
in this minimal KPZ equation, in the last two decades.
If the nonlinear parameter λ is to remain an invariant, i.e.
independent of b in eq. (6), then χ + z = 2, a relation which
need not be satisfied by the equilibrium growth. It is this
relation connecting the two exponents of the scaling
function of eq. (2) that distinguishes nonequilibrium growth
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
from equilibrium, the former requiring one less exponent
than the latter. We wonder if such an exponent relation is
generally true for all nonequilibrium systems.
Though we are far away from a complete understanding
of all the nuances and details of the KPZ equation, the RG
analysis has been very successful in identifying different
phases, nature of phase transitions, and, in certain cases,
relevant exponents. In brief, the various results obtained
from RG analysis are as follows. In one-dimension and for
d < 2, even a small nonlinearity, as already mentioned, being
relevant in the RG sense, leads to new values of roughening
and dynamic exponents, and is characterized by a RG fixed
point. Beyond d = 2, there is a phase transition demarcating
two different types of surfaces. A small nonlinearity is
irrelevant around EW model and the surface is almost flat
with χ = 0 and z = 2. A strong nonlinear growth, however,
drives the system to a different phase with rougher surface
where χ ≠ 0. Several aspects of this phase transition can be
studied from RG but the strong λregime is still out of reach,
because of the absence of any RG fixed point.
The KPZ equation in d = 1 has distinct nonequilibrium
behaviour, and the scaling behaviour is the same no matter
how small or large λ is. More peculiar is the existence of a
stationary probability distribution of the height in onedimension which is the same as for the linear EW model.
This is not just an accident but a consequence of certain
subtle relations valid only in one-dimension. We do not go
into those issues here. The same stationary distribution
implies that the nonlinearity does not affect the stationary
state solution, and χ = 1/2. The two models, however, differ
in the dynamic exponent which, in the case of KPZ growth,
has to satisfy χ + z = 2. This leads to an exact answer
z = 3/2. Its significance can be grasped if we compare
various known cases. For ballistic motion, distance goes
linearly with time so that the dynamic exponent is z = 1
while for diffusive motion or in quantum mechanics (e.g. a
nonrelativistic free quantum particle), z = 2 as is also the
case for EW. Here is an example where the nonequilibrium
nature of the problem leads to a completely new exponent
connecting the scaling of space and that of time.
Dynamic renormalization group analysis
A dynamic RG analysis is a more general approach
applicable for dynamics which e.g. may be governed by the
Langevin equation for the appropriate dynamical variable.
For our problem, it is easier to work in Fourier coordinates q,
and ωconjugate to space and time. Long distance, long time
implies q → 0 and ω→ 0, and q can be taken as the inverse
wavelength at which the height variable is probed. The
magnitude of wave vector q varies from 0 to Λ, where the
upper cutoff is determined by the underlying microscopic
length scale like lattice spacing or size of particles, etc. In
the Fourier space, different Fourier modes in the linear EW
model get decoupled so that each h(q, ω) for each (q, ω)
behaves independently. It is this decoupling that allows the
397
SPECIAL SECTION:
simple rescaling analysis of eq. (6) or dimensional analysis
to give the correct exponents. For the KPZ equation, the
nonlinear term couples heights of various wavelengths and
therefore any attempt to integrate out the large (q, ω) modes
will affect h with low values of (q, ω). This mixing is taken
into account in the RG analysis which is implemented in a
perturbative way. One thinks of the noise and the nonlinear
term as disturbances affecting the EW-like surface. If we
know the response of such a surface to a localized
disturbance we may recover the full response by summing
over the disturbances at all the points and times. However,
this disturbance from the nonlinear term itself depends on
the height, requiring an iterative approach that generates
successively a series of terms. By averaging over the noise,
one then can compute any physical quantity. At this stage
only degrees of freedom with q in a small shell e–lΛ < q < Λ
is integrated out. In real space this corresponds to
integrating out the small scale fluctuation. The contribution
from this integration over the shell is absorbed by
redefining the various parameters ν, λ and D. These are the
coupling constants for a similar equation as eq. (5) but with
a smaller cutoff Λe–l. A subsequent rescaling then restores
the original cutoff to Λ. Following this procedure, the flow
equations for different parameters ν, D, and λ can be
obtained5. Using the exponent relation predicted from the
Galilean invariance and the RG invariance of ν, the flow
equations for all the parameters can be combined into a
single flow equation for λ2 = λ2D/ν3 (with Λ = 1). This is
the only dimensionless parameter that can be constructed
from λ, ν, D, and Λ, and it is always easier to work with
dimensionless quantities. Its recursion relation is
dλ 2 − d
2d − 3 3
=
λ + Kd
λ ,
dl
2
4d
(7)
where Kd is the surface area of a d-dimensional sphere
divided by (2π)d. The invariance of ν under the RG
2−d
transformation implies z = 2 – Kd λ 2 4 d , and the Galilean
invariance provides the value of χ = 2 – z once the value of
z is known. To be noted here is that the dynamic exponent
is different from 2 by a term that depends on λ coming from
the renormalization effects.
A few very important features are apparent from eq. (7).
From the fixed point requirement dλ /dl = 0, we find that at
d = 1, there is a stable fixed point λ2 = 2/K1. At this fixed
point z = 3/2 and χ = 1/2 supporting the results predicted
from the symmetry analysis. At d = 2, the coupling is
marginally relevant, indicating a strong coupling phase not
accessible in a perturbation scheme. At d > 2, the flow
equation indicates two different regimes, namely a weak
coupling regime where λ asymptotically vanishes leading
to a flat EW phase with χ = 0, z = 2, and a strong coupling
rough phase, the fixed point of which cannot be reached by
perturbation analysis.
Owing to this limitation of the RG analysis based on the
398
perturbation expansion, the scaling exponents in this strong
coupling phase cannot be determined by this RG scheme.
Different numerical methods yield z = 1.6 at d = 2. The
phase transition governed by the unstable fixed point of λ
is well under control with z = 2 for all d > 2. To explore the
strong coupling phase, techniques like self-consistent mode
coupling approach, functional RG, etc. have been employed,
but even a basic question whether there is an upper critical
dimension at which z will again become 2 remains
controversial.
Relation with other systems
The relation of the KPZ equation with other quite unrelated
topics in equilibrium and nonequilibrium statistical
mechanics is impressive. Here, we provide a very brief
account of these systems.
Noisy Burgers equation: By defining a new variable
v = ∇h, we obtain an equation
∂v
= D∇2v+λv · ∇ν + f(x, t),
∂t
(8)
where the noise term f = ∇η. The above equation
represents the noisy Burgers equation for vortex free
(∇ × v = 0) fluid flow with a random force. This equation is
very important in studies of turbulence. The tilt invariance
of the KPZ equation turns out to be the conventional
Galilean invariance for the Burgers equation (for λ= 1), and
that is how the name stayed on.
Directed polymer in a random medium: A directed
polymer, very frequently encountered in different problems
in statistical mechanics, is a string-like object which has a
preferred longitudinal direction along which it is oriented,
with fluctuations in the transverse direction. The flux lines
in type II or high Tc superconductors are examples of such
directed polymers in 3-dimensions, while the steps on a
vicinal or miscut crystal surface or the domain walls in a
uniaxial two-dimensional system are examples in twodimensions. The formal mathematical mapping to such
objects
follows
from
a
simple
(Cole–
Hopf) transformation of the KPZ equation
using
W(x, t) = exp
The Cole–Hopf transformation linearizes
the nonlinear KPZ equation and the resulting linear
diffusion equation (or imaginary time Schrödinger equation)
is identical to that satisfied by the partition function of a
directed polymer in a random potential. For such random
problems, one is generally interested in the averages of
thermodynamic quantities like the free energy and we see
that the noise averaged height ⟨h(x, t)⟩ gives the average
free energy of a directed polymer of length t with one end at
the origin and the other end at x. This is a unique example of
a system where the effect of such quenched averaging of
free energy can be studied without invoking any tricks (like
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
the replica method). This has led to many important results
and enriched our understanding of equilibrium statistical
mechanics. Recently, this formulation has been extended to
study details of the properties of the random system near
the phase transition point and overlaps in lower
dimensions6,7. It turns out that one needs an infinite number
of exponents to describe the statistical behaviour of the
configurations of the polymer in the random medium8. We
do not go into this issue as this is beyond the scope of this
article.
An interesting connection between the 1 + 1dimensional KPZ equation and the equilibrium statistical
mechanics of a two-dimensional smectic-A liquid crystal
has been recently established by Golubovich and Wang9.
This relationship further provides exact approach to study
the anomalous elasticity of smectic-A liquid crystals.
Apart from these, there are a number of other relations
between KPZ equation and kinetics of annihilation
processes with driven diffusion, the sine–Gordon chain, the
driven diffusion equation and so on.
Beyond KPZ
Conservation condition: The situation encountered in
molecular-beam epitaxy (MBE) for growth of thin films is
quite different from the mechanism prescribed by the KPZ
equation2. In MBE, surface diffusion takes place according
to the chemical potential gradient on the surface, respecting
the conservation of particles. If the particle concentration
does not vary during growth, then a mass conservation
leads to a volume conservation and the film thickness is
governed by an underlying continuity equation
∂h
+∇ · j = η,
∂t
(9)
where j is the surface diffusion current which states that the
change of height at one point is due to flow into or out from
that point. The current is then determined by the gradient of
the chemical potential, and since the chemical potential has
already been argued to be proportional to the curvature
∇2h, the growth equation thus becomes a simple linear
equation involving ∇4h which, like the EW model, is exactly
solvable. Taking into account the effect of nonlinearity the
full equation can be written as
∂h
λ


= –∇2 ν∇ 2 h + (∇ h ) 2  +η(x, t),
∂t
2


(10)
where the noise correlation is ⟨η(x, t)η(x′, t′)⟩ = 2D∇2
δ(x – x′)δ(t – t′), if the noise also maintains conservation (if
it originates from the stochasticity of diffusion) or would be
the same white noise as in the KPZ equation, if the noise is
from random deposition. It goes without saying that the
exponents are different from the EW model even for the
linear theory. The invariance of λ in this case leads to a
different relation between χ and z. At the dimension of
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
physical interest d = 2, this growth equation leads to an
enhanced roughness than the KPZ case and may explain
the results of experiments of high temperature MBE.
Quenched noise: A different type of generalization of the
KPZ equation was to explore the motion of domain walls or
interfaces in a random medium. In this case, the noise is not
explicitly dependent on time but on the spatial position and
the height variable. Such a noise has been called quenched
noise because the noise is predetermined and the interface
or the surface moves in this random system. The simple
features of the KPZ equation and the EW model are lost.
Functional RG analysis and numerical studies are attempted
to clarify the question of universality classes and details of
dynamics in such cases. The important concept that
emerged in this context is the depinning transition so that
the surface remains pinned by the randomness until the
drive exceeds a certain critical value. Interface depinning is
an example of a nonequilibrium phase transition. The
velocity of the surface near this depinning transition also
has critical-like behaviour with long-range correlations.
Below the threshold, the dynamics is sluggish, while just
above the threshold, the velocity is in general not
proportional to the drive but obeys a power law with a
universal exponent. For a very strong drive (or large
velocity of the interface) the moving surface encounters
each site only once, and therefore the noise is effectively
like a space–time dependent noise rather than the quenched
one. The nature of the surface would then be like KPZ.
Coloured noise: In the previous section we discussed the
KPZ equation with white noise. If the noise is coloured in
the sense that there is correlation in space or time or both,
the universal behaviour, the phase transitions and the
properties are different but still can be studied by the same
RG technique. Several aspects of the problem, especially the
role of noise correlation, have been explored10.
All of the above seem to suggest that if there is no
conservation law, then the KPZ equation is the equation to
describe any nonlinear or nonequilibrium growth process
and all phenomena can be put in one of the known
universality classes. However, experimentally KPZ
exponents seem to elude us so far1,2,11. Since results are
known exactly in one-dimension, special one-dimensional
experiments were conducted like paper burning, interface
motion in paper, colloid suspension, etc., but KPZ
exponents have not been seen. In the colloid experiment11,
the surface formed by the depositing colloids on the
contact line (d = 1) between the colloid latex film and a glass
slide was measured from video images. This method yields
χ = 0.71 but cannot determine the dynamic exponent. A
recent analysis12 of tropical cumulus clouds in the upper
atmosphere, from satellite and space shuttle data from 0.1 to
1000 km, seems to agree with the KPZ results in d = 2.
Kinetic roughening with nonlocality
399
SPECIAL SECTION:
In spite of a tremendous conceptual and quantitative
success of the KPZ equation in describing the
nonequilibrium growth mechanism, the agreement with
experimentally observed exponents is rather unsatisfactory.
One wonders whether there is any relevant perturbation
that drives the systems away from the KPZ strong coupling
perturbation. One goal of this section is to point out that
indeed there can be long-range interactions that may give
rise to non-KPZ fixed points.
In many recently studied systems involving proteins,
colloids, or latex particles the medium-induced interactions
are found to play an important role13. This nonlocal
interaction can be introduced by making a modification of
the nonlinear term in the KPZ equation. Taking the gradient
term as the measure of the local density of deposited
particles, the long-range effect is incorporated by coupling
these gradients at two different points. The resulting
growth equation is a KPZ-like equation with the nonlinear
term
modified
as14
1
2 ∫ dr′ V (r′)∇h(r + r′, t) · ∇h(r – r′, t). For generality, we
take V (r′) to have both short- and long-range parts with a
specific form in Fourier space as V (k) = λ0 + λρk – ρ such that
in the limit λρ → 0, KPZ results are retrieved. The aim is to
observe whether the macroscopic properties are governed
by only λ0 and hence KPZ-like or the behaviour is
completely different from KPZ due to the relevance of λρ
around the KPZ fixed points.
A scaling analysis as done in eq. (6) clearly indicates
different scaling regimes and the relevance of λ0 and λρ for
d > 2 at the EW fixed point. For any λρ(≠ 0) with ρ > 0, the
local KPZ theory (i.e. λρ = 0 and χ + z = 2) is unstable under
renormalization and a non-KPZ behaviour is expected. For
2 < d < 2 + 2ρ, only λρ is relevant at the EW fixed point. The
exponents of the non-KPZ phases can be obtained by
performing a dynamic RG calculation14. By identifying the
phases with the stable fixed points, we then see the
emergence of a new fixed point where the long-range
features dominate (χ + z = 2 + ρ). Most importantly, at
d = 2, the marginal relevance of λ is lost and there is a
stable fixed point (LR) for ρ > 0.0194.
On the experimental side, there are experiments on
colloids with χ = 0.71 which is the value also obtained from
paper burning exponents. For colloids, hydrodynamic
interactions are important. Similar long-range interactions
could also play a role in paper burning experiment due to
the microstructure of paper. With this χ our exponents
suggest ρ = – 0.12 at d = 1 at the long-range fixed point.
Further experiments on deposition of latex particles or
proteins yielding the roughness of growing surface have
not been performed. Probably such experiments may reveal
more insights on this growth mechanism.
More recently, the effect of coloured noise in presence of
nonlocality has been studied15 and the nature of the phases
and the various phase transitions clarified. A conserved
version of the nonlocal equation has also been considered
and it shows rich behaviour16.
400
Roughening transition in nonequilibrium
It is interesting to study the impact of equilibrium phase
transitions on the nonequilibrium growth of a surface. This
is the situation observed experimentally in growth of solid
4
He in contact with the superfluid phase17. There is an
equilibrium roughening transition at TR = 1.28 K. For T > TR
the growth velocity is linear in the driving force F (chemical
potential difference), but for T < TR the velocity is
exponentially small in the inverse of the driving force. For
infinitesimal drive, the mobility which is the ratio of the
growth velocity and F vanishes with a jump from a finite
value at the transition. With a finite force the transition is
blurred and the flat phase below TR in equilibrium becomes
rougher over large length scale.
