Appendix E
Fokker-Planck Equation
with Detailed Balance
A stochastic process is simply a function of two variables, one is the time,
the other is a stochastic variable X, defined by specifying:
a: the set {x} of possible values for X;
b: the probability distribution, PX (x), over this set, or briefly P (x)
The set of values {x} for X may be discrete, or continuous. If the set of
values is continuous, then PX (x) is a probability density so that PX (x)dx
is the probability that one finds the stochastic variable X to have values
in the range [x, x + dxi.
An arbitrary number of other stochastic variables may be derived from
X. For example, any Y given by a mapping of X, is also a stochastic
variable. The mapping may also be ‘time’ dependent, that is, the mapping
depends on an additional variable t:
YX (t) = f (X, t) .
(E.1)
The quantity Y (t) is called a random function, or, since t often is time,
a stochastic process. A stochastic process is a function of two variables,
one is the time, the other is a stochastic variable X. Let x be one of the
possible values of X then
y(t) = f (x, t) ,
224
(E.2)
Fokker-Planck Equation with Detailed Balance 225
Appendix E.
is a function of t, called a sample function or realization of the process.
In physics one considers the stochastic process to be an ensemble of such
sample functions.
For many physical systems initial distributions of a stochastic variable y tend to equilibrium distributions: P (y, t) → P0 (y) as t → ∞. In
equilibrium detailed balance constrains the transition rates:
W (y|y 0 )P0 (y 0 ) = W (y 0 |y)P0 (y) ,
(E.3)
wehere W (y 0 |y) is the probability, per unit time, that the system changes
from a state | y i, characterized by the value y for the stochastic variable
Y , to a state | y 0 i.
Note that for a system in equilibrium the transition rate W (y|y 0 ) and
the reverse W (y 0 |y), may be very different. Consider, for instance, a simple
system that has only two energy levels 0 = 0 and 1 = ∆E. Then we find
that
W (1 |0 ) exp(−0 /kT ) = W (0 |1 ) exp(−1 /kT ) .
(E.4)
Therefore W (1 |0 )/W (0 |1 ) = exp(−∆E/kT ) → 0 when ∆E/kT → ∞,
that is, T tends to zero.
Assume that W (y|y 0 ) is finite only for small jumps, and that it varies
slowly with y. It is convenient to introduce the transition rate R+ for
positive (or forward) jumps:
W (y 0 |y) = R+ (ρ; y) ,
for y 0 = y + ρ ,
and ρ > 0 .
(E.5)
The forward jump rate is assumed to be sharply peaked so that R+ (ρ; y) '
0 for r > δ, and R+ (ρ; y + ∆y) ' R+ (ρ; y) for ∆y < δ. The change in the
probability distribution is then given by the Master equation
∂
P (y, t) =
∂t
Z
∞
{W (y|y − ρ)P (y − ρ, t) − W (y − ρ|y)P (y, t)} dρ
Z 0∞
. (E.6)
{W (y|y + ρ)P (y + ρ, t) − W (y + ρ|y)P (y, t)} dρ
+
0
Here we note that the detailed balance equation (E.3) may be written
R+ (r; y − ρ)P0 (y − ρ) = W (y|y − ρ)P0 (y − ρ) = W (y − ρ|y)P0 (y) ,
W (y|y + ρ)P0 (y + ρ) = W (y + ρ|y)P0 (y)
= R+ (ρ; y)P0 (y) .
(E.7)
226 Appendix E.
Fokker-Planck Equation with Detailed Balance
With these expressions equation (E.6) take the form
Z ∞
∂
P (y + ρ, t) P (y, t)
P (y, t) =
dρ R+ (ρ; y)P0 (y)
−
∂t
P0 (y + ρ)
P0 (y)
0
(E.8)
Z ∞
P (y − ρ, t) P (y, t)
+
dρ R+ (ρ; y − ρ)P0 (y − ρ)
−
P0 (y − ρ)
P0 (y)
0
Now we may expand the terms in the parentheses to give
Z ∞
∂
∂ P (y, t)
P (y, t) =
dρ ρ R+ (ρ; y)P0 (y)
∂t
∂y P0 (y) y
0
Z ∞
∂ P (y, t)
dρ ρ R+ (ρ; y − ρ)P0 (y − ρ)
−
∂y P0 (y) y−ρ
0
(E.9)
The two terms in the integral differ only slightly and we expand the last
term around y and obtain
∂
∂ P (y, t)
∂
P (y, t) =
D(y)P0 (y)
.
(E.10)
∂t
∂y
∂y P0 (y)
Here the generalized diffusion constant D is given by:
Z
Z ∞
1
(y 0 − y)2 W (y 0 |y)dy 0 ,
D(y) =
ρ2 R+ (ρ; y)dρ '
2
0
(E.11)
where the second expression uses that we assumed that P0 (y) varies little
over the range where R+ has a significant value. Equation (E.10) is a
consequence of detailed balance.
