Long-range correlations in driven systems
David Mukamel
Outline
Will discuss two examples where long-range correlations show up
and consider some consequences
Example I: Effect of a local drive on the steady state of a system
Example II: Linear drive in two dimensions: spontaneous symmetry
breaking
Example I :Local drive perturbation
T. Sadhu, S. Majumdar, DM, Phys. Rev. E 84, 051136 (2011)
Local perturbation in equilibrium
Particles diffusing (with exclusion) on a grid
occupation number 𝜏𝑖 = 0,1
N particles
V sites
Prob. of finding a particle at site k
𝑁
𝑝 𝑘 =
𝑉
Add a local potential u at site 0
1
1
𝑒 𝛽𝑢
N particles
V sites
1
The density changes only locally.
1
1
0
1
Effect of a local drive: a single driving bond
Main Results
In d ≥ 2 dimensions both the density corresponds to a potential of
a dipole in d dimensions, decaying as 𝜙 𝑟 ~1/𝑟 𝑑−1 for large r.
The current satisfies 𝑗(𝑟)~1/𝑟 𝑑 .
The same is true for local arrangements of driven bonds.
The power law of the decay depends on the specific
configuration.
The two-point correlation function corresponds to a quadrupole
In 2d dimensions, decaying as 𝐶(𝑟, 𝑠)~1/ 𝑟 2 + 𝑠 2 𝑑 for 𝜌 = 1/2
The same is true at other densities to leading order in 𝜖
(order 𝜖 2 ).
Density profile (with exclusion)
The density profile
1/𝑟 2
𝜙(𝑟)~
1/𝑟
along the y axis
in any other direction
Non-interacting particles
• Time evolution of density:
𝜕𝑡 𝜙 𝑟, 𝑡 = 𝛻 2 𝜙 𝑟, 𝑡 + 𝜖𝜙 0, 𝑡 [𝛿𝑟,0 − 𝛿𝑟,𝑒1 ]
𝛻 2 = 𝜙 𝑚 + 1, 𝑛 + 𝜙 𝑚 − 1, 𝑛 + 𝜙 𝑚, 𝑛 + 1 + 𝜙 𝑚, 𝑛 − 1 − 4𝜙(𝑚, 𝑛)
The steady state equation
𝛻 2 𝜙 𝑟 = −𝜖𝜙(0)[𝛿𝑟,0 − 𝛿𝑟,𝑒1 ]
particle density → electrostatic potential of an electric dipole
𝛻 2 𝜙 𝑟 = −𝜖𝜙(0)[𝛿𝑟,0 − 𝛿𝑟,𝑒1 ]
Green’s function
solution
𝛻 2 𝐺 𝑟, 𝑟𝑜 = −𝛿𝑟,𝑟𝑜
𝜙 𝑟 = 𝜌 + 𝜖𝜙 0 [𝐺 𝑟,0 − 𝐺(𝑟,𝑒1 )]
Unlike electrostatic configuration here the strength of
the dipole should be determined self consistently.