The equilibrium roughening transition is an effect of
discrete translational symmetry of the lattice. The
equilibrium dynamics in this case is essentially governed by
the Langevin equation
∂h
 2π

= K∇ 2 h (r , t) − V sin 
h (r, t )  + ζ (r, t ) ,
∂t
 a

(11)
where the sine term favours a periodic structure of spacing
a. Extensive investigations have been done on this
equilibrium model. At low temperature, this periodic
potential is relevant and it ensures that minimum energy
configuration is achieved when φ is an integer multiple of
lattice periodicity. In this phase, the surface is smooth and
the roughness is independent of length. In the high
temperature phase the equilibrium surface is thermally
rough and the roughness is logarithmic
C(L, τ) ~ ln[Lf(τ/Lz)].
(12)
The critical point is rather complicated and goes by the
name of Kosterlitz–Thouless transition, first discussed in
the context of defect mediated transitions in twodimensional XY magnets18.
For a nonequilibrium crystal growth problem, one needs
to introduce the KPZ nonlinear term in eq. (11). There is no
longer any roughening transition. The fact that away from
equilibrium the roughening transition is blurred is
manifested by the domination of the nonlinear term and the
suppression of the pinning potential in the asymptotic
regime19.
A very nontrivial situation arises when the surface
contains quenched disorder which shifts the position of the
minima of the pinning potential in an arbitrary random
fashion20,21. In this case, there is a new phase transition
which is drastically different from the equilibrium
roughening transition. This transition is called super
roughening. Above the transition temperature, i.e. for
T > Tsr, the surface is logarithmically rough as it is in the
high temperature phase of the pure problem. However in the
low temperature phase, i.e. for T < Tsr, the surface is no
longer flat and is even rougher than the high temperature
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
phase. Recent numerical treatments suggest that the surface
roughness behaves as (ln L)2. In the nonequilibrium
situation, the linear response mobility vanishes
continuously at the transition temperature unlike the jump
discontinuity in the pure case. A general treatment with a
correlated disorder elucidates the connection between the
roughening and super roughening transition and one
observes that the roughening turns into a super roughening
transition if the disorder correlation decays sufficiently fast.
Away from equilibrium, the super roughening transition is
essentially dominated by the KPZ nonlinearity and instead
of the logarithmic roughness, an asymptotic power law
behaviour of the roughness is found over all temperature
ranges.
In a similar situation in the nonequilibrium case, one
needs to study the role of the KPZ nonlinearity with longrange disorder correlation22. A functional renormalization
scheme with an arbitrary form of the disorder correlation
turns out to be useful, though a detailed solution is not
available. It is found that the flow of the KPZ nonlinearity
under renormalization, with power law form of the disorder
correlation, is such that it decays with length. This implies
that nonequilibrium feature does not set in over a certain
length scale. Over this scale one would then expect usual
roughening transition. However, there is generation of a
driving force due to the nonlinearity, and the growth of this
force with length scale would invalidate use of perturbative
analysis. For large length scales, one expects a KPZ-type
power law roughness of the surface. Nevertheless, the
initial decay of the nonlinearity with the length scale due to
the long-range correlation of the disorder is an interesting
conclusion that seems to be experimentally detectable.
Remarks
In this brief overview, we attempted to focus on the
difference between equilibrium and nonequilibrium growth
problems with an emphasis on the scaling behaviour and
RG approach. Many details with references to pre-1995
papers can be found in Halpin-Healy and Zhang1, and
Barbási and Stanley2, which should be consulted for more
detailed analysis. Though the success story of the KPZ
equation is rather impressive, there are still many
unresolved, controversial issues. In fact for higher
dimensions, the behaviour is not known with as much
confidence as for lower dimensions. Developments in this
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
direction are awaited.
Note added in proof:
1. The growth mechanism of metal-organic films deposited
by the Langmuir–Blodgett technique has been studied in
ref. 23 by X-ray scattering and atomic force microscopy.
The results have been interpreted by a combination of 1dimensional EW equation (eq. (4)) and 2-dimensional linear
conserved equation (eq. (10)) with conserved noise.
2. For effects of nonlocality in equilibrium critical dynamics,
see ref. 24.
1. Halpin-Healy, T. and Zhang, Y. C., Phys. Rep., 1995, 254, 215.
2. Barbási, A.-L. and Stanley, H. E., Fractal Concepts in Surface
Growth, Cambridge Univ. Press, Cambridge, 1995.
3. Edwards, S. F. and Wilkinson, D. R., Proc. R. Soc. London A,
1982, 381, 17.
4. Kardar, M., Parisi, G. and Zhang, Y.-C., Phys. Rev. Lett., 1986,
57, 1810.
5. Medina, E. et al., Phys. Rev. A, 1989, 39, 3053.
6. Mezard, M., J. Phys. (Paris), 1990, 51, 1831.
7. Mukherji, S., Phys. Rev. E, 1994, 50, R2407.
8. Mukherji, S. and Bhattacharjee, S. M., Phys. Rev. B, 1996, 53,
R6002.
9. Golubovic, L. and Wang, Z.-G., Phys. Rev. E, 1994, 49, 2567.
10. Chekhlov, A. and Yakhot, V., Phys. Rev. E, 1995, 51, R2739;
ibid, 1995, 52, 5681; Hayat, F. and Jayaprakash, C., Phys. Rev.
E, 1996, 54, 681; Chattopadhyay, A. K. and Bhattacharjee, J.
K., Euro. Phys. Letts., 1998, 42, 119.
11. Lei, X. Y., Wan, P., Zhou, C. H. and Ming, L. B., Phys. Rev. E,
1996, 54, 5298.
12. Pelletier, J. D., Phys. Rev. Lett., 1997, 78, 2672.
13. Feder, J. and Giaever, I., J. Colloid Interface Sci., 1980, 78, 144;
Ramsden, J. J., Phys. Rev. Lett., 1993, 71, 295; Pagonabarraga, I.
and Rubi, J. M., Phys. Rev. Lett., 1994, 73, 114; Wojtaszczyk, P.
and Avalos, J. B., Phys. Rev. Lett., 1998, 80, 754.
14. Mukherji, S. and Bhattacharjee, S. M., Phys. Rev. Lett., 1997,
79, 2502.
15. Chattopadhyay, A. K., Phys. Rev. E, cond-mat/9902194 (to
appear).
16. Jung, Y., Kim, I. and Kim, J. M., Phys. Rev. E, 1998, 58, 5467.
17. Noziers, P. and Gallet, F., J. Phys. (Paris), 1987, 48, 353.
18. Chui, S. T. and Weeks, J. D., Phys. Rev. Lett., 1976, 38, 4978.
19. Rost, M. and Sphon, H., Phys. Rev. E, 1994, 49, 3709.
20. Tsai, Y.-C. and Shapir, Y., Phys. Rev. Lett., 1992, 69, 1773;
Phys. Rev. E , 1994, 50, 3546; ibid, 1994, 50, 4445.
21. Scheidl, S., Phys. Rev. Lett., 1995, 75, 4760.
22. Mukherji, S., Phys. Rev. E, 1997, 55, 6459.
23. Basu, J. K., Hazra, S. and Sanyal, M. K., Phys. Rev. Lett., 1999,
82, 4675.
24. Sen, P., J. Phys. A., 1999, 32, 1623.
401
SPECIAL SECTION:
Suspensions far from equilibrium
Sriram Ramaswamy
Centre for Condensed-Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560 012, India
A review is presented of recent experimental and
theoretical work on the dynamics of suspensions of
particles in viscous fluids, with emphasis on phenomena
that should be of interest to experimenters and
theoreticians working on the statistical mechanics of
condensed matter. The article includes a broad introduction to the field, a list of references to important papers,
and a technical discussion of some recent theoretical
progress in which the author was involved.
Equilibrium and nonequilibrium suspensions
SUSPENSIONS of particles in a fluid medium are all around
us 1,2. Examples include river water, smog, blood, many liquid
foods, many medicines, cosmetics, paints, and so on. The
particles, which we shall call the solute, are generally
submicron to several microns in size, and the suspending
fluid, which we shall call the solvent, is frequently less
dense than the solute. The viscosity of the solvent could
range from that of water (or air) to several thousands of
times higher. The field of suspension science has
distinguished origins: Einstein’s3 interest in Brownian
motion as evidence for the existence of molecules led
him to calculate the viscosity and diffusivity of a
dilute suspension (before diversions such as relativity
and quantum mechanics took him over completely);
Smoluchowskii’s4,5 studies of sedimentation and aggregation in colloids led to major advances in the theory of
stochastic processes; somewhat more recently, the
challenging many-body nature of the dynamics of
suspensions was highlighted in the work of Batchelor6.
Today, the study of the static and dynamic properties of
suspensions from the point of view of statistical mechanics
is a vital part of the growing field of soft condensed matter
science.
In applications and in industrial processing, suspensions
are usually subjected to strongly nonequilibrium
conditions. By nonequilibrium I mean that the system in
question is driven by an external agency which does work
on it – stirring, pumping, agitation – which the system
dissipates internally. The bulk of interesting and, by and
large, incompletely understood phenomena in suspension
science, and in the area of complex fluids in general, are also
those that occur far from equilibrium. Problems in which I
have been or am currently interested are: the melting of
e-mail: sriram@physics.iisc.ernet.in
402
colloidal crystals when they are sheared7–13; spontaneous
segregation in sheared hard-sphere suspensions 14; the
collapse of elastic colloidal aggregates under gravity15, and
its possible relation to the instability of sedimenting
crystalline suspensions12,16,17; the enhancement of redblood-cell sedimentation rates in the blood of a very sick
person18; and the puzzle19–23 of the statistics of velocity
fluctuations in ultraslow fluidized beds. (References 24–31
should give the reader an idea of the range of this field.)
None of the observations in the papers I have mentioned
can be understood purely with the methods used to study
hydrodynamic instabilities: they are fluctuation phenomena,
and therefore belong in this special issue on nonequilibrium
statistical physics.
A suspension can be out of equilibrium in a number of
ways: in particular, it could be in a nonstationary state (in
the process of settling or aggregating or crystallizing, for
example), or it could be stuck in a metastable amorphous
state 32 or it could be held, by the application of a driving
force, in a time-independent but not time-reversal invariant
state with characteristics different from the equilibrium
state. In this article we shall mainly be concerned with these
nonequilibrium steady states, characterized by a constant
mean throughput of energy. These are to be contrasted
with thermal equilibrium states which have a constant
mean budget of energy, i.e. a temperature. A suspension of
charged Brownian particles with precisely the same density
as the solvent, such as are discussed in the review article
by Sood33 is the standard example of an equilibrium
suspension. The two most common ways of driving a
suspension out of equilibrium are shear34, wherein the
solute and solvent are jointly subjected to a velocity
gradient, and sedimentation or fluidization35, where the
velocity of solute relative to solvent is on average nonzero
and spatially uniform. This latter class of problems is very
close to the currently rather active area of driven diffusive
systems 36, which has provided much insight into statistical
physics far from equilibrium.
Accordingly, this review will focus largely on sedimentation and fluidized beds, although a brief summary of
shear-flow problems with relevant references will be
provided. Even with this restriction, the field is much too
vast to allow anything like representative coverage, so my
choice of topics will be dictated by familiarity, in the hope
that the problems I highlight will attract the reader to the
area. It is in that sense not a true review article, but an
advertisement for a field and therefore includes as an
integral part a reasonably large list of references. The aim is
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
to give a general idea of the richness of phenomena in
nonequilibrium suspensions, as well as a technical
understanding of some of my theoretical work in this field.
To theoreticians, especially in India, this article argues that
the vast literature on suspension dynamics is a largely
untapped lode of problems in the statistical physics of
driven systems. Elegant models of the sort popular among
practitioners of the more mathematical sort of statistical
mechanics could acquire a greater meaning and relevance if
born out of an attempt to understand phenomena in these
systems (for an example of such an attempt, see the section
on the stability of steadily drifting crystals). I hope in
addition that this article persuades condensed-matter
experimenters in India to look for problems in these and
related soft-matter systems (including powders, on which I
am not competent to write), which are as rich as traditional
solid-state systems, without the complications of low
temperatures, high vacuum, etc.
I shall work exclusively in the limit of slow motions
through highly viscous fluids, i.e. the limit of low Reynolds
number Re ≡ Ua/ν, where U is a typical velocity, a the
particle
size,
and
the
kinematic
viscosity
ν ≡ µ/ρ, µ and ρ being respectively the shear viscosity
and mass density of the solvent. Re, which measures the
relative importance of inertia and viscosity in a flow, can
here be thought of simply as the fraction of a particle’s own
size that it moves if given an initial speed U, before
viscosity brings it to a halt. For bacteria (sizes of order 1 to
10 microns) swimming at, say, 1 to 10 microns a second in
an aqueous medium, Re ~ 10–6 to 10–4, and for polystyrene
spheres (specific gravity 1.05, radius 1 to 10 µm)
sedimenting in water, Re ~ 10–7 to 10–4. So we are amply
justified in setting Re to zero. Recall that in the low Re limit,
things move at a speed proportional to how hard they are
pushed, and stop moving as soon as you stop pushing.
While this corresponds very poorly to our experience of
swimming, it is an accurate picture of how water feels to a
bacterium or to a colloidal particle. A good general
introduction to the subject of zero Reynolds-number flow is
given in Happel and Brenner37, although standard fluid
dynamics texts38,39 also discuss it. A very detailed treatment
of the dynamics of hydrodynamically interacting particles
can be found in ref. 40.
The dimensionless control parameter relevant to zero
Reynolds number suspensions is the Peclet number
Pe ≡ Ua/D (ref. 1), where D = T/6πµa is the Stokes–Einstein
diffusivity of the particle at temperature T. In a shear flow
with velocity gradient γ&, clearly U ~ γ&
a, so that Pe ~
2
which isγ&
athe
/Dratio
, of a diffusion time to a shearing time. In
sedimentation, Pe measures the relative importance of
settling (gravitational forces) and diffusion (thermal
fluctuations), and can be expressed as mRga/T, where mRg is
the buoyancy-reduced weight of the particle, g being the
acceleration due to gravity. Since Pe ∝ a 4, it is clear that
small changes in the particle radius make a large difference.
For polystyrene spheres in water, Pe ranges from 0.5 to 5000
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
as the radius is varied from 1 to 10 µm. Suspensions with
Pe >> 1 are termed non-Brownian, since their behaviour is
dominated by the gravity-induced drift and the resulting
solvent flow, not by thermal fluctuations. This Pe = ∞,
Re = 0 limit turns out to be a very interesting one.
The most trivial example of a nonequilibrium suspension
is a single solute particle, heavier than its solvent, so that in
the presence of gravity it settles to the bottom of the
container. If we wish to study the steady-state properties of
this settling process, we must make the particle settle
forever. There are two ways of doing this: either use a very
tall container and study the flow around the particle as it
drifts past the middle, or arrange to be in the rest frame of
the particle. The latter is realized in practice by imposing an
upward flow so that the viscous drag on the particle
balances its buoyancy-reduced weight, suspending the
particle stably. The nature of the flow around such an
isolated particle, without Brownian motion and in the limit of
low velocity, was studied about a century and a half ago41.