In the case of multivariate stochastic processes we have more than one
stochastic variable and if we write r = (y1 , y2 , . . . , yn ), then the FokkerPlanck equation for stationary Markov processes with narrow transition
rates takes the convenient form:
P (r, t)
∂
(E.12)
P (r, t) = ∇· D(r)P0 (r)·∇
∂t
P0 (r)
where the ∇ = ( ∂y∂ 1 , ∂y∂ 2 , . . . ∂y∂n ). The Fokker-Planck equation in this form
makes explicit that there is no time dependence if P (r, t) = P0 (r).
The diffusion tensor D is given in terms of an expression similar to
equation (E.11)
Z
1
D(r) =
(r 0 − r)·W (r 0 |r)·(r 0 − r)dn r 0 ,
(E.13)
2
Z
1
Dij (r) =
(yi0 − yi )W (r 0 |r)(yj0 − yj )dn r 0 ,
(E.14)
2
E.1
E.1
227
The Einstein Relations
The Einstein Relations
In a system of Brownian particles undergoing diffusion, the stochastic variable describing particle is its position r. The probability density P (r, t) is
proportional to the concentration of particles, c(r, t). Therefore the FokkerPlanck equation (E.12) becomes an equation for the concentration of the
Brownian particles:
∂
c(r, t)
c(r, t) = ∇· D(r)c0 (r)·∇
(E.15)
∂t
c0 (r)
We have assumed that the concentration at position r is proportional to
P (r, t), that the Brownian particle positions are well approximated by a
Markov process, and that the jumps are short ranged. However, the Brownian particles need not be at a low concentration, in fact they may interact
strongly. The equilibrium concentration has the general form
c0 (r) ∼ exp(−∆G(r)/kT )
(E.16)
for a system at constant temperature T , pressure P , and number of particles N . Here, ∆G(r) is the change in Gibbs free energy, from some
reference state. For systems at constant volume V , one uses the Helmholtz
free energy change ∆F (r) instead of the Gibbs free energy. Other system
constrains replaces the appropriate free energy for ∆G.
The diffusion tensor D has components
Z
1
Dij (r) =
(yi0 − yi )W (r 0 |r)(yj0 − yj )dn r 0 ,
(E.17)
2
If we define the jump rate Γ as
Z
Γ=
W (r 0 |r)dn r 0
Then we may define the mean square jump distances as
Z
(yi0 − yi )W (r 0 |r)(yj0 − yj )dn r 0
0
0
Z
h(yi − yi )(yj − yj ) iW =
,
W (r 0 |r)dn r 0
(E.18)
(E.19)
and we arrive at the Einstein relation for the diffusion constant
Dij =
1
Γh(yi0 − yi )(yj0 − yj ) iW
2
1st Einstein relation
(E.20)
228 Appendix E.
Fokker-Planck Equation with Detailed Balance
That is, the diffusion constant is one half the mean square jump distance
times the jump rate.
The Fokker-Planck equation for the concentration may also be written
as a continuity equation:
∂
c(r, t) + ∇·J = 0
∂t
(E.21)
Where the probability flux, or rather Brownian particle flux, J is given by
J (r) = − D(r)c0 (r)·∇[c(r, t)/c0 (r)]
= − D(r)·∇c(r, t) + c(r, t)D(r)·∇ ln c0 (r)
= − D(r)·∇c(r, t) + c(r, t)µ·F
(E.22)
Here F is the driving force that generates a drift velocity
v = µ·F
(E.23)
The diving force is
F = ∇ ln c0 (r) = kT ∇(−∆G(r)/kT ) = −∇(∆G(r))
(E.24)
whereas the mobility µ is given by the second Einstein relation
µ=
1
D
kT
2nd. Einstein relation
(E.25)
The driving force is just the negative gradient of the related potential—as
driving forces should be. The second Einstein relation is a relation between
the mobility of a particle and the diffusion constant. One of the best known
uses of this relation is for a single spherical particle radius a in a fluid of
viscosity µ in this case the Stokes equation gives the mobility of the particle
to be
µ = (6πµa)−1
(E.26)
and therefore, by the second Einstein relation (E.25), we find the diffusion
constant of a Brownian particle
D=
kT
6πµa
Stokes-Einstein relation
(E.27)
This expression for the diffusion constant of Brownian particles, in terms
of the Stokes expression for the mobility, is valid only for non-interacting
particles. For sedimenting particles at a finite density, which allows the
Brownian particles to interact, the Stokes expression (E.26) is no longer
valid, however, the second Einstein relation between mobility and Diffusion
constant still holds.