Green’s function of the discrete Laplace equation
p\q
0
0
0
1
2
1
−
4
2
−1
𝜋
1
1
−
4
1
−
𝜋
1
−
4
2
2
−1
𝜋
1 2
−
4 𝜋
4
−
3𝜋
𝐺 𝑟, 𝑟𝑜
1
≈−
ln |𝑟 − 𝑟0 |
2𝜋
determining 𝜙(0)
To find 𝜙(0) one uses the values 𝐺 0, 0 = 0 , 𝐺 0, 𝑒1 =
𝜙 0 =
𝜌
1−
𝜖
4
1
−
4
at large 𝑟
density:
current:
𝜖𝜙 0 𝑒1 𝑟
1
𝜙 𝑟 =𝜌−
+ 𝑂( 2 )
2
2𝜋 𝑟
𝑟
𝜖𝜙 0 1
2 𝑒1 𝑟 𝑟
1
𝑗 𝑟 =
[𝑒1 −
+ 𝑂( 3 )
2
2
2𝜋 𝑟
𝑟
𝑟
Multiple driven bonds
𝜙 𝑟 = 𝜌 + 𝜖𝜙 𝑖1 𝐺 𝑟, 𝑖1 − 𝐺 𝑟, 𝑖1 + 𝑒1
+ 𝜖𝜙 𝑖2 𝐺 𝑟, 𝑖2 − 𝐺 𝑟, 𝑖2 + 𝑒1
Using the Green’s function one can solve for 𝜙 𝑖1 , 𝜙 𝑖2 …
by solving the set of linear equations for 𝑟 = 𝑖11 , 𝑖2 , ⋯
+⋯
Two oppositely directed driven bonds – quadrupole field
The steady state equation: 𝛻 2 𝜙 𝑟 = −𝜖𝜙(0)[2𝛿𝑟,0 − 𝛿𝑟,𝑒1 −𝛿𝑟,−𝑒1 ]
𝜖𝜙 0
𝜙 𝑟 =𝜌−
2𝜋
1
𝑒1 𝑟
−2 2
2
𝑟
𝑟
2
1
+ 𝑂( 4 )
𝑟
𝑑 ≠ 2 dimensions
𝑑=1
𝜙 𝑥 =𝜌−
𝐺 𝑥, 𝑥𝑜 = −
𝑑≥2
𝜖
2
𝜙 0 𝑠𝑔𝑛(𝑥)
𝑥−𝑥𝑜
2
𝜙 𝑟 ∼
1
𝑟 𝑑−1
The model of local drive with exclusion
Here the steady state measure is not known however one can
determine the behavior of the density.
𝜕𝑡 𝜙 𝑟, 𝑡 = 𝛻 2 𝜙 𝑟, 𝑡 + 𝜖 𝜏 0 1 − 𝜏 𝑒1
[𝛿𝑟,0 − 𝛿𝑟,𝑒1 ]
𝜏 = 0,1 is the occupation variable
𝜖 𝜏 0 1 − 𝜏 𝑒1
𝜙 𝑟 =𝜌−
2𝜋
𝑒1 𝑟
1
+ 𝑂( 2 )
2
𝑟
𝑟
𝜖 𝜏 0 1 − 𝜏 𝑒1
𝜙 𝑟 =𝜌−
2𝜋
𝑒1 𝑟
1
+ 𝑂( 2 )
2
𝑟
𝑟
The density profile is that of the dipole potential with a dipole
strength which can only be computed numerically.
Simulation results
Simulation on a 200 × 200 lattice with 𝜌 = 0.6
For the interacting case the strength of the dipole was
measured separately .
Two-point correlation function
𝐶 𝑟, 𝑠 =< 𝜏 𝑟 𝜏 𝑠 >- 𝜙(r) 𝜙(𝑠)
In d=1 dimension, in the hydrodynamic limit
𝐶𝐿 𝑖, 𝑗 =
1 𝑖
g(
𝐿 𝐿
,
𝑗
)
𝐿
T. Bodineau, B. Derrida, J.L. Lebowitz, JSP, 140 648 (2010).
In higher dimensions local currents do not vanish for large
L and the correlation function does not vanish in this limit.
T. Sadhu, S. Majumdar, DM, in progress
Symmetry of the correlation function:
𝐶 𝑟, 𝑠 =< 𝜏 𝑟 𝜏 𝑠 >- 𝜙(r) 𝜙(𝑠)
𝐶𝜖 r, s = C−𝜖 (−𝑟, −𝑠)
inversion
particle-hole
𝐶𝜖,𝜌 r, s = C−𝜖,1−𝜌 (r, s)
at 𝜌 = 1/2
𝐶𝜖 𝑟, 𝑠 = 𝐶−𝜖 (𝑟, 𝑠)
𝑑𝑑 𝑟 𝐶 𝑟, 𝑠 = 0
Δ𝑟 + Δ𝑠 𝐶 𝑟, 𝑠 = 𝜎
𝑟, 𝑠
𝐶(𝑟, 𝑠) corresponds to an electrostatic potential in 2𝑑 induced by 𝜎
Consequences of the symmetry:
The net charge =0
At 𝜌 = 0.5 𝐶𝜖 𝑟, 𝑠 is even in 𝜖
Thus the charge cannot support a dipole and the
leading contribution in multipole expansion is
that of a quadrupole (in 2d dimensions).