As soon as the number of particles is three or greater, the
motion becomes random even in the absence of a thermal
bath (this can be seen either in Stokesian dynamics
simulations42 or simply by using particles large enough that
Brownian motion due to thermal fluctuations is negligible)
because the flow produced by each particle disturbs the
others, leading to chaos43. In the limit of a large number of
particles, the problem can thus be treated only by the
methods of statistical mechanics (suitably generalized to the
highly nonequilibrium situations we have in mind). A
collection of particles stabilized against sedimentation by an
imposed upflow is called a fluidized bed.
It is useful here to distinguish two classes of nonequilibrium systems. In thermal driven systems the source
of fluctuations is temperature, and the driving force (shear,
for example) simply advects the fluctuations injected by the
thermal bath. An example of this type would be a
suspension of submicron particles, whose Brownian motion
is substantial, subjected to a shear flow. Non-thermal
driven systems are more profoundly nonequilibrium, in that
the same agency is responsible for the driving force and the
fluctuations. A sedimenting non-Brownian many-particle
suspension is thus an excellent example of a non-thermal
nonequilibrium system.
The structure of this article is as follows. In the next
section I summarize our present state of understanding of
the shear-induced melting of colloidal crystals. Then, I
discuss the physics of steadily sedimenting colloidal
crystals, including the construction of an appropriate driven
diffusive model for which exact results for many quantities
of physical interest can be obtained. The next section
introduces the reader to the seemingly innocent problem of
the steady sedimentation of hard spheres interacting only
via the hydrodynamics of the solvent. Puzzles are
introduced and partly resolved. The last section is a
summary.
403
SPECIAL SECTION:
Shear-melting of colloidal crystals
It has been possible for some years now to synthesize
spheres of polystyrene sulphonate and other polymers,
with precisely controlled diameter in the micron and
submicron range. These ‘polyballs’ have become the
material of choice for systematic studies in colloid science33.
In aqueous suspension, the sulphonate or similar acid end
group undergoes ionization. The positive ‘counterions’ go
into solution, leaving the polyballs with a charge of several
hundred electrons. The counterions together with any ionic
impurities partly screen the Coulomb repulsion between the
polyballs, yielding, to a good approximation in many
situations, a collection of polymer spheres interacting
through a Yukawa potential exp(– κ r)/r with a screening
length κ–1 in the range of a micron, but controllable by the
addition of ionic impurities (NaCl, HCl, etc.). Not
surprisingly, when the concentration of particles is large
enough that the mean interparticle spacing a s ~ κ–1, these
systems order into crystalline arrays with lattice spacings
~ a s comparable to the wavelength of visible light. The
transition between colloidal crystal and colloidal fluid is a
perfect scale model of that seen in conventional simple
atomic liquids33. It is first-order and the fluid phase at
coexistence with the crystal has a structure factor whose
height is about 2.8. The transition is best seen by varying
not temperature but ionic strength. These colloidal
crystals33 are very weak indeed, with shear moduli of the
order of a few tens of dyn/cm2. The reason for this is clear:
the interaction energy between a pair of nearest neighbour
particles is of order room temperature, but their separation is
about 5000 Å. Thus on dimensional grounds the elastic
moduli, which have units of energy per unit volume, should
be scaled down from those of a conventional crystal such
as
copper
by
a
factor
of
5000–3 ~ 10–11. It is thus easy to subject a colloidal crystal to
stresses which are as large as or larger than its shear
modulus, allowing one to study a solid in an extremely
nonlinear regime of deformation. One can in fact make a
colloidal crystal flow if the applied stress exceeds a modest
minimum value (the yield stress) required to overcome the
restoring forces of the crystalline state.
When a colloidal crystal is driven into such a steadily
flowing state, it displays at least two different kinds of
behaviour, with a complex sequence of intermediate stages
which appear to be crossovers rather than true
nonequilibrium phase transitions, and whose nature
depends strongly on whether the system is dilute and
charge-stabilized or concentrated and hard-sphere-like7–9. I
shall focus on the two main ‘phases’ seen in the
experiments, not the intermediate ones. At low shear-rates,
it flows while retaining its crystalline order: crystalline
planes slide over one another, each well-ordered but out of
registry with its neighbours. At large enough shear-rates, all
order is lost, through what appears to be a nonequilibrium
phase transition from a flowing colloidal crystal to a flowing
404
colloidal liquid. The shear-rate required to produce this
transition depends on the ionic strength n i, and appears to
go to zero as n i approaches the value corresponding to the
melting transition of a colloidal crystal at equilibrium. This
connection to the equilibrium liquid–solid transition
prompted us to extend the classical theory44 of this
transition to include the effects of shear flow. I will not
discuss our work on that problem here: the interested reader
may read about them in Ramaswamy and Renn10 and Lahiri
and Ramaswamy11. The problem remains incompletely
understood, and work on it especially in experiments and
simulations continues13. I mention it here as an outstanding
problem in nonequilibrium statistical physics on which I
should be happy to see further progress.
The stability of steadily drifting crystals
The crystalline suspensions of the earlier section are
generally made of particles heavier than water. Left to
themselves, they will settle slowly, giving slightly
inhomogeneous, bottom-heavy crystals with unit cells
shorter at the bottom of the container than at the top. To
get a truly homogeneous crystal, one must counteract
gravity. This is done, as remarked earlier, in the fluidized
bed geometry, where the viscous drag of an imposed
upflow balances the buoyant weight of the particles. As a
result, one is in the rest frame of a steadily and perpetually
sedimenting, spatially uniform crystalline suspension,
whose steady-state statistical properties we can study.
Although most crystalline suspensions are made of heavierthan-water particles, and therefore do sediment, there have
been only a few studies45 that focus on this aspect. Rather
than summarizing the experiments, I shall let the reader read
about them45. Our work was in fact inspired by attempts to
understand drifting crystals in a different context, namely
flux-lattice motion46, and the interest in crystalline fluidized
beds arose when we chanced upon some papers by
Crowley47, to whose work I shall return later in this review.
As with crystals at equilibrium, the first thing to
understand was the response to weak, long-wavelength
perturbation. For a crystal at thermal equilibrium, elastic
theory48 and broken-symmetry dynamics49,50 provide, in
principle, a complete answer. Our crystalline fluidized bed,
however, is far from equilibrium, in a steady state in which
the driving force of gravity is balanced by viscous
dissipation. We must therefore simply guess the correct
form for the equations of motion, based on general
symmetry arguments16. This general form should apply in
principle to any lattice moving through a disspative medium
without static inhomogeneities. Apart from the steadily
sedimenting colloidal crystal, another example is a flux-point
lattice moving through a thin slab of ultraclean type II
superconductor under the action of the Lorentz force due to
an applied current51. We will restrict our attention to the
former case alone in this review.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
One technical note is in order before we construct the
equations of motion. A complete analysis of the sedimentation dynamics of a three-dimensional crystalline
suspension requires the inclusion of the hydrodynamic
velocity field as a dynamical variable. This is because the
momentum of a local disturbance in the crystalline fluidized
bed cannot decay locally but is transferred to the nearby
fluid and thence to more and more remote regions. We get
around this difficulty by considering an experimental
geometry in which a thin slab of crystalline suspension
(particle size a << interparticle spacing l ) is confined in a
container with dimensions Lx , Lz >> Ly ~ l (gravity is along
− ẑ ). The local hydrodynamics that (see below) leads to
the configuration-dependent mobilities16,47 is left unaffected
by this, but the long-ranged hydrodynamic interaction is
screened in the xz plane on scales >> Ly by the no-slip
boundary condition at the walls, so that the velocity field of
the fluid can be ignored.
Instead of keeping track of individual particles, we work
on scales much larger than the lattice spacing l, treating the
crystal as a permeable elastic continuum whose distortions
at point r and time t are described by the (Eulerian)
displacement field u(r, t). Ignoring inertia as argued above,
the equation of motion must take the general form
velocity = mobility × force, i.e.
∂
u = M ( ∇u ) ( K ∇∇ u + F + f ) .
∂t
(1)
In eq. (1), the first term in parentheses on the right-hand
side represents elastic forces, governed by the elastic
tensor K, the second (F) is the applied force (gravity, for the
colloidal crystal and the Lorentz force for the flux lattice),
and f is a noise source of thermal and/or hydrodynamic
origin. Note that in the absence of the driving force F the
dynamics of the displacement field in this overdamped
system is purely diffusive: ∂t u ~∇2u, with the scale of the
diffusivities set by the product of a mobility and an elastic
constant. All the important and novel physics in these
equations, when the driving force is nonzero, lies in the
local mobility tensor M, which we have allowed to depend
on gradients of the local displacement field. The reason for
this is as follows: The damping in the physical situations we
have mentioned above arises from the hydrodynamic
interaction of the moving particles with the medium, and will
in general depend on the local configuration of the particles.
If the structure in a given region is distorted relative to the
perfect lattice, the local mobility will depart from its ideallattice value as well, through a dependence on the
distortion tensor ∇u. Assuming, as is reasonable, that M
can be expanded in a power series in ∇u, eq. (1) leads to
u&x = λ1 ∂ z u x + λ2 ∂ x u z
+ O(∇∇u ) + O(∇u∇u) + f x ,
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
(2)
u&z = λ3∂x u x + λ4∂zu z
+ O(∇∇u) + O(∇u∇u) + f z ,
(3)
where the nonlinear terms as well as those involving the
coefficients λi are a consequence of the sensitivity of the
mobility to local changes on the concentration and
orientation, and are proportional to the driving force
(gravity). They are thus absent in a crystal at equilibrium.
The {λi} terms dominate the dynamics at sufficiently long
wavelength for a driven crystal, since they are lower order
in gradients than the O(∇∇u) elastic terms. It is immediately
obvious that the sign of the product α≡ λ2λ3 decides the
long-wavelength behaviour of the steadily moving crystal.
If α> 0, small disturbances of the sedimenting crystal
should travel as waves with speed ~ λi, while if α< 0, the
system is linearly unstable: disturbances with wave vector
k with k x ≠ 0 should grow at a rate
Symmetry
cannot tell
~ us
| αwhich
| k x . of these happens: the sign of α
depends on the system in question, and is determined by a
more microscopic calculation than those presented above.
This is where the work of Crowley47 comes in. He studied
the settling of carefully prepared ordered arrays of steel
balls in turpentine oil, and found they were unstable. He
also calculated the response of such arrays to small
perturbations, taking into account their hydrodynamic
interaction, and found that the theory said that they should
indeed be unstable. Thus, one would expect crystalline
fluidized beds to be linearly unstable (the analogous
calculation for drifting flux lattices in type II superconductors finds linearly stable behaviour51) and is hence
described by eqs (2) and (3) with α< 0. However,
Crowley’s47 arguments and experiments were for an array of
particles merely prepared in the form of an ordered lattice.
Unlike in the case of a charge-stabilized suspension, there
were no forces that favoured such order at equilibrium in
the first place. Our model eqs (2) and (3), however, contain
such forces as well as nonlinearities and noise. While the
elastic terms are of course subdominant in a linearized
treatment at long wavelength to the {λi} terms, which are
O(∇u), we asked whether these terms, in combination with
nonlinearities and noise could undo or limit the Crowley
(α< 0) instability.
We answer this question not by studying eq. (3) with
α< 0, but instead by building a discrete Ising-like dynamical
model embodying the essential physics of those equations.
The crucial features of the dynamics implied by eq. (3) (see
Happel and Brenner37 and Crowley47) are that a downtilt
favours a drift of material to the right, an uptilt does the
opposite, an excess concentration tends to sink, and a
deficit in the concentration to float up. Let us implement this
dynamics in a one-dimensional system with sites labelled by
an integer i, and describe the state of the lattice of spheres
in terms of an array of two types of two-state variables:
ρi = ±, which tells us if the region around site i is
compressed (+) or dilated (–) relative to the mean, and
405
b
E
θi = ± which tells us if the local tilt is up (+, which we will
call ‘/’) or down (–, which we will denote ‘\’). Let us put the
two types of variables on the odd and even sublattices. A
valid
configuration
could
then
look
like
ρ1θ1ρ2θ2ρ3θ3ρ4θ4 = + \ + / – \ + \ – / – /. An undistorted
lattice is then a statistically homogeneous admixture of +
and – for both the variables (a ‘paramagnet’). The timeevolution of the model is contained in a set of transition
rates for passing from one configuration to another, as
follows16,17,52:
W(+ \ – → – \ +) = D + a
W(– \ + → + \ –) = D – a
W(– \ + → + \ –) = D′ + a′
W(+ \ – → – \ +) = D′ – a′
(4)
W(/ + \ → \ + / ) = E + b
W( \ + / → / + \ ) = E – b
W( \ – / → / – \ ) = E′ + b′
W( / – \ → \ – / ) = E′ – b′,
where the first line, for example, represents the rate of + –
going to – + in the presence of a downtilt \, and so on. D, E,
D′, E′ (all positive) and a, b, a′, b′ are all in principle
independent parameters, but it turns out to be sufficient on
grounds of physical interest, relevance to the sedimentation
problem, and simplicity to consider the case D = D′, E = E′,
a = a′, b = b′, with γ ≡ ab > 0 corresponding to α= λ2λ3 < 0
in eqs (2) and (3).
Associating ρi with ∂x u x and θi with ∂x u z we expect that
the above stochastic dynamics yields, in the continuum
limit, the same behaviour as a one-dimensional version of
eqs (2) and (3) in which all z derivatives are dropped, and
appropriate nonlinearities are included to ensure that the
instability seen in the linear approximation is controlled.
This mapping provides a convincing illustration of the close
connection between quite down-to-earth problems in
suspension science and the area of driven diffusive
systems which is currently the subject of such intense
study 36.
Initial numerical studies16 of eq. (4) suggested a tendency
towards segregation into macroscopic domains of + and –,
as well of / and \ with interfaces between + and – are shifted
with respect to those between / and \ by a quarter of the
system size. This arrangement is just such as to make it
practically impossible, given the dynamical rules eq. (4), for
the domains to remix. The numerics seemed to suggest that
enough interparticle repulsion or a high enough temperature
could undo the phase separation, but this turned out to be a
finite size effect. We have shown52 that the model always
phase-separates for γ > 0. More precisely, we have shown
exactly, for the symmetric case Σiρi = Σiθi = 0, if
that the steady state of our model obeys detailed balance,
(i.e. acts like a thermal equilibrium system) with respect to
the
energy
function
406
= εΣ Nk=1[Σ kj=1θ j ] ρk .
=
a
D
,
SPECIAL SECTION:
H
A little reflection will convince the
reader that this is like the energy of particles (the {ρi}) on a
hill-and-valley landscape with a height profile whose local
slope is θi. Since the dynamics moves particles downhill,
and causes occupied peaks of the landscape to turn into
valleys, it is clear that the final state of the model will be one
with a single valley, the bottom of which is full of pluses
(and the upper half full of minuses). It is immediately clear
that this phase separation is very robust, and will persist at
any finite temperature (i.e. any finite value of the base rates
D and E). Figure 1 demonstrates this graphically.
Many properties of the model (eq. (4)) can be obtained
exactly in the detailed-balance limit mentioned above. These
include the prediction that the phase separation, while
robust, is anomalously slow: domain sizes grow with time t
as log t. It is important to note that the behaviour obtained
exactly for the above special values of parameters can be
shown to apply much more generally. This phase
separation, in terms applicable to a real crystalline
suspension, leads to macroscopic particle-rich and particlepoor regions. In the middle of each such region, one expects
a fracture separating regions of opposite tilt. There are
preliminary reports of such behaviour in experiments53, and
it is tempting to think that some recent observations15 of the
collapse of elastic aggregates in suspension are related to
the above ideas, but this is mere speculation at this point.
Detailed experiments on large, single-crystalline fluidized
beds are needed to test the model.
Velocity fluctuations in fluidized beds
Hard-sphere suspensions, in which the only interactions are
hydrodynamic, are a subject of continuing interest to fluid
dynamicists, as should be clear from a glance at current
issues of fluid mechanics journals (or physics journals, for
that matter). In practice, these suspensions are chargestabilized, with ionic strength so large that electrostatics is
screened within a tiny distance of the particle surface33,
making the particles effectively hard spheres. Up to volume
fractions of about 0.5, the particles in these suspensions
Figure 1. A schematic picture of the final state of phase
separation of the one-dimensional model: The tilt field has been
summed to give the height of the profile at each point, the filled
circles denote pluses and the open circles minuses. It is clear that for
the phase separation to undo itself a plus, for example, must climb a
distance of order 1/4 of the system size. The time for this to happen
diverges exponentially with system size.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
have only short-ranged positional correlations, and move
freely. They are thus fluid suspensions unlike the
crystalline suspensions we discussed earlier. The
suspensions of our interest are made of large particles
(several microns in size), so that Brownian motion is
unimportant, but (as discussed earlier) there is nonetheless
a substantial random component to the motion if the system
is sedimenting or sheared. Single-component suspensions
are themselves the home of many mysteries, as the
references in earlier sections will bear out, while two (or
more) -component suspensions31,54 are understood even
less. I will focus on just one puzzle in monodisperse suspensions, and on what I believe is its resolution using
methods imported from time-dependent statistical
mechanics.
Until recently, theory and experiment disagreed rather
seriously on the nature of velocity fluctuations in these
suspensions in steady-state sedimentation. A simple
theory19 predicted that σv(L), the standard deviation of the
velocity of the particles sedimenting in a container of size L,
L
,
should diverge as
while
experiments20,21 saw no such
dependence (For a dissenting voice, see Tory et al.55). More
precisely, Segrè et al.21 see size dependence for L smaller
than a ‘screening length’ ξ~ 30 interparticle spacings, but
none for L > ξ. Further confusion is provided by direct
numerical simulations which do seem to see the sizedependence56, although these contain about 30,000
particles, which, while that sounds large, means they are
only about 30 particles on a side, which probably just
means that these huge simulations have not crossed the
scale ξof Segrè et al.21!
It is important to note that experiments in this area are
frequently done by direct imaging of the particles at each
time (Particle Imaging Velocimetry or PIV, see Adrian57).
Since motions are slow (velocities of a few µm/s), this is
relatively easy to do. Once the data on particle positions are
stored, they can be analysed in detail on a computer and
objects of interest such as correlation functions extracted.
This is one of the nice things about the field of suspension
science: problems of genuine interest to practitioners of
statistical physics arise and can be studied in relatively
inexpensive experiments, which measure quite directly the
sorts of quantities a theoretician can calculate.
Our contribution to this issue22,23 is to formulate the
problem in the form of generalized Langevin equations, and
then use methods well-known in areas such as dynamical
critical phenomena to construct a phase diagram for
sedimenting suspensions. Our approach is close in spirit to
that of Koch and Shaqfeh58, but differs in detail and
predicitive power. We find that there are two possible
nonequilibrium ‘phases’, which we term ‘screened’ and
‘unscreened’, for a steadily sedimenting suspension. In the
screened phase, σv (L) is independent of L, while in the
unscreened phase, it diverges as in Caflisch and Luke19. The
two phases are separated by a boundary which has the
characteristics of a continuous phase transition, in that a
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
certain correlation length diverges there in a power-law
manner. Although the relation between the parameters in
terms of which our phase diagram is drawn and
conventional suspension properties is still somewhat
uncertain, we believe that the screened and unscreened
phases should occur respectively at large and small Peclet
numbers. Properties such as particle shape (aspect ratio)
and Reynolds number also probably play a role in
determining where in our phase diagram a given system lies.
Let us begin by reconstructing the prediction of Caflisch
and Luke19. Let δ v(r) and δ c(r) be respectively the local
velocity and concentration fluctuations about the mean in a
steadily sedimenting suspension, each of whose particles
has a buoyancy-reduced weight mRg. Then, ignoring inertia,
the balance between gravitational (acceleration g) and
viscous (viscosity η) force densities is expressed in the
relation η∇2δv ~ mRgδc(r, t). This implies that a
concentration fluctuation at the origin leads to a velocity
field ~ g/(ηr) a distance r away. Now, if you assume, with
Caflisch and Luke19, that the δc’s are completely random in
space or have at best a finite range of correlations, the
variance ⟨|δv|2⟩ due to all the concentration fluctuations is
obtained by simply squaring and adding incoherently
giving, for a container of finite volume L3,
(g/η)2 ∫d 3r (1/r2) ~ L. What this tells us, really, is that either
⟨|δv|2⟩ diverges or the concentration fluctuations are
strongly anticorrelated at large length scales. What we
should do, therefore, is not assume a spectrum of
concentration fluctuations and calculate the velocity
variance, but calculate both of them.
To this end, let me first summarize the construction of the
equations of motion for the suspension. Our description is
phenomenological, to precisely the same extent as a
continuum Ginzburg–Landau model or its time-dependent
analogue for an equilibrium phase transition problem such
as phase separation in a binary fluid. Our construction is
constrained by the following general principles, each of
which plays an indispensable role: (i) We need to keep track
only of the slowest variables in the problem. (ii) Assuming
our suspension has not undergone a phase transition into a
state where some invariance (translation, rotation) is
spontaneously broken, the only slow variables are the local
densities of conserved quantities. For an incompressible
suspension, these are just the particle concentration and
the suspension momentum density (effectively, for a dilute
suspension, the fluid velocity field). (iii) To get the long
wavelength physics right, we can work at leading order in a
gradient expansion. (iv) We must keep all terms not
explicitly forbidden on grounds of symmetry, and impose
no relations amongst the phenomenological parameters
other than those forced on us by the symmetries of the
problem. (v) Since the microscopic Stokesian dynamics
shows chaotic behaviour and diffusion (see earlier
sections), our coarse-grained model, since it is an effective
description for the long-wavelength degrees of freedom,
should contain stochastic terms (a direct effect of the
407
SPECIAL SECTION:
eliminated fast degrees of freedom) as well as diffusive
terms (an indirect effect) but with no special (fluctuation–
dissipation) relation between them. Since only a limited
range of modes (say, with wave numbers larger than a
cutoff scale Λ, of order an inverse interparticle spacing)
have been eliminated, the resulting noise can have
correlations only on scales smaller than Λ–1. As far as a
description on scales >> Λ–1 is concerned, the noise can be
treated as spatially uncorrelated. This approach, we argue,
should yield a complete, consistent description of the longtime, long-wavelength properties of the system in question.
These premises accepted, one is led inevitably to a
stochastic advection–diffusion equation
∂δc
2
+ δv ·∇δc = [D⊥∇⊥ + Dz∇2z ]δc + ∇ · f(r, t),
∂t
(5)
for the concentration field δc and the Stokes equation
η∇2δvi(r, t) = mR gPizδc(r, t).
S (q ) =
(6)
for the velocity field δv. Equation (6) simply says that a local
concentration fluctuation, since the particles are heavier
than the solvent, produces a local excess force density
which is balanced by viscous damping. In eq. (5),
are
respectively
the projectors along and normal to the zδ ijz and
δ ij⊥
axis, which is the direction of sedimentation.
Incompressibility (∇ · δ v = 0) has been used to eliminate the
pressure field in eq. (6) by means of the transverse projector
Pij (in Fourier space Pij(q) = δij –q iq j/q 2). Equation (5)
contains the advection of the concentration by the velocity,
an anisotropic hydrodynamic diffusivity (D⊥, Dz), and a
noise or random particle current f(r, t). The last two are of
course a consequence of the eliminated small-scale chaotic
modes. The noise is taken, reasonably, to have Gaussian
statistics with mean zero and covariance
⟨ fi (r , t) f j (r ′, t ′)⟩ = c0 ( N ⊥δ ⊥j + N z δijz )δ( r − r ′)δ (t − t ′) ,
(7)
where c0 is the mean concentration. In an ideal nonBrownian system, the noise variances (N⊥, Nz) and the
diffusivities (D⊥, Dz) should, on dimensional grounds, scale
(at fixed volume fraction) as the product of the Stokesian
settling speed and the particle radius. However, no further
relation between the noise and the diffusivities may be
assumed, since this is a nonequilibrium system. In
laboratory systems there will of course be in addition a
thermal contribution to both noise and diffusivities. In
either case, what matters, and indeed plays a crucial role in
our nonequilibrium phase diagram, is that the parameter
K ≡ N⊥Dz – D⊥Nz
(8)
is in general nonzero. Since at equilibrium the ratio of noise
408
strength to kinetic coefficient is a temperature, K measures
the anisotropy of the effective ‘temperature’ for this driven
system.
As a first step towards extracting the behaviour of
correlation functions and hence the velocity variance, note
that eqs (5) and (6) can be solved exactly if the nonlinear
term in eq. (5) is ignored. This amounts to allowing
concentration fluctuations to produce velocity fluctuations
while forgetting that the latter must then advect the former.
If we do this, then it is straightforward to see by Fouriertransforming eq. (5) that the the static structure factor
S(q) ≡ c0–1 ∫r e–iq·r ⟨δc(0)δc(r)⟩ (where the angle brackets
denote an average over the noise) for concentration
fluctuations with wave vector q = (q⊥, qz) in a suspension
with mean concentration c0 is
N ⊥ q 2⊥ + N z q 2z
D ⊥ q 2⊥ + D z q 2z
,
(9)
and is hence independent of the magnitude of q. In
particular, it is therefore non-vanishing at small q. This can
quickly be seen to imply, through eq. (6), that the velocity
variance diverges exactly as in Caflisch and Luke19. In other
words, Caflisch and Luke19 fail to take into account the
hydrodynamic
interaction
between
concentration
fluctuations. If nonlinearities are to change this, they must
leave the noise relatively unaffected while causing the
relaxation rate (D⊥q ⊥2 + Dzq 2z in the linearized theory) to
become nonzero at small q. This, if it happens, would be
called ‘singular diffusion’ since diffusive relaxation rates
normally vanish at small wave number. We shall see below
that this does happen, in a substantial part of the parameter
space of this problem.
More precisely, we have been able to show22,23, in a fairly
technical and approximate ‘self-consistent’ calculation
which I shall not present here, that the behaviour once the
advection term is included depends on the temperature
anisotropy parameter K defined in eq. (8). If K = 0, i.e. if the
noise and the diffusivities happen to obey a fluctuation–
dissipation relation, then the static structure factor is totally
unaltered by the advective nonlinearity, and therefore the
velocity variance diverges as in Caflisch and Luke19. If K is
larger than a critical value Kc , i.e. the fluctuations injected
by the noise are substantially more abundant for wave
vectors in the xy plane than for those along z, then there is a
length scale ξ such that S(q) → 0 for qξ<< 1 with q ⊥ >> q z.
Thus the velocity variance is finite, and independent of
system size L for L > ξ. We call this ‘screening’, and the
regions K > Kc and K < Kc as the screened and unscreened
phases, respectively. A schematic phase diagram is given in
Figure 2. However, ξ diverges as K → Kc . For K < Kc ,
according to preliminary calculations, the long-wavelength
behaviour is the same as that at K = 0, although a detailed
renormalization-group calculation to establish this is still in
progress.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
Let me try to provide a qualitative understanding of these
results. The basic question is: how does an imposed longwavelength inhomogeneity in the solute concentration
decay in a steadily sedimenting suspension? It can, of
course, always do so by hydrodynamic diffusion. In
addition, it can scatter off the background of chaos-induced
fluctuations, which I will call noise-injected fluctuations
(NIF). This scattering is best thought of as the advection of
the imposed inhomogeneity by the velocity field produced
by the NIF. So consider two cases: (a) where the NIF has
wave vector predominantly along z, and (b) where the wave
vector is mainly in the xy plane. In (a), the induced flow has
a z-velocity which alternates in sign as a function of z. The
advection of the imposed inhomogeneity by this flow will
concentrate it further, in general, thus enhancing the
perturbation. In case (b), i.e. when the NIF has variation
mainly along xy, the resulting z-velocity will alternate in sign
along x and y, which will break up the inhomogeneity. Thus,
a noise with Fourier components only with wave vector
along z would give a negative contribution to the damping
rate due to scattering, while one with Fourier components
with wave vector only orthogonal to z would give a purely
positive contribution. In general, it is thus clear that this
mechanism gives a correction to the damping rate
proportional to N⊥ – Nz (assuming, for simplicity, the same
diffusivity in all directions). In addition, the long-ranged
nature of the hydrodynamic interaction means that no
matter how long-wavelength the NIF, it will produce
macroscopic flows on scales comparable to its wavelength
(and instantaneously, in the Stokesian approximation),
hence the singular diffusion.
In addition, we predict the form of static and dynamic
correlation functions of the concentration or the velocity
fields in detail, in the screened and unscreened phases as
well as at the transition between them22,23. Very careful
experiments, in particular some light scattering measurements at very small angles, are currently underway to test
our predictions.
We close this section by remarking that ours is not the
only candidate theory of the statistics of fluctuations in
zero Reynolds number fluidized beds. An earlier, nominally
more microscopic approach58 had some similar conclusions;
we, however, disagree with that work in several details.
There are some who criticize the experiments of Segrè21
because they are done in narrow cells which could
introduce finite size effects. There is also a very qualitative
set of arguments59 based on an analogy with high Prandtl
number turbulence; it is unclear at this stage whether that
work is a recasting of our theory of the screened phase or
distinct from it.
Conclusion
This review has tried to summarize experimental and
theoretical work in the area of suspension hydrodynamics,
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
Figure 2. Schematic phase diagram for steadily sedimenting hard
spheres (after Levine et al.22 ). On the dashed line and, we conjecture,
near it, the structure factor is as in the linearized theory, and
screening fails. The solid curve is the phase boundary between the
screened and unscreened regions.
from the point of view of a physicist interested in
nonequilibrium statistical mechanics. The major aim has
been to convince condensed matter physicists in India that
this is a field which merits their attention. Apart from listing
a large number of general references, I have tried to support
my case by describing some problems on which I have
worked, which have their origins squarely within
suspension science, but whose solutions required all the
machinery of statistical mechanics and used in a crucial way
the fact that the systems concerned were not at thermal
equilibrium. I hope this article will win some converts to this
wonderful field.
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20. Nicolai, H. and Guazzelli, E., Phys. Fluids, 1995, 7, 3.
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Lett., 1997, 79, 2574.
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24. Davis, R. H., in Sedimentation of Small Particles in a Viscous
Fluid (ed. Tory, E. M.), Computational Mechanics Publications,
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30. Van Saarloos, W. and Huse, D. A., Europhys. Lett., 1990, 11,
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Commun., 1996, 150, 123.
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K., J. Phys. C, 1988, 21, 4737.
33. Sood, A. K., in Solid State Physics (eds Ehrenreich, H. and
Turnbull, D.), Academic Press, 1991, 45, 1.
34. Sheared suspensions are an old subject: see Reynolds, O., Philos.
Mag., 1885, 20, 46.
35. Blanc, R. and Guyon, E., La Recherche, 1991, 22, 866.
36. Schmittmann, B. and Zia, R. K. P., in Phase Transitions
and Critical Phenomena (eds Domb, C. and Lebowitz, J. L.),
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1992.
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111.
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2775; Ramakrishnan, T. V., Pramana, 1984, 22, 365.