𝐶(𝑟, 𝑠)~1/ 𝑟 2 + 𝑠 2
𝑑
For 𝜌 ≠ 0.5 one can expand 𝜎 in powers of 𝜖
One finds:
The leading contribution to 𝜎 is of order 𝜖 2 implying
no dipolar contribution, with the correlation decaying as
𝐶(𝑟, 𝑠)~1/ 𝑟 2 + 𝑠 2
𝑑
𝐶 𝑟, 𝑠 = 𝜖𝛼1 𝑟, 𝑠 + 𝜖 2 𝛼2 𝑟, 𝑠 + ⋯
𝛼2𝑝−1 −𝑟, −𝑠 = −𝛼2𝑝−1 𝑟, 𝑠
𝛼2𝑝 −𝑟, −𝑠 = 𝛼2𝑝 (𝑟, 𝑠)
Since 𝛼1 = 0 (no dipole) and the net charge is zero
the leading contribution is quadrupolar
Δ𝑟 + Δ𝑠 𝐶 𝑟, 𝑠 = 𝜎1 𝑟, 𝑠 + 𝜎2 𝑟, 𝑠 + 𝜎3 (𝑟, 𝑠)
𝜎1 𝑟, 𝑠 = 𝜖 𝑄 𝑛 𝑟 𝛿𝑠,0 − 𝛿𝑠+𝑒1 ,0 1 − 𝛿𝑟,0 − 𝛿𝑟+𝑒1 ,0
+ 𝜖 𝑄 𝑛 𝑠 𝛿𝑟,0 − 𝛿𝑟+𝑒1 ,0 1 − 𝛿𝑠,0 − 𝛿𝑠+𝑒1 ,0
1 2 2
2
𝜎2 𝑟, 𝑠 = − 𝜖 𝑄 𝛿𝑟,−𝑒1 𝛿𝑠,0 +
𝜙 𝑟 + 𝑒𝜈 − 𝜙 𝑟 𝛿𝑠,𝑟+𝑒𝜈
𝑑
𝜈
1 2 2
2
− 𝜖 𝑄 𝛿𝑠,−𝑒1 𝛿𝑟,0 +
𝜙 𝑠 + 𝑒𝜈 − 𝜙 𝑠 𝛿𝑠,𝑟−𝑒𝜈
𝑑
𝜈
𝜎3 𝑟, 𝑠 =
𝐶 𝑟 + 𝑒𝜈 , 𝑠 − 2𝐶 𝑟, 𝑠 + 𝐶 𝑟, 𝑠 − 𝑒𝜈
𝛿𝑠,𝑟+𝑒𝜈 + 𝛿𝑠,𝑟
𝜈
+
𝐶 𝑟, 𝑠 + 𝑒𝜈 − 2𝐶 𝑟, 𝑠 + 𝐶 𝑟 − 𝑒𝜈 , 𝑠
𝛿𝑠,𝑟−𝑒𝜈 + 𝛿𝑠,𝑟
𝜈
𝑄 = 𝑛 0 1 − 𝑛 −𝑒1
+ 𝑛(−𝑒1 ) 1 − 𝑛(0)
Summary
Local drive in 𝑑 > 1 dimensions results in:
Density profile corresponds to a dipole in d dimensions
𝜙 𝑟 ~ 1/𝑟 𝑑−1
Two-point correlation function corresponds to a quadrupole
in 2d dimensions
𝐶(𝑟, 𝑠)~1/ 𝑟 2 + 𝑠 2 𝑑
At density 𝜌 = 0.5 to all orders in 𝜖
At other densities to leading (𝜖 2 ) order
Example II: a two dimensional model with a driven line
The effect of a drive on a fluctuating interface
T. Sadhu, Z. Shapira, DM PRL 109, 130601 (2012)
Motivated by an experimental study of the effect of shear on
colloidal liquid-gas interface.