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Chaikin, P. M., Phys. Rev. E, 1995, 51, 4674.
46. See, e.g. Balents, L., Marchetti, M. C. and Radzihovsky, L.,
Phys. Rev. B, 1998, 57, 7705.
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submitted to Phys. Rev. E.
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1986, 166, 379.
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ACKNOWLEDGEMENT.
with figures.
I thank R. A. Simha and C. Das for help
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
Vehicular traffic: A system of interacting
particles driven far from equilibrium§
Debashish Chowdhury*,†, Ludger Santen** and Andreas Schadschneider**
*Department of Physics, Indian Institute of Technology, Kanpur 208 016, India and Institute for Theoretical Physics,
University of Cologne, D-50923 Köln, Germany
**Institute for Theoretical Physics, University of Cologne, D-50923 Köln, Germany
In recent years statistical physicists have developed
discrete ‘particle-hopping’ models of vehicular traffic,
usually formulated in terms of cellular automata, which
are similar to the microscopic models of interacting
charged particles in the presence of an external electric
field. Concepts and techniques of non-equilibrium
statistical mechanics are being used to understand the
nature of the steady states and fluctuations in these socalled microscopic models. In this brief review we explain,
primarily to nonexperts, these models and the physical
implications of the results.
A RE you surprised to see an article on vehicular traffic in
this special section of Current Science where physicists are
supposed to report on some recent developments in the
area of dynamics of nonequilibrium statistical systems?
Aren’t civil engineers (or, more specifically, traffic
engineers) expected to work on traffic? Solving traffic
problems would become easier if one knows the
fundamental laws governing traffic flow and traffic jam. For
almost half a century physicists have been trying to
develop a theoretical framework of traffic science,
extending concepts and techniques of statistical physics1–6.
The main aim of this brief review is to show how these
attempts, particularly the recent ones, have led to deep
insight in this frontier area of inter-disciplinary research.
The dynamical phases of systems driven far from
equilibrium are counterparts of the stable phases of systems
in equilibrium. Let us first pose some of the questions that
statistical physicists have been addressing in order to
discover the fundamental laws governing vehicular traffic.
For example,
(i) What are the various dynamical phases of traffic? Does
traffic exhibit phase-coexistence, phase transition, criticality
or self-organized criticality and, if so, under which
circumstances?
(ii) What is the nature of fluctuations around the steadystates of traffic?
†
For correspondence (e-mail: debch@thp.Uni-koeln.DE)
This is a modified and shortened version of a longer detailed review
article to be published elsewhere.
§
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
(iii) If the initial state is far from a stationary state of the
driven system, how does it evolve with time to reach a true
steady-state?
(iv) What are the effects of quenched disorder (i.e. timeindependent disorder) on the answers to the questions
posed in (i)–(iii) above?
The microscopic models of vehicular traffic can find
practical applications in on-line traffic control systems as
well as in the planning and design of transportation
network.
There are two different conceptual frameworks for
modelling vehicular traffic. In the ‘coarse-grained’ fluiddynamical description, the traffic is viewed as a compressible fluid formed by the vehicles but these individual
vehicles do not appear explicitly in the theory. In contrast,
in the ‘microscopic’ models traffic is treated as a system of
interacting ‘particles’ driven far from equilibrium where
attention is explicitly focused on individual vehicles, each
of which is represented by a ‘particle’; the nature of the
interactions among these particles is determined by the way
the vehicles influence each others’ movement. Unlike the
particles in a gas, a driver is an intelligent agent who can
‘think’, make individual decisions and ‘learn’ from
experience. Nevertheless, many phenomena in traffic can be
explained in general terms with these models provided the
behavioural effects of the drivers are captured by only a few
phenomenological parameters.
The conceptual basis of the older theoretical approaches
is explained briefly in the next section. Most of the
‘microscopic’ models developed in the recent years are
‘particle-hopping’ models which are usually formulated
using the language of cellular automata (CA)7. The Nagel–
Schreckenberg (NaSch)8 model and the Biham–Middleton–
Levine (BML)9 model, which are the most popular CA
models of traffic on idealized highways and cities,
respectively, have been extended by several authors to
develop more realistic models. Some of the most interesting
aspects of these recent developments are discussed here.
The similarities between various particle-hopping models of
traffic and some other models of systems, which are also far
from equilibrium, are pointed out followed by the
concluding section.
411
SPECIAL SECTION:
Older theories of vehicular traffic
Fluid-dynamical theories of vehicular traffic
In traffic engineering, the fundamental diagram depicts the
relation between density c and the flux J, which is defined
as the number of vehicles crossing a detector site per unit
time10. Because of the conservation of vehicles, the local
density c(x; t) and local flux J(x; t) satisfy the equation of
continuity which is the analogue of the equation of
continuity in the hydrodynamic theories of fluids. In the
early works12 it was assumed that (i) the flux (or,
equivalently, the velocity) is a function of the density; and
(ii) following any change in the local density, the local
speed instantaneously relaxes to a magnitude consistent
with the new density at the same location. However, for a
more realistic description of traffic, in the recent fluiddynamical treatments13–15 of traffic an additional equation
(the analogue of the Navier–Stokes equation for fluids),
which describes the time-dependence of the velocity V(x; t),
has been considered. This approach, however, has its
limitations; for example, viscosity of traffic is not a directly
measurable quantity.
Kinetic theory of vehicular traffic
In the kinetic theory of traffic, one begins with the basic
quantity g(x, v, w; t) dx dv dw which is the number of
vehicles, at time t, located between x and x + dx, having
actual velocity between v and v + dv and desired velocity
between w and w + dw. In this approach, the fundamental
dynamical equation is the analogue of the Boltzmann
equation in the kinetic theory of gases3. Assuming
reasonable forms of ‘relaxation’ and ‘interaction’, the
problem of traffic is reduced to that of solving the
Boltzmann-like equation, a formidable task, indeed16–18!
Car-following theories of vehicular traffic
In the car-following theories one writes, for each individual
vehicle, an equation of motion which is the analogue of the
Newton’s equation for each individual particle in a system
of interacting classical particles. In Newtonian mechanics,
the acceleration may be regarded as the response of the
particle to the stimulus it receives in the form of force which
includes both the external force as well as those arising from
its interaction with all the other particles in the system.
Therefore, the basic philosophy of the car-following
theories1,2 can be summarized by the equation
[Response]n ∝ [Stimulus]n,
Cellular-automata models of highway-traffic
In the car-following models, space is treated as a continuum
and time is represented by a continuous variable t, while
velocities and accelerations of the vehicles are also real
variables. However, most often, for numerical manipulations
of the differential equations of the car-following models,
one needs to discretize the continuous variables with
appropriately chosen grids. In contrast, in the CA models of
traffic not only time but also the position, speed, and
acceleration of the vehicles are treated as discrete variables.
In this approach, a lane is represented by a one-dimensional
lattice. Each of the lattice sites represents a ‘cell’ which can
be either empty or occupied by at most one ‘vehicle’ at a
given instant of time (see Figure 1).
At each discrete time step t → t + 1, the state of the
system is updated following a well-defined prescription.
Nagel–Schreckenberg model of highway traffic
In the NaSch model, the speed V of each vehicle can take
one of the Vmax + 1 allowed integer values V = 0, 1, . . ., Vmax .
Suppose, Xn and Vn denote the position and speed,
respectively, of the nth vehicle. Then, d n = Xn + 1 – Xn, is the
gap in between the nth vehicle and the vehicle in front of it
at time t. At each time step t → t + 1, the arrangement of the
N vehicles on a finite lattice of length L is updated in
parallel according to the following ‘rules’:
Step 1: Acceleration. If Vn < Vmax , the speed of the nth
vehicle is increased by one, but Vn remains unaltered if
Vn = Vmax , i.e. Vn → min(Vn + 1, Vmax ).
Step 2: Deceleration (due to other vehicles). If d n ≤ Vn,
the speed of the nth vehicle is reduced to d n – 1, i.e.
(1)
for the nth vehicle (n = 1, 2, . . .). The constant of
proportionality in eq. (1) can be interpreted as a measure of
the sensitivity coefficient of the driver; it indicates how
412
strongly the driver responds to unit stimulus. Each driver
can respond to the surrounding traffic conditions only by
accelerating or decelerating the vehicle. The stimulus and
the sensitivity factor are assumed to be functions of the
position and speed of the vehicle under consideration and
those of its leading vehicle. Different forms of the equations
of motion of the vehicles in the different versions of the carfollowing models arise from the differences in their
postulates regarding the nature of the stimulus. In general,
the dynamical equations for the vehicles in the carfollowing theories are coupled non-linear differential
equations19–25 and thus, in this ‘microscopic’ approach, the
problem of traffic flow reduces to problems of nonlinear
dynamics.
Figure 1. Typical configuration in a CA model. The number in the
upper right corner is the speed of the vehicle.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
Vn → min(Vn, d n – 1).
Step 3: Randomization. If Vn > 0, the speed of the nth
vehicle is decreased randomly by unity with probability p
but Vn does not change if Vn = 0, i.e. Vn → max(Vn – 1, 0)
with probability p.
Step 4: Vehicle movement. Each vehicle is moved forward
so that Xn → Xn + Vn.
The NaSch model is a minimal model in the sense that all
the four steps are necessary to reproduce the basic features
of real traffic; however, additional rules need to be
formulated to capture more complex situations. Step 1
reflects the general tendency of the drivers to drive as fast
as possible, if allowed to do so, without crossing the
maximum speed limit. Step 2 is intended to avoid collision
between the vehicles. The randomization in step 3 takes into
account the different behavioural patterns of the individual
drivers, especially, nondeterministic acceleration as well as
overreaction while slowing down; this is crucially important
for the spontaneous formation of traffic jams. So long as
p ≠ 0, the NaSch model may be regarded as stochastic CA7.
For a realistic description of highway traffic8, the typical
length of each cell should be about 7.5 m and each time step
should correspond to approximately 1 sec of real time when
Vmax = 5.
The update scheme of the NaSch model is illustrated with
a simple example in Figure 2.
Space-time diagrams showing the time evolutions of the
NaSch model demonstrate that no traffic jam is present at
sufficiently low densities, but spontaneous fluctuations
give rise to traffic jams at higher densities (Figure 3 a). From
Figure 3 b it should be obvious that the intrinsic
stochasticity of the dynamics8, arising from non-zero p, is
essential for triggering the jams8,26.
The use of parallel dynamics is also important. In
contrast to a random sequential update, it can lead to a
chain of overreactions. Suppose, a vehicle slows down due
the randomization step. If the density of vehicles is large
enough this might force the following vehicle also to brake
in the deceleration step. In addition, if p is not too small, it
might brake even further in step 3. Eventually this can lead
to the stopping of a vehicle, thus creating a traffic jam. This
mechanism of spontaneous jam formation is rather realistic
and cannot be modelled by the random sequential update.
Relation between NaSch model and asymmetric
simple exclusion process
In the NaSch model with Vmax = 1, every vehicle moves
forward with probability q = 1 – p in the time step t + 1 if the
site immediately in front of it were empty at the time step t;
this is similar to the fully asymmetric simple exclusion
process (ASEP)27,29, where a randomly chosen particle can
move forward with probablity q if the site immediately in
front is empty. But, updating is done in parallel in the NaSch
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
model whereas that in the ASEP is done in a random
sequential manner. Nevertheless, the special case of
Vmax = 1 for the NaSch model derives special importance
from the fact that so far it has been possible to derive exact
analytical results for the NaSch model only in the special
limits (a) Vmax = 1 and arbitrary p and (b) p = 0 and arbitrary
Vmax .
NaSch model in the deterministic limits
If p = 0, the system can self-organize so that at low densities
every vehicle can move with Vmax and the corresponding
flux
is
cVmax ;
this
is,
however,
possible only if enough empty cells are available in front
of every vehicle, i.e. for c ≤ cmdet = (Vmax + 1)–1 and the corresponding maximum flux is Jmdeatx= Vmax /(Vmax + 1). On the other
hand, for c >
the flow is limited by the density of
holes. Hence, the fundamental diagram in the deterministic
limit p = 0 of the NaSch model (for any arbitrary Vmax ) is
(a)
(b)
(c)
(d)
Figure 2. Step-by-step example for the application of the update
rules. We have assumed Vmax = 2 and p = 1/3. Therefore, on average
one-third of the cars qualifying will slow down in the randomization
step.
413
c det
m ,
SPECIAL SECTION:
given
by
the
exact
expression
J = min
(cVmax , (1 – c)).
Aren’t the properties of the NaSch model with maximum
allowed speed Vmax , in the deterministic limit p = 1, exactly
identical to those of the same model with maximum allowed
speed Vmax – 1? The answer to the question posed above is
‘No’; if p = 1, all random initial states lead to J = 0 in the
stationary state of the NaSch model irrespective of Vmax and
c!
Analytical theory for the NaSch model
In the ‘site-oriented’ theories one describes the state of the
finite system of length L by completely specifying the state
of each site. In contrast, in the ‘car-oriented’ theories the
state of the traffic system is described by specifying the
positions and speeds of all the N vehicles in the system. In
the naive mean-field approximation, one treats the
probabilities of occupation of the lattice sites as
independent of each other. In this approximation, for
example, the steady-state flux for the NaSch model with
Vmax = 1 and periodic boundary conditions, one gets30
J = qc (1 – c).
(2)
It turns out30 that the naive mean-field theory
underestimates the flux for all Vmax . Curiously, if instead of
parallel updating one uses the random sequential updating,
the NaSch model with Vmax = 1 reduces to the ASEP for
which the eq. (2) is known to be the exact expression for the
corresponding flux (see, e.g. Nagel and Schreckenberg8)!
What are the reasons for these differences arising from
parallel updating and random sequential updating? There
are ‘Garden of Eden’ (GoE) states (dynamically forbidden
states) 31 of the NaSch model which cannot be reached by
the parallel updating, whereas no state is dynamically
forbidden if the updating is done in a random sequential
manner.
For example, the configuration shown in Figure 4 is a GoE
state (The configuration shown in Figure 1 is also a GoE
state!) because it could occur at time t only if the two
vehicles occupied the same cell simultaneously at time
t – 1. The naive mean-field theory mentioned above does
not exclude the GoE states. The exact expression, given in
the next subsection, for the flux in the steady-state of the
NaSch model with Vmax = 1 can be derived by merely
excluding these states from consideration in the naive
mean-field theory31, thereby indicating that the only source
of correlation in this case is the parallel updating. But, for
Vmax > 1, there are other sources of correlation because of
which exclusion of the GoE states merely improves the
naive mean-field estimate of the flux but does not yield exact
results31.
A systematic improvement of the naive mean-field theory
of the NaSch model has been achieved by incorporating
short-ranged correlations through cluster approximations.
We define a n-cluster to be a collection of n successive
sites. In the general n-cluster approximation, one divides the
lattice into ‘clusters’ of length n such that two
neighbouring clusters have n – 1 sites in common (see
Figure 5).
If n = 1, then the 1-cluster approximation can be regarded
Figure 3. Typical space–time diagrams of the NaSch model with Vmax = 5. a, p = 0.25, c = 0.20; b, p = 0.0, c = 0.5. Each horizontal row of
dots represents the instantaneous positions of the vehicles moving towards right
Figure
while
4. theA successive
GoE state for
rows
theofNaSch
dots represent
model with
theVmax
positions
≥ 2. of
the same vehicles at the successive time steps.