D. Derks, D. G. A. L. Aarts, D. Bonn, H. N. W. Lekkerkerker, A. Imhof,
PRL 97, 038301 (2006).
T.H.R. Smith, O. Vasilyev, D.B. Abraham, A. Maciolek, M. Schmidt,
PRL 101, 067203 (2008).
What is the effect of a driving line on an interface?
-
+
In equilibrium- under local attractive potential
+
-
Local potential localizes the interface at any temperature 𝑇 < 𝑇𝑐
Transfer matrix: 1d quantum particle in a local attractive potential,
the wave-function is localized.
no localizing potential: < Δℎ2 > ~ 𝐿
with localizing potential: < 𝐽 < Δℎ2 > ~𝑐𝑜𝑛𝑠𝑡
Schematic magnetization profile
my
1.0
0.5
20
10
10
20
y
0.5
1.0
The magnetization profile is antisymmetric with respect to the zero
line with 𝑚 𝑦 = 0 = 0
Consider now a driving line
+-→-+
with rate min(1, 𝑒 −𝛽Δ𝐻+𝛽𝐸 )
-+→+-
with rate min(1, 𝑒 −𝛽Δ𝐻−𝛽𝐸 )
𝐻 = −𝐽
𝑆𝑖 𝑆𝑗
<𝑖𝑗>
Ising model with Kawasaki dynamics which is biased on the middle row
Main results
The interface width is finite (localized)
A spontaneous symmetry breaking takes place
by which the magnetization of the driven line
is non-zero and the magnetization profile is not
symmetric.
The fluctuation of the interface are not symmetric around
the driven line.
These results can be demonstrated analytically in certain limit.
Results of numerical simulations
Example of configurations in the two mesoscopic states for a 100X101 with
fixed boundary at T=0.85Tc
Schematic magnetization profiles
my
my
20
1.0
1.0
0.5
0.5
10
10
20
y
20
10
10
0.5
0.5
1.0
1.0
20
my
1.0
0.5
unlike the equilibrium antisymmetric profile
20
10
10
0.5
1.0
20
y
y
Averaged magnetization profile in the two states
L=100 T=0.85Tc
Time series of Magnetization of driven lane for a
100X101 lattice at T= 0.6Tc.
Switching time on a square LX(L+1) lattice with Fixed
boundary at T=0.6Tc.
Analytical approach
In general one cannot calculate the steady state measure of this system.
However in a certain limit, the steady state distribution (the large
deviations function) of the magnetization of the driven line can be calculated.
Typically one is interested in calculating −𝐹 𝑚 𝑥, 𝑦
the large deviation function of a magnetization profile
𝑃(𝑚 𝑥, 𝑦 )~𝑒
−𝐿2 𝐹(𝑚 𝑥,𝑦 )
We show that in some limit a restricted large deviation function,
that of the driven line magnetization, 𝑚0 , can be computed
𝑃 𝑚0 = 𝑒 −𝐿𝑈(𝑚0)
The following limit is considered
Slow exchange rate between the driven line and the rest of the system
Large driving field 𝐸 ≫ 𝐽
Low temperature
In this limit the probability distribution of m0 is 𝑃 𝑚0 = 𝑒 −𝐿𝑈(𝑚0)
where the potential (large deviations function) 𝑈 𝑚0 can be computed.