414
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
as the naive mean-field approximation. You can easily
verify, for example, in the special case of Vmax = 1, that the
state of the 2-cluster at time t + 1 depends on the state of
the 4-cluster at time t, which, in turn, depends on the state
of a larger cluster at time t – 1 and, so on. Therefore, one
needs to make an approximation to truncate this hierarchy in
a sensible manner. For example, in the 2-cluster
approximation for the NaSch model with Vmax = 1, the 4cluster probabilities are approximated in terms of an
appropriate product of 2-cluster probabilities. Thus, in the
n-cluster approximation30 a cluster of n neighbouring cells
are treated exactly and the cluster is coupled to the rest of
the system in a self-consistent way.
Carrying out the 2-cluster calculation30 for Vmax = 1, one
not only finds an effective particle-hole attraction (particle–
particle repulsion), but also obtains the exact result
J (c, p ) = 12 [1 − 1 − 4 qc(1 − c) ] ,
(3)
for the corresponding flux. But one gets only approximate
results from the 2-cluster calculations for all Vmax > 1 (see
Schadschneider32 for higher order cluster calculations for
Vmax = 2 and comparison with computer simulation data).
Let us explain the physical origin of the generic shape of
the fundamental diagrams shown in Figure 6. At sufficiently
low density of vehicles, practically ‘free flow’ takes place
whereas at higher densities traffic becomes ‘congested’ and
traffic jams occur. So long as c is sufficiently small, the
average speed ⟨V⟩ is practically independent of c as the
vehicles are too far apart to interact mutually. However, a
faster monotonic decrease of ⟨V⟩ with increasing c takes
place when the forward movement of the vehicles is
strongly hindered by others because of the reduction in the
average separation between them. Because of this trend of
variation of ⟨V⟩ with c, the flux J = ⟨cV⟩ exhibits a maximum10
at cm; for c < cm, increasing c leads to increasing J whereas
for c > cm sharp decrease of ⟨V⟩ with increase of c leads to
the overall decrease of J.
An interesting feature of the eq. (3) is that the flux is
invariant under charge conjugation, i.e. under the operation
c → (1 – c) which interchanges particles and holes.
Therefore, the fundamental diagram is symmetric about
c = 1/2 when Vmax = 1 (see Figure 6 a). Although this
symmetry breaks down for all Vmax > 1 (see Figure 6 b), the
corresponding fundamental diagrams appear more realistic.
Moreover, for given p, the magnitude of cm decreases with
increasing Vmax as the higher is the Vmax the longer is the
effective range of interaction of the vehicles (see Figure
6 b). Furthermore, for Vmax = 1, flux merely decreases with
increasing p (see eq. (3)), but remains symmetric about
c = 1/2 = cm. On the other hand, for all Vmax > 1, increasing p
not only leads to smaller flux but also lowers cm.
Spatio-temporal organization of vehicles
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
The distance from a selected point on the lead vehicle to the
same point on the following vehicle is defined as the
distance-headway (DH)10. In order to get information on the
spatial organization of the vehicles, one can calculate the
DH distribution P dh(∆X) by following either a site-oriented
approach33 or a car-oriented approach34 if ∆Xj = Xj – Xj – 1, i.e.
if the number of empty lattice sites in front of the jth vehicle
is identified as the corresponding DH. At moderately high
densities, P dh(∆X) exhibits two peaks; the peak at ∆X = 1 is
caused by the jammed vehicles while that at a larger ∆X
corresponds to the most probable DH in the free-flowing
regions.
The time-headway is defined as the time interval between
the departures (or arrivals) of two successive vehicles
recorded by a detector placed at a fixed position on the
highway10. The time-headway distribution contains
information on the temporal organization. Suppose, P m (t1) is
the probability that the following vehicle takes time t1 to
reach the detector, moving from its initial position where it
was located when the leading vehicle just left the detector
site. Suppose, after reaching the detector site, the following
vehicle waits there for τ – t1 time steps, either because of the
presence of another vehicle in front of it or because of its
own random braking; the probability for this event is
denoted
by
Q(τ – t1|t1).
The
distribution
P th(τ), of the time-headway τ, can be obtained from35,36
P th(τ) =
P m (t1) Q(τ– t1|t1). The most-probable timeheadway, when plotted against the density, exhibits a
minimum36; this is consistent with the well-known exact
relation J = 1/Tav between flux and the average timeheadway, Tav.
a
b
c
Figure 5. Decomposition of a lattice into: a, 1-clusters; b, 2clusters; and c, 3-clusters in the cluster-theoretic approach to the
NaSch model.
415
Στt1−=11
SPECIAL SECTION:
Is there a phase transition from ‘free-flowing’ to
‘congested’ dynamical phase of the NaSch model? No
satisfacory order parameter has been found so far37,38,
except in the deterministic limit39. The possibility of the
existence of any critical density in the NaSch model is ruled
out by the observations 32,37,38,40 that, for all non-zero p, (a)
the equal-time correlation function decays exponentially
with separation, and (b) the relaxation time and lifetimes of
the jams remain finite. This minimal model of highway traffic
also does not exhibit any first order phase transition and
two-phase co-existence35.
Extensions of the NaSch model and practical
applications
In recent years, other minimal models of traffic on highways
have been developed by modifying the updating rules of
the NaSch model41–43. In the cruise control limit of the NaSch
model44, the randomization step is applied only to vehicles
which have a velocity V < Vmax after step 2 of the update
rule. Vehicles moving with their desired velocity Vmax are
not subject to fluctuations. This is exactly the effect of a
cruise-control which automatically keeps the velocity
constant at a desired value. Interestingly, the cruise-control
limit of the NaSch model exhibits self-organized
criticality45,46. Besides, a continuum limit of the NaSch model
has also been considered47.
The vehicles which come to a stop because of hindrance
from the leading vehicle may not be able to start as soon as
the leading vehicle moves out of its way; it may start with a
probability q s < 1. When such possibilities are incorporated
in the NaSch model, the ‘slow-to-start’ rules48–52 can give
rise to metastable states of very high flux and hysteresis
effects as well as phase separation of the traffic into a ‘freeflowing’ phase and a ‘mega-jam’.
The bottleneck created by quenched disorder of the
a
highway usually slows down traffic and can give rise to
jams35,53 and phase segregation54,55. However, a different
type of quenched disorder, introduced by assigning
randomly different braking probabilities p to different
drivers in the NaSch model, can have more dramatic
effects56,57 which are reminiscent of ‘Bose–Einstein-like
condensation’ in the fully ASEP where particle-hopping
rates are quenched random variables58,59. In such Bose–
Einstein-like condensed states, finite fraction of the empty
sites are ‘condensed’ in front of the slowest vehicle (i.e. the
driver with highest p).
Several attempts have been made to generalize the NaSch
model to describe traffic on multi-lane highways and to
simulate traffic on real networks in and around several
cities60. For planning and design of the transportation
network61,
for
example,
in
a
metropolitan
area62–64, one needs much more than just micro-simulation
of how vehicles move on a linear or square lattice under a
specified set of vehicle–vehicle and road–vehicle interactions. For such a simulation, to begin with, one needs to
specify the roads (including the number of lanes, ramps,
bottlenecks, etc.) and their intersections. Then, times and
places of the activities, e.g. working, shopping, etc. of
individual drivers are planned. Micro-simulations are carried
out for all possible different routes to execute these plans;
the results give information on the efficiency of the different
routes and these informations are utilized in the designing
of the transportation network61,65,66. Some socio-economic
questions as well as questions on the environmental
impacts of the planned transportation infrastructure also
need to be addressed during such planning and design.
Cellular-automata models of city-traffic
The Biham–Middleton–Levin model of city traffic
b
Figure 6. Fundamental diagram in the NaSch model: a, Vmax = 1, and b, Vmax > 1, both for p = 0.25. The data for all Vmax > 1 have been
obtained through computer simulations.
416
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
and its generalizations
In the BML model9, each of the sites of a square lattice
represents the crossing of a east–west street and a north–
south street. All the streets parallel to the X̂ - direction of a
Cartesian coordinate system are assumed to allow only
single-lane east-bound traffic while all those parallel to the
direction allow only single-lane northbound traffic. In the initial state of the system, vehicles are
randomly distributed among the streets. The states of eastbound vehicles are updated in parallel at every odd discrete
time step whereas those of the north-bound vehicles are
updated in parallel at every even discrete time step
following a rule which is a simple extension of the fully
ASEP: a vehicle moves forward by one lattice spacing if and
only if the site in front is empty, otherwise the vehicle does
not move at that time step.
Computer simulations demonstrate that a first-order
phase transition takes place in the BML model at a finite
non-vanishing density c*, where the average velocity of the
vehicles vanishes discontinuously signalling complete
jamming; this jamming arises from the mutual blocking of
the flows of east-bound and north-bound traffic at various
crossings 67,68. Note that the dynamics of the BML model is
fully deterministic and the randomness arises only from the
random initial conditions69.
As usual, in the naive mean-field approximation one
neglects the correlations between the occupations of
different sites70. However, if you are not interested in
detailed information on the ‘structure’ of the dynamical
phases, you can get a mean-field estimate of c* by carrying
out
a
back-of-the-envelope
calculation71–73.
In the symmetric case cx = cy , for which v x = v y = v,
c = c* ~ 0.343.
The BML model has been extended to take into account
the effects of (i) asymmetric distribution of the vehicles71,
i.e. cx ≠ cy , (ii) overpasses or two-level crossings72 that are
represented by specifically identified sites, each of which
can accommodate up to a maximum of two vehicles
simultaneously, (iii) faulty traffic lights74, (iv) static
hindrances or road blocks or vehicles crashed in traffic
accident, i.e. stagnant points75,76, (v) stagnant street where
the local density cs of the vehicles is initially higher than
that in the other streets77, (vi) jam-avoiding drive78 of
vehicles to a neighbouring street, parallel to the original
direction, to avoid getting blocked by other vehicles in
front, (vii) turning of the vehicles from east-bound (northbound) to north-bound (east-bound) streets79, (viii) a single
north-bound street cutting across east-bound streets80, (ix)
more realistic description of junctions of perpendicular
streets 81,82, and (x) green-waves83.
Marriage of NaSch and BML models
At first sight, the BML model may appear very unrealistic
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
because the vehicles seem to hop from one crossing to the
next. However, it may not appear so unrealistic if each unit
of discrete time interval in the BML model is interpreted as
the time for which the traffic lights remain green (or red)
before switching red (or green) simultaneously in a
synchronized manner, and over that time scale each vehicle,
which faces a green signal, gets an opportunity to move
from jth crossing to the j + 1-th (or, more generally84, to the
j + rth where r > 1).
However, if one wants to develop a more detailed ‘finegrained’ description then one must first decorate each
bond 85 with D – 1 (D > 1) sites to represent D – 1 cells
between each pair of successive crossings, thereby
modelling each segment of the streets between successive
crossings in the same manner in which the entire highway is
modelled in the NaSch model. Then, one can follow the
prescriptions of the NaSch model for describing the
positions, speeds and accelerations of the vehicles82,86 as
well as for taking into account the interactions among the
vehicles moving along the same street. Moreover, one
should flip the colour of the signal periodically at regular
interval of T (T >> 1) time steps where, during each unit of
the discrete time interval every vehicle facing green signal
should get an opportunity to move forward from one cell to
the next. Such a CA model of traffic in cities has, indeed,
been proposed very recently87 where the rules of updating
have been formulated in such a way that, (a) a vehicle
approaching a crossing can keep moving, even when the
signal is red, until it reaches a site immediately in front of
which there is either a halting vehicle or a crossing; and (b)
no grid-locking would occur in the absence of random
Figure 7. Typical jammed configuration of the vehicles. The eastbound and north-bound vehicles are represented by the symbols →
and ↑, respectively. (N = 5, D = 8.
417
SPECIAL SECTION:
braking.
A phase transition from the ‘free-flowing’ dynamical
phase to the completely ‘jammed’ phase has been observed
in this model at a vehicle density which depends on D and
T. The intrinsic stochasticity of the dynamics, which
triggers the onset of jamming, is similar to that in the NaSch
model, while the phenomena of complete jamming through
self-organization as well as the final jammed configurations
(Figure 7) are similar to those in the BML model. This model
also provides a reasonable time-dependence of the average
speeds of the vehicles in the ‘free-flowing’ phase87.
Relation with other systems and phenomena
You must have noticed in the earlier sections that some of
the models of traffic are non-trivial generalizations or
extensions of the ASEP, the simplest of the driven–
dissipative systems which are of current interest in nonequilibrium statistical mechanics28. Some similarities
between these systems and a dynamical model of protein
synthesis have been pointed out 88. Another drivendissipative system, which is also receiving wide attention
from physicists in recent years, is granular material flowing
through a pipe4,5. There are some superficial similarities
between the clustering of vehicles on a highway and
particle–particle
(and
particle–cluster)
aggregation
process 89.
The NaSch model with Vmax = 1 can be mapped on to
stochastic growth models of one-dimensional surfaces in a
two-dimensional medium. Particle (hole) movement to the
right (left) corresponds to local forward growth of the
surface via particle deposition. In this scenario, a particle
evaporation would correspond to a particle (hole)
movement to the left (right) which is not allowed in the
NaSch model. It is worth pointing out that any quenched
disorder in the rate of hopping between two adjacent sites
would correspond to columnar quenched disorder in the
growth rate for the surface55.
Inspired by the recent success in theoretical studies of
traffic, some studies of information traffic on the computer
network (internet) have also been carried out90,92.
Summary and conclusion
Nowadays the tools of statistical mechanics are
increasingly being used to study self-organization and
emergent collective behaviour of complex systems many of
which, including vehicular traffic, fall outside the traditional
domain of physical systems. However, as we have shown in
this article, a strong theoretical foundation of traffic science,
can be built on the basic principles of statistical mechanics.
In this brief review, we have focused attention mainly on
the progress made in the recent years using ‘particlehopping’ models, formulated in terms of cellular automata,
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ACKNOWLEDGEMENTS. It is our pleasure to thank R. Barlovic,
J. G. Brankov, B. Eisenblätter, K. Ghosh, N. Ito, K. Klauck, W.
Knospe, D. Ktitarev, A. Majumdar, K. Nagel, V. B. Priezzhev, M.
Schreckenberg, A. Pasupathy, S. Sinha, R. B. Stinchcombe and D. E.
Wolf for enjoyable collaborations, the results of some of which have
been reviewed here. We also thank M. Barma, J. Kertesz, J. Krug, G.
Schütz, D. Stauffer and J. Zittartz for useful discussions and
encouragements. This work is supported by SFB341 Köln-AachenJülich.
419
SPECIAL SECTION:
Dynamical transitions in network models of
collective computation
Sitabhra Sinha* and Bikas K. Chakrabarti†§
*Department of Physics, Indian Institute of Science, Bangalore 560 012, India and Condensed Matter Theory Unit,
Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560 064, India
†
Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Calcutta 700 064, India
The field of neural network modelling has grown up on the
premise that the massively parallel distributed processing
and connectionist structure observed in the brain is the
key behind its superior performance. The conventional
network paradigm has mostly centered around a static
approach – the dynamics involves gradient descent of the
network state to stable fixed-points (or, static attractors)
corresponding to some desired output. Neurobiological
evidence however points to the dominance of nonequilibrium activity in the brain, which is a highly
connected, nonlinear dynamical system. This has led to a
growing interest in constructing nonequilibrium models of
brain activity –several of which show extremely interesting
dynamical transitions. In this paper, we focus on models
comprising elements which have exclusively excitatory or
inhibitory synapses. These networks are capable of a wide
range of dynamical behaviour, including high period
oscillations and chaos. Both the intrinsic dynamics of such
models and their possible role in information processing
are examined.
SINCE the development of the electronic computer in the
1940s, the serial processing computational paradigm has
successfully held sway. It has developed to the point where
it is now ubiquitous. However, there are many tasks which
are yet to be successfully tackled computationally. A case
in point is the multifarious activities that the human brain
performs regularly, including pattern recognition,
associative recall, etc. which are extremely difficult, if not
impossible to do using traditional computation.
This problem has led to the development of non-standard
techniques to tackle situations at which biological
information processing systems excel. One of the more
successful of such developments aims at ‘reverseengineering’ the biological apparatus itself to find out why
and how it works. The field of neural network models has
grown up on the premise that the massively parallel
distributed processing and connectionist structure
observed in the brain is the key behind its superior
performance. By implementing these features in the design
of a new class of architectures and algorithms, it is hoped
that machines will approach human-like ability in handling
real-world situations.
§
For correspondence. (e-mail: bikas@emp.saha.ernet.in)
420
The complexity of the brain lies partly in the multiplicity
of structural levels of organization in the nervous system.
The spatial scale of such structures span about ten orders
of magnitude – starting from the level of molecules and
synapses, going all the way up to the entire central nervous
system (Figure 1).
The unique capabilities of the brain to perform cognitive
tasks are an outcome of the collective global behaviour of
its constituent neurons. This is the motivation for
investigating the network dynamics of model neurons.
Depending upon one’s purpose, such ‘neurons’ may be
either, extremely simple binary threshold-activated elements,
or, described by a set of coupled partial differential
equations incorporating detailed knowledge of cellular
physiology and action potential propagation. However,
both simplifying and realistic neural models are based on
the theory of nonlinear dynamical systems in highdimensional spaces1. The development of nonlinear
dynamical systems theory – in particular, the discovery of
‘deterministic chaos’ in extremely simple systems – has
furnished the theoretical tools necessary for analysing nonequilibrium network dynamics. Neurobiological studies
indicating the presence of chaotic dynamics in the brain and
the investigation of its possible role in biological
information processing has provided further motivation.
Figure 1. Structural levels of organization of the nervous system
(from Churchland and Sejnowski1 ).
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
Actual networks of neuronal cells in the brain are
extremely complex (Figure 2). In fact, even single neurons
(Figure 3) are much more complicated than the ‘formal
neurons’ usually used in modelling studies, and are capable
of performing a large amount of computation2,3. To gain
insight into the network properties of the nervous system,
researchers have focused on artificial neural networks.
These usually comprise of binary neurons (i.e. neurons
capable of being only in one of two states), S i (= ± 1; i = 1,
2, . . ., N), whose temporal evolution is determined by the
equation:
S i = F (Σj Wij S j – θi),
(1)
where θi is an internal threshold, Wij is the connection
weight from element j to element i, and F is a nonlinear
function, most commonly taken as a sign or tanh (for
continuous value S i) function. Different neural network
models are specified by
• network topology, i.e. the pattern of connections
between the elements comprising the network;
• characteristics of the processing element, e.g. the explicit
form of the nonlinear function F, and the value of the
threshold θ;
• learning rule, i.e. the rules for computing the connection
weights Wij appropriate for a given task, and,
• updating rule, e.g. the states of the processing elements
may be updated in parallel (synchronous updating),
sequentially or randomly.
One of the limitations of most network models at present
is that they are basically static, i.e. once an equilibrium state
is reached, the network remains in that state, until the arrival
of new external input4. In contrast, real neural networks
show a preponderance of dynamical behaviour. Once we
recall a memory, our minds are not permanently stuck to it,
but can also roll over and recall other associated memories
without being prompted by any additional external stimuli.
This ability to ‘jump’ from one memory to another in the
absence of appropriate stimuli is one of the hallmarks of the
brain. It is an ability which one should try to recreate in a
network model if it is ever to come close to human-like
performance in intellectual tasks. One of the possible ways
of simulating such behaviour is through models guided by
non-equilibrium dynamics, in particular, chaos. This is
because of the much richer dynamical possibilities of such
networks, compared to those in systems governed by
convergent dynamics5.
The focus in this work will be on ‘simple’ network
models: ‘simple’ not only in terms of the size of the
networks considered when compared to the brain
(consisting of ~ 1011 neurons and ~ 1015 synapses), but
‘simple’ also in terms of the properties of the constituent
elements (i.e. the ‘neurons’) themselves, in that, most of the
physiological details of real neurons are ignored. The
objective is to see and retain what is essential for a
particular function performed by the network, treating other
details as being of secondary importance for the task at
hand. To do that one has to discard as much of the
complexity as possible to make the model tractable –
while at the same time retaining those features of the system
which make it interesting. So, while this kind of modelling is
indeed inspired by neuroscience, it is not exclusively
concerned with actually mimicking the activity of real
neuronal systems.
The Hopfield model
The foundation for computational neural modelling can be
traced to the work of McCulloch and Pitts in 1943 on the
universal computing capabilities of logic circuits akin to
neural nets. However, the interest of physicists was drawn
much later, mostly due to the work of Hopfield6 who
showed the equivalence between the problem of associative
Figure 2. Neuronal network of purkinje cells in the cerebellum of a
hedgehog (image obtained through golgi staining of neurons). (From
http:// weber.u.washington.edu/ chudler/).
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
Figure 3. Schematic diagram of a neuron (from http://www.
utexas.edu/research/asrec/neuron.html).
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SPECIAL SECTION:
memory – where, one of many stored patterns has to be
retrieved which most closely matches a presented input
pattern – and the problem of energy minimization in spin
glass systems. In the proposed model, the 2-state neurons,
S i (i = 1, . . ., N), resemble Ising spin variables and interact
among each other with symmetric coupling strengths, Wij. If
the total weighted input to a neuron is above a specified
threshold, it is said to be ‘active’, otherwise it is ‘quiescent’.
The static properties of the model have been well
understood from statistical physics. In particular, the
memory loading capacity (i.e. the ratio of the number of
patterns, p, stored in the network, to the total number of
neurons), α= p/N, is found to have a critical value at
αc ~ 0.138, where the overlap between the asymptotic state
of the network and the stored patterns show a
discontinuous transition. In other words, the system goes
from having good recall performance (α< αc ) to becoming
totally useless (α> αc ).
It was observed later that dynamically defined networks
with asymmetric interactions, Wij, have much better recall
performance. In this case, no effective energy function can
be defined and the use of statistical physics of spin glasslike systems is not possible. Such networks have, therefore,
mostly been studied through extensive numerical
simulations. One such model is a Hopfield-like network with
a single-step delay dynamics with some tunable weight λ:
S i(n + 1) = sign [ΣjWij(S j(n) + λS j(n – 1))].
(2)
Here, S i(n) refers to the state of the i-th spin at the n-th time
interval. For λ> 0, the performance of the model improved
enormously over the Hopfield network, both in terms of
recall and overlap properties7. The time-delayed term seems
to be aiding the system in coming out of spurious minimas
of the energy landscape of the corresponding Hopfield
model. It also seems to have a role in suppressing noise. For
λ< 0, the system shows limit cycle behaviour. These limit–
cycle attractors have been used to store and associatively
recall patterns8. If the network is started off in a state close
to one of the stored memories, it goes into a limit cycle in
which the overlap of the instantaneous configuration of the
network with the particular stored pattern shows large
amplitude oscillations with time, while overlap with other
memories remains small. The model appears to have a larger
storage capacity than the Hopfield model and better recall
performance. It also performs well as a pattern classifier if
the memory loading level and the degree of corruption
present in the input are high.
The travelling salesman problem
To see how collective computation can be more effective
than conventional approaches, we can look at an example
from the area of combinatorial optimization: the Travelling
Salesman Problem (TSP). Stated simply, TSP involves
422
finding the shortest tour through N cities starting from an
initial city, visiting each city once, and returning at the end
to the initial city. The non-triviality of the problem lies in the
fact that the number of possible solutions of the problem
grows as (N – 1)!/2 with N, the number of cities. For N = 10,
the number of possible paths is 181,440 – thus, making it
impossible to find out the optimal path through exhaustive
search (brute-force method) even for a modest value of N. A
‘cost function’ (or, analogously, an energy function) can be
defined for each of the possible paths. This function is a
measure of the optimality of a path, being lowest for the
shortest path. Any attempt to search for the global solution
through the method of ‘steepest descent’ (i.e. along a
trajectory in the space of all possible paths that minimizes
the cost function by the largest amount) is bound to get
stuck at some local minima long before reaching the global
minima. The TSP has also been formulated and studied on a
randomly dilute lattice9. If all the lattice sites are occupied,
the desired optimal path is easy to find; it is just a Hamilton
walk through the vertices. If however, the concentration p
of the occupied lattice sites (‘cities’) is less than unity, the
search for a Hamilton walk through only the randomly
occupied sites becomes quite nontrivial. In the limit p → 0,
the lattice problem reduces to the original TSP (in
continuum).
A neural network approach to solving the TSP was first
suggested by Hopfield and Tank10. A more effective
solution is through the use of Boltzmann machines11, which
are recurrent neural networks implementing the technique of
‘simulated annealing’12. Just as in actual annealing, a
material is heated and then made to cool gradually, here, the
system dynamics is initially made noisy. This means, that
the system has initially some probability of taking up higher
energy configurations. So, if the system state is a local
optima, because of fluctuations, it can escape a sufficiently
small energy barrier and resume its search for the global
optima. As the noise is gradually decreased, this probability
becomes less and less, finally becoming zero. If the noise is
decreased at a sufficiently slow rate, convergence to the
global optima is guaranteed. This method has been applied
to solve various optimization problems with some measure
of success. A typical application of the algorithm to obtain
an optimal TSP route through 100 specific European cities is
shown in Figure 4 (ref. 13).
Nonequilibrium dynamics and excitatory–
inhibitory networks
The Hopfield network is extremely appealing owing to its
simplicity, which makes it amenable to theoretical analysis.
However, these very simplifications make it a
neurobiologically implausible model. For these reasons,
several networks have been designed incorporating known
biological facts – such as, the Dale’s principle, which states
that a neuron has either exclusively excitatory or exclusively
inhibitory synapses. In other words, if the i-th neuron is
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
excitatory (inhibitory), then Wji > 0 (< 0) for all j. It is
observed that, even connecting only an excitatory and an
inhibitory neuron with each other leads to a rich variety of
behaviour, including high period oscillations and chaos14–16.
The continuous-time dynamics of pairwise connected
excitatory–inhibitory neural populations have been studied
before17. However, an autonomous two-dimensional system
(i.e. one containing no explicitly time-dependent term),
evolving continuously in time, cannot exhibit chaotic
phenomena, by the Poincare–Bendixson theorem (see e.g.
Strogatz18). Network models updated in discrete time, but
having binary-state excitatory and inhibitory neurons, also
cannot show chaoticity, although they have been used to
model various neural phenomena, e.g. kindling, where
epilepsy is generated by means of repeated electrical
stimulation of the brain19. In the present case, the resultant
system is updated in discrete-time intervals and the
continuous-state (as distinct from a binary or discrete-state)
neuron dynamics is governed by a nonlinear activation
function, F. This makes chaotic behaviour possible in the
model, which is discussed in detail below.
If X and Y be the mean firing rates of the excitatory and
inhibitory neurons, respectively, then their time evolution is
given by the coupled difference equations:
Xn + 1 = Fa(Wxx Xn – Wxy Yn),
(3)
Yn + 1 = Fb(Wyx Xn – Wyy Yn).
The network connections are shown in Figure 5. The Wxy
and Wyx terms represent the synaptic weights of coupling
between the excitatory and inhibitory elements, while Wxx
and Wyy represent self-feedback connection weights.
Although a neuron coupling to itself is biologically
implausible, such connections are commonly used in neural
network models to compensate for the omission of explicit
terms for synaptic and dendritic cable delays. Without loss
of generality, the connection weightages Wxx and Wyx can be
Figure 4.
An optimal solution for a 100-city TSP (from Aarts et
al.13 ).
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
absorbed into the gain parameters a and b and the
correspondingly rescaled remaining connection weightages,
Wxy and Wyy , are labelled k and k′, respectively. For
convenience, a transformed set of variables, zn = Xn – kY n
and zn′ = Xn –k′Yn, is used. Now, if we impose the restriction
k = k′, then the two-dimensional dynamics is reduced
effectively
to
that of an one-dimensional difference equation (i.e. a ‘map’),
zn + 1= F (zn) = Fa(zn) – kF b(zn),
(4)
simplifying the analysis. The dynamics of such a map has
been investigated for both piecewise linear and smooth, as
well as asymmetric and anti-symmetric, activation functions.
The transition from fixed point behaviour to a dynamic one
(asymptotically having periodic or chaotic trajectory) has
been found to be generic across the different forms of F.
Features specific to each class of functions have also been
observed. For example, in the case of piecewise linear
functions, border-collision bifurcations and multifractal
fragmentation of the phase space occur for a range of
parameter values16. Anti-symmetric activation functions
show a transition from symmetry-broken chaos (with
multiple coexisting but disconnected attractors) to
symmetric chaos (when only a single chaotic attractor
exists). This feature has been used to show noise-free
‘stochastic resonance’ in such neural models20, as
discussed in the following section.
Stochastic resonance in neuronal assemblies
Stochastic resonance (SR) is a recently observed
cooperative phenomena in nonlinear systems, where the
ambient noise helps in amplifying a sub threshold signal
(which would have been otherwise undetected) when
the signal frequency is close to a critical value21 (see
Gammaitoni et al.22 for a recent review). A simple scenario
for observing such a phenomenon is a heavily damped
bistable dynamical system (e.g. a potential well with two
minima) subjected to an external periodic signal. As a result,
each of the minima is alternately raised and lowered in the
course of one complete cycle. If the amplitude of the forcing
is less than the barrier height between the wells, the system
Figure 5. The pair of excitatory (x) and inhibitory (y) neurons.
The arrows and circles represent excitatory and inhibitory synapses,
respectively.
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SPECIAL SECTION:
cannot switch between the two states. However, the
introduction of noise can give rise to such switching. This
is because of a resonance-like phenomenon due to
matching of the external forcing period and the noiseinduced (average) hopping time accross the finite barrier
between the wells, and as such, it is not a very sharp
resonance. As the noise level is gradually increased, the
stochastic switchings will approach a degree of
synchronization with the periodic signal until the noise is so
high that the bistable structure is destroyed, thereby
overwhelming the signal. So, SR can be said to occur
because of noise-induced hopping between multiple stable
states of a system, locking on to an externally imposed
periodic signal.
These results assume significance in light of the
observation of SR in the biological world. It has been
proposed that the sensory apparatus of several creatures
use SR to enhance their sensitivity to weak external
stimulus, e.g. the approach of a predator. Experimental
studies involving crayfish mechanoreceptor cells23 and
even, mammalian brain slice preparations24, have provided
evidence of SR in the presence of external noise and
periodic stimuli. Similar processes have been claimed to
occur for the human brain also, based on the results of
certain psychophysical studies25. However, in neuronal
systems, a non-zero signal-to-noise ratio is found even
when the external noise is set to zero26. This is believed to
be due to the existence of ‘internal noise’. This phenomenon has been examined through neural network
modelling, e.g. in Wang and Wang27, where the main source
of such ‘noise’ is the effect of activities of adjacent
neurons. The total synaptic input to a neuron, due to its
excitatory and inhibitory interactions with other neurons,
turns out to be aperiodic and noise-like. The evidence of
chaotic activity in neural processes of the crayfish28
suggests that nonlinear resonance due to inherent chaos
might be playing an active role in such systems. Such
noise-free SR due to chaos has been studied before in a
non-neural setting29. As chaotic behaviour is extremely
common in a recurrent network of excitatory and inhibitory
neurons, such a scenario is not entirely unlikely to have
occurred in the biological world. There is also a possible
connection of such ‘resonance’ to the occurrence of
epilepsy, whose principal feature is the synchronization of
activity among neurons.