The large deviations function
𝑈
𝑃 𝑚0 = 𝑒 −𝐿𝑈(𝑚0)
Slow exchange between the line and the rest of the system
𝑟𝑤(Δ𝐻)
+−
⇆
𝑟𝑤(−Δ𝐻)
− +
1
𝑟 < 𝑂( 3 )
𝐿
𝑤 Δ𝐻 = min(1, 𝑒 −𝛽Δ𝐻 )
In between exchange processes the systems is
composed of 3 sub-systems evolving independently
Fast drive 𝐸 ≫ 𝐽
the coupling 𝐽 within the lane can be ignored. As a result
the spins on the driven lane become uncorrelated and they are
randomly distributed (TASEP)
The driven lane applies a boundary field 𝐽𝑚0 on the two other
parts
Due to the slow exchange rate with the bulk, the two bulk
sub-systems reach the equilibrium distribution of an Ising model
with a boundary field 𝐽𝑚0
Low temperature limit
In this limit the steady state of the bulk sub systems can
be expanded in T and the exchange rate with the driven line can
be computed.
𝑚𝑜 ⟶ 𝑚𝑜 +
𝑚𝑜 ⟶ 𝑚𝑜 −
2
𝐿
2
𝐿
𝑞(𝑚0 )
with rate
𝑝(𝑚𝑜 )
with rate
𝑞(𝑚𝑜 )
𝑝(𝑚0 )
𝑚𝑜 performs a random walk with a rate which depends on 𝑚𝑜
2 𝑘−1
𝑝(0)
⋯
⋯
𝑝(
)
2𝑘
𝐿
𝑃 𝑚𝑜 =
=
≡ 𝑒 −𝑈(𝑚𝑜)
2
2𝑘
𝐿
𝑞( ) ⋯ ⋯ 𝑞( )
𝐿
𝐿
𝑚𝑜 𝐿
2 −1
𝑈 𝑚𝑜 = −
𝑘=0
2𝑘
ln 𝑝
+
𝐿
𝑚𝑜 𝐿
2
𝑘=1
2𝑘
𝑞( )
𝐿
Calculate p 𝑚𝑜 at low temperature
- - +
+ +
+ +
+
+
+
+
+
+
+
- - +
+ +
+ +
contribution to p 𝑚𝑜 :
+
+
1
8
+
+
+
+
+
1 − 𝑚𝑜 1 + 𝑚𝑜 2 𝑒 −2𝛽𝐽 𝑒 −2𝛽𝐽1
𝐽1 is the exchange rate between the driven line and the adjacent lines
The magnetization of the driven lane 𝑚∘ changes in steps of 2/𝐿
Expression for rate of increase, p(𝑚𝑜 )
𝑝 𝑚0
1
= 1 + m0
8
2
1
+ [2 1 + 𝑚0 1 − 𝑚0
8
1 − m0 𝑒 −2𝛽
2
𝐽+𝐽1
𝑒 −2𝛽𝐽1 + 𝑒 2𝛽𝐽1 𝑚0
+ 1 + 𝑚0 2 (1 − 𝑚0 )𝑒 2𝛽𝐽1 𝑚0 + 1 − 𝑚0 3 𝑒 2𝛽𝐽1 𝑚0 ]𝑒 −6𝛽𝐽 + 𝑂(𝑒 −8𝛽𝐽 )
𝑞 𝑚𝑜 = 𝑝(−𝑚𝑜 )
𝑚
𝑈 𝑚 =−
0
𝑚
ln 𝑝 𝑘 𝑑𝑘 + Type
ln 𝑞equation
𝑘 𝑑𝑘
0
here.
𝑈
This form of the large deviation function demonstrates
the spontaneous symmetry breaking. It also yield the
exponential flipping time at finite 𝐿. (𝑇 = 0.6𝑇𝑐 , 𝐽1 = 𝐽)
𝑃 𝑚0 = 𝑒 −𝐿𝑈(𝑚0)
< 𝑚𝑜 > 1 − 𝑂(𝑒 −4𝛽𝐽 )
Summary
Simple examples of the effect of long range correlations in driven
models have been presented.
A limit of slow exchange rate is discussed which enables
the evaluation of some large deviation functions far from
equilibrium.