The simplest neural model20 which can use its inherent
chaotic dynamics to show SR-like behaviour is a pair of
excitatory–inhibitory neurons with anti-symmetric piecewise
linear
activation
function,
viz.
Fa(z) = – 1,
if
z < – 1/a, Fa(z) = az, if – 1/a ≤ z ≤ 1/a, and Fa(z) = 1, if
z > 1/a. From eq. (4), the discrete time evolution of the
effective neural potential is given by the map,
where I is an external input. The design of the network
ensures that the phase space [– 1 +(kb/a),1 –(kb/a)] is
divided into two well-defined and segregated sub-intervals
L : [–1 +(kb/a), 0] and R : [0, 1 –(kb/a)]. For a < 4, there is no
dynamical connection between the two sub-intervals and
the trajectory, while chaotically wandering over one of the
sub intervals, cannot enter the other sub interval. For a > 4,
in a certain range of (b, k) values, the system shows both
symmetry-broken and symmetric chaos, when the trajectory
visits both sub intervals in turn. The chaotic switching
between the two sub-intervals occurs at random. However,
the average time spent in any of the sub-intervals before a
switching event, can be exactly calculated for the present
model as
⟨ n⟩ =
1
 bk 
bk1 −
 −1
a 

.
(5)
As a complete cycle would involve the system switching
from one sub-interval to the other and then switching back,
the ‘characteristic frequency’ of the chaotic process is
ωc = 1/(2⟨n⟩). For example, for the system to have a
characteristic frequency of ω= 1/400 (say), the above
relation provides the value of k ~ 1.3811 for a = 6, b = 3.42.
If the input to the system is a sinusoidal signal of amplitude
δ and frequency ~ ωc , we can expect the response to the
signal to be enhanced, as is borne out by numerical
simulations. The effect of a periodic input, In = δsin (2πωn),
is to translate the map describing the dynamics of the neural
pair, to the left and right, periodically. The presence of
resonance is verified by looking at the peaks of the
residence time distribution30, where the strength of the j-th
peak is given by
Pj = ∫
n j +α n 0
n j −α n 0
N (n ) dn ( 0 < α < 0 .25 ).
(6)
For maximum sensitivity, α is set as 0.25. As seen in Figure
6, the dependence of Pj( j = 1, 2, 3) on external signal
frequency, ω, exhibits a characteristic non-monotonic
profile, indicating the occurrence of resonance at
ω~ 1/(2⟨n⟩). For the system parameters used in the
simulation, ⟨n⟩ = 200. The results clearly establish that the
switching between states is dominated by the subthreshold periodic signal close to the resonant frequency.
This signal enhancement through intrinsic dynamics is an
example of how neural systems might use noise-free SR for
information processing.
Formation of neural assemblies via activity
synchronization
zn+1 = F (zn + In) = Fa(zn + In) – kF b(zn + In),
424
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
Dynamical transitions leading to coherence in brain activity,
in the presence of an external stimulus, have received
considerable attention recently. Most investigations of
these phenomena have focussed on the phase
synchronization of oscillatory activity in neural assemblies.
An example is the detection of synchronization of ‘40 Hz’
oscillations within and between visual areas and between
cerebral hemispheres of cats31 and other animals.
Assemblies of neurons have been observed to form and
separate depending on the stimulus. This has led to the
speculation that, phase synchronization of oscillatory
neural activity is the mechanism for ‘visual binding’. This is
the process by which local stimulus features of an object
(e.g. colour, motion, and shape), after being processed in
parallel by different (spatially separate) regions of the
cortex, are correctly integrated in higher brain areas, forming
a coherent representation (‘gestalt’).
Recent neurobiological studies32 have shown that many
cortical neurons respond to behavioural events with rapid
modulations of discharge correlation. Epochs with a
particular correlation may last from ~ 10–2 to 10 secs. The
observed modulation of correlations may be associated with
changes in the individual neuron’s firing rates. This
supports the notion that a single neuron can intermittently
participate in different computations by rapidly changing its
coupling to other neurons, without associated changes in
firing rate. The mechanisms of such dynamic correlations
are unknown. The correlation could probably arise from
changes in the pattern of activity of a large number of
neurons, interacting with the sampled neurons in a
correlated manner. This modification of correlations
between two neurons in relation to stimulation and
behaviour most probably reflects changes in the organi-
zation of spike activity in larger groups of neurons. This
immediately suggests the utilization of synchronization by
neural assemblies for rapidly forming a correlated spatial
cluster. There are indeed indications that such binding
between neurons occurs and the resultant assemblies are
labelled by synchronized firing of the individual elements
with millisecond precision, often associated with
oscillations in the so-called gamma-frequency range,
centered around 40 Hz.
Mostly due to its neurobiological relevance as described
above, the synchronization of activity has also been
investigated in network models. In the case of the
excitatory–inhibitory neural pair described before, even
N = 2 or 3 such pairs coupled together give rise to novel
kinds of collective behaviour15. For N = 2, synchronization
occurs for both unidirectional and bidirectional coupling,
when the magnitude of the coupling parameter is above a
certain critical threshold. An interesting feature observed is
the intermittent occurrence of desynchronization (in
‘bursts’) from a synchronized situation, for a range of
coupling values. This intermittent synchronization is a
plausible mechanism for the fast creation and destruction of
neural assemblies through temporal synchronization of
activity. For N = 3, two coupling arrangements are possible
for both unidirectional and bidirectional coupling: local
coupling, where nearest neighbours are coupled to each
other, and global coupling, where the elements are coupled
in an all-to-all fashion. In the case of bidirectional, local
coupling, we observe a new phenomenon, referred to as
mediated synchronization. The equations governing the
dynamics of the coupled system are given by:
z1n +1 =
(z1n + λ z 2n ),
z 2n +1 =
(z n2 + λ [z1n + z 3n ]),
z 3n +1 = F (z 3n + λ z 2n ).
Figure 6. Peak strengths of the normalized residence time
distribution, P1 (circles), P2 (squares) and P3 (diamonds), for periodic
stimulation of the excitatory–inhibitory neural pair (a = 6, b = 3.42
and k = 1.3811). Peak amplitude of the periodic signal is δ = 0.0005.
P1 shows a maximum at a signal frequency ωc ~ 1/400. Averaging is
done over 18 different initial conditions, the error bars indicating the
standard deviation.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
For the set ofF activation parameters a = 100, b = 25 (where
F is of anti-symmetric, sigmoidal nature), we observe the
following feature
F
over a range of values of the coupling
parameter, λ: the neural pairs, z1 and z3 which have no direct
connection between themselves synchronize, although z2
synchronizes with neither. So, the system z2 appears to be
‘mediating’ the synchronization interaction, although not
taking part in it by itself. This is an indication of how longrange synchronization might occur in the nervous system
without long-range connections.
For a global, bidirectional coupling arrangement, the
phenomenon of ‘frustrated synchronization’ is observed.
The phase space of the entire coupled system is shown in
Figure 7. None of the component systems is seen to be
synchronized. This is because the three systems, each
trying to synchronize the other, frustrate all attempts at
collective synchronization. Thus, the introduction of
structural disorder in chaotic systems can lead to a kind of
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‘frustration’33, similar to that seen in the case of spin
glasses. These features were of course sudied for very small
systems (N = 2 or 3), where all the possible coupling
arrangements could be checked. For larger N values, the set
of such combinations quickly becomes a large one, and was
not checked systematically. We believe, however, that the
qualitative behaviour remains unchanged.
Image segmentation in an excitatory–inhibitory
network
Sensory segmentation, the ability to pick out certain objects
by segregating them from their surroundings, is a prime
example of ‘binding’. The problem of segmentation of
sensory input is of primary importance in several fields. In
the case of visual perception, ‘object-background’
discrimination is the most obvious form of such sensory
segmentation: the object to be attended to, is segregated
from the surrounding objects in the visual field. This
process is demonstrated by dynamical transitions in a
model comprising excitatory and inhibitory neurons,
coupled to each other over a local neighbourhood. The
basic module of the proposed network is a pair of excitatory
and inhibitory neurons coupled to each other. As before,
imposing restrictions on the connection weights, the
dynamics can be simplified to that of the following onedimensional map:
zn + 1 = Fa(zn + In) – kF b(zn + I′n ),
(7)
where the activation function F is of asymmetric, sigmoidal
nature:
Fa(z) = 1– e–az, if z > 0,
= 0, otherwise.
Without loss of generality, we can take k = 1. In the
Figure 7. Frustrated synchronization: Phase space for three
bidirectional, globally coupled neural pairs (z1 , z2 , z3 ) with coupling
magnitude λ = 0.5 (a = 100, b= 5 for all the pairs).
426
following, only time-invariant external stimuli will be
considered, so that:
In = In′ = I.
The autonomous behaviour (i.e. I, I′ = 0) of the isolated
pair of excitatory–inhibitory neurons show a transition from
fixed point to periodic behaviour and chaos with the
variation of the parameters a, b, following the ‘perioddoubling’ route, universal to all smooth, one-dimensional
maps. The introduction of an external stimulus of magnitude
I has the effect of horizontally displacing the map to the left
by I, giving rise to a reverse period-doubling transition from
chaos to periodic cycles to finally, fixed-point behaviour.
The critical magnitude of the external stimulus which leads
to a transition from a period-2 cycle to fixed point behaviour
is given as34.
1−
Ic =
( µa )
1/ µ
2
µ
− ( a /µ)
+
1
[ln( µa ) − 1].
µa
(8)
To make the network segment regions of different
intensities (I1 < I2, say), one can fix µ and choose a suitable
a, such that I1 < Ic < I2. So elements, which receive input of
intensity I1, will undergo oscillatory behaviour, while
elements receiving input of intensity I2, will go to a fixedpoint solution.
The response behaviour of the excitatory–inhibitory
neural pair, with local couplings, has been utilized in
segmenting images and the results are shown in Figure 8.
The initial state of the network is taken to be totally random.
The image to be segmented is presented as external input to
the network, which undergoes 200–300 iterations. Keeping
a fixed, a suitable value of µ is chosen from a consideration
of the histogram of the intensity distribution of the image.
This allows the choice of a value for the critical intensity
(Ic ), such that, the neurons corresponding to the ‘object’
converge to fixed-point behaviour, while those belonging to
the ‘background’ undergo period-2 cycles. In practice, after
the termination of the specified number of iterations, the
neurons which remain unchanged over successive
iterations (within a tolerance value) are labelled as the
‘object’, the remaining being labelled the ‘background’.
The image chosen is that of a square of intensity I2 (the
object) against a background of intensity I1 (I1 < I2). Uniform
noise of intensity ε is added to this image. The signal-tonoise ratio is defined as the ratio of the range of grey levels
in the original image to the range of noise added (given by
ε). Figure 8 shows the results of segmentation for unit
signal-to-noise ratio. Figure 8 a shows the original image
while segmentation performance of the uncoupled network
is presented in Figure 8 b. As is clear from the figure, the
isolated neurons perform poorly in identifying the
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
NONEQUILIBRIUM STATISTICAL SYSTEMS
‘background’ in the presence of noise. The segmentation
performance improves remarkably when spatial interactions
are included in the model. We have considered discrete
approximations of circular neighbourhoods of excitatory
and inhibitory neurons with radii rex and rin(r = 1, 2),
respectively, in our simulations.
Results for rex = 1, rin = 2 and rex = rin = 2 are shown in
Figure 8 c, d respectively. The two architectures show very
similar segmentation results, at least up to the iterations
considered here, although the latter is unstable. Excepting
for the boundary of the ‘object’, which is somewhat broken,
the rest of the image has been assigned to the two different
classes quite accurately. More naturalistic images have also
been considered, such as a 5-bit ‘Lincoln’ image, and
satisfactory results have been obtained34. Note that, a
single value of a (and hence Ic ) has been used for the entire
image. This is akin to ‘global thresholding’. By
implementing local thresholding and choosing a on the
basis of local neighbourhood information, the performance
of the network can be improved.
Outlook
We have pointed out some of the possible uses of
dynamical transitions in a class of network models of
computation, namely excitatory–inhibitory neural networks
updated at discrete time-intervals. Dynamics however plays
an important role in a much broader class of systems
implementing collective computation – cellular automata35,
lattices of coupled chaotic maps36, ant-colony models37, etc.
Other examples may be obtained from the ‘Artificial Life’38
genre of models. However, even in the restricted region that
we have focused on, several important issues are yet to be
addressed.
One important point not addressed here is the issue of
a
b
c
d
learning. The connection weights {Wij} have been assumed
constant, as they change at a much slower time scale
compared to that of the neural activation states. However,
modification of the weights due to learning will also cause
changes in the dynamics. Such bifurcation behaviour,
induced by weight changes, will have to be taken into
account when devising learning rules for specific purposes.
The interaction of chaotic activation dynamics at a fast time
scale and learning dynamics on a slower time scale might
yield richer behaviour than that seen in the present models.
The first step towards such a programme would be to
incorporate time-varying connection weights in the model.
Such time-dependence of a system parameter has been
shown to give rise to interesting dynamical behaviours, e.g.
transition between periodic oscillations and chaos. This
suggests that varying the environment can facilitate
memory retrieval if dynamic states are used for storing
information in a neural network. The introduction of
temporal variation in the connection weights, independent
of the neural state dynamics, should allow us to develop an
understanding of how the dynamics at two time-scales
interact with each other.
Parallel to this, one has also to look at the learning
dynamics itself. Freeman39, among others, has suggested an
important role of chaos in the Hebbian model of learning40.
This is one of the most popular learning models in the
neural network community and is based on the following
principle postulated by Hebb40 in 1949:
When an axon of cell A is near enough to excite cell B
and repeatedly or consistently takes part in firing it, some
growth process or metabolic change takes place in one or
both cells such that A’s efficiency, as one of the cells
firing B, is increased.
According to the principle known as synaptic plasticity, the
synapse between neurons A and B increase its ‘weight’, if
the neurons are simultaneously active. By invoking an
‘adiabatic approximation’, we can separate the time scale of
updating the connection weights from that of neural state
updating. This will allow us to study the dynamics of the
connection weights in isolation.
The final step will be to remove the ‘adiabatic
approximation’, so that the neural states will evolve, guided
by the connection weights, while the connection weights
themselves will also evolve, depending on the activation
states of the neurons, as:
Wij(n + 1) = F ε (Wij(n), Xi(n), Xj (n)),
Figure 8. Results of implementing the proposed segmentation
method on noisy synthetic image: a, original image; b, output of the
uncoupled network; c, output of the coupled network (rex = 1,
rin = 2); and d, output of the coupled network (rex = rin = 2), after
200 iterations (a = 20, b/a = 0.25 and tolerance = 0.02).
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999
where X(n) and W(n) denote the neuron state and
connection weight at the n-th instant, F is a nonlinear
function that specifies the learning rule, and ε is related to
the time-scale of the synaptic dynamics. The cross-level
effects of such synaptic dynamics interacting with the
chaotic network dynamics might lead to significant
departure from the overall behaviour of the networks
427
SPECIAL SECTION:
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ACKNOWLEDGEMENTS. We thank R. Siddharthan (IISc) for
assistance during preparation of the electronic version of the
manuscript.
CURRENT SCIENCE, VOL. 77, NO. 3, 10 AUGUST 1999