Long-range correlations in driven systems
David Mukamel
Outline
Will discuss two examples where long-range correlations show upand 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)
N particlesV sites
Particles diffusing (with exclusion) on a grid
π (π)= ππProb. of finding a particle at site k
Local perturbation in equilibrium
occupation number
N particlesV sites
Add a local potential u at site 0
The density changes only locally.
0 1ππ½π’
1
1
1
1
1
Effect of a local drive: a single driving bond
In d β₯ 2 dimensions both the density corresponds to a potential ofa dipole in d dimensions, decaying as for large r.The current satisfies .
The same is true for local arrangements of driven bonds.The power law of the decay depends on the specificconfiguration.
The two-point correlation function corresponds to a quadrupole In 2d dimensions, decaying as for
The same is true at other densities to leading order in (order ).
Main Results
Density profile (with exclusion)
The density profile along the y axisin any other direction
β’ Time evolution of density:
π»2=π (π+1 ,π )+π (πβ1 ,π)+π (π ,π+1 )+π (π ,πβ1 )β4π (π ,π)
π»2π (π )=βππ( 0β)[πΏπ ,0ββπΏοΏ½βοΏ½ ,π1]
Non-interacting particles
The steady state equation
particle density electrostatic potential of an electric dipole
π»2π (π )=βππ( 0β)[πΏπ ,0ββπΏοΏ½βοΏ½ ,π1]
π»2πΊ (π , οΏ½βοΏ½π )=βπΏοΏ½βοΏ½ ,π πGreenβs function
solution
Unlike electrostatic configuration here the strength ofthe dipole should be determined self consistently.
p\q 0 1 2
0 0
1
2
Greenβs function of the discrete Laplace equation
πΊ (π ,π π )ββ 12π ln β¨ οΏ½βοΏ½β π0β¨ΒΏ
To find one uses the values
π (0β )= π
1β π4
determining
π ( οΏ½βοΏ½ )=πβ ππ (0β )2π
οΏ½βοΏ½1ππ 2
+π ( 1π2
)
π (π )=ππ (0β )2π
1π2
ΒΏ
density:
current:
at large
Multiple driven bonds
Using the Greenβs function one can solve for , β¦by solving the set of linear equations for
The steady state equation:
Two oppositely directed driven bonds β quadrupole field
π ( οΏ½βοΏ½ )=πβ ππ (0β )2π [ 1π 2 β2( οΏ½βοΏ½1π
π2 )2]+π (
1π 4
)
dimensions
ππ‘ π (π , π‘ )=π»2π (π ,π‘ )+π β¨π ( 0β ) {1βπ ( οΏ½βοΏ½1 ) }β© [πΏπ ,0ββπΏοΏ½βοΏ½ ,π1]
The model of local drive with exclusion
Here the steady state measure is not known however one candetermine the behavior of the density.
is the occupation variable
π ( οΏ½βοΏ½ )=πβπ β¨π (0β ) {1βπ (οΏ½βοΏ½1 )}β©
2ποΏ½βοΏ½1ππ2
+π(1π 2
)
π ( οΏ½βοΏ½ )=πβπ β¨π (0β ) {1βπ (οΏ½βοΏ½1 )}β©
2ποΏ½βοΏ½1ππ2
+π(1π 2
)
The density profile is that of the dipole potential with a dipolestrength which can only be computed numerically.
Simulation on a lattice with
For the interacting case the strength of the dipole was measured separately .
Simulation results
T. Bodineau, B. Derrida, J.L. Lebowitz, JSP, 140 648 (2010).
Two-point correlation function
- (r)
g( , )
In d=1 dimension, in the hydrodynamic limit
In higher dimensions local currents do not vanish for largeL and the correlation function does not vanish in this limit.
T. Sadhu, S. Majumdar, DM, in progress
Symmetry of the correlation function:
( Ξπ +Ξπ )πΆ (π ,π )=πβ (π , π )
πΆπ (r , s )=Cβπ(βπ ,βπ )
πΆπ ,π (r , s )=Cβπ ,1βπ(r , s )inversionparticle-hole
β«πππ πΆ (π ,π )=0at πΆπ (π ,π )=πΆβπ (π ,π )
corresponds to an electrostatic potential in induced by
- (r)
Consequences of the symmetry:
The net charge =0
At is even in
Thus the charge cannot support a dipole and the leading contribution in multipole expansion is that of a quadrupole (in 2d dimensions).
πΆ (π ,π ) 1 / (π2+π 2 )π
For one can expand in powers of
One finds: The leading contribution to is of order implying no dipolar contribution, with the correlation decaying as
πΆ (π ,π ) 1 / (π2+π 2 )π
πΆ (π ,π )=π πΌ1 (π ,π )+π2πΌ2 (π ,π )+β¦
Since (no dipole) and the net charge is zero the leading contribution is quadrupolar
( Ξπ +Ξπ )πΆ (π ,π )=π 1 (π ,π )+π2 (π ,π )+π3(π ,π )
+
π=π (0 ) (1βπ (βπ1 ))+π(βπ1)(1βπ (0))
Summary
Local drive in dimensions results in:
Density profile corresponds to a dipole in d dimensions
Two-point correlation function corresponds to a quadrupolein 2d dimensions
πΆ (π ,π ) 1 / (π2+π 2 )π
At density to all orders in At other densities to leading order
Example II: a two dimensional model with a driven line
T. Sadhu, Z. Shapira, DM PRL 109, 130601 (2012)
The effect of a drive on a fluctuating interface
Motivated by an experimental study of the effect of shear oncolloidal 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?
Local potential localizes the interface at any temperature Transfer matrix: 1d quantum particle in a local attractive potential, the wave-function is localized.
no localizing potential: with localizing potential:
+
-
In equilibrium- under local attractive potential
2 0 1 0 1 0 2 0 y
1 .0
0 .5
0 .5
1 .0
m y
Schematic magnetization profile
The magnetization profile is antisymmetric with respect to the zeroline with
+ - - + with rate
- + + - with rate
π»=β π½ βΒΏ π π>ΒΏ πππ πΒΏ
ΒΏ
Consider now a driving line
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 placeby which the magnetization of the driven lineis non-zero and the magnetization profile is notsymmetric.
The fluctuation of the interface are not symmetric aroundthe driven line.
These results can be demonstrated analytically in certain limit.
Example of configurations in the two mesoscopic states for a 100X101 with fixed boundary at T=0.85Tc
Results of numerical simulations
20 10 10 20 y
1 .0
0 .5
0 .5
1 .0
m y
20 10 10 20 y
1 .0
0 .5
0 .5
1 .0
m y
Schematic magnetization profiles
2 0 1 0 1 0 2 0 y
1 . 0
0 . 5
0 . 5
1 . 0
m y
unlike the equilibrium antisymmetric profile
L=100 T=0.85Tc
Averaged magnetization profile in the two states
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.
Typically one is interested in calculating the large deviation function of a magnetization profile
We show that in some limit a restricted large deviation function,that of the driven line magnetization, , can be computed
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.
π (π ( π₯ , π¦ )) πβπΏ2πΉ (π ( π₯ , π¦ ) )
π (π0 )=πβπΏπ (π0)
In this limit the probability distribution of is where the potential (large deviations function) can be computed.
Large driving field
Slow exchange rate between the driven line and the rest of the system
Low temperature
The following limit is considered
π
The large deviations function
π (π0 )=πβπΏπ (π0)
Slow exchange between the line and the rest of the system
In between exchange processes the systems iscomposed of 3 sub-systems evolving independently
+βββ+ΒΏππ€ (Ξπ» )
ππ€ (β Ξπ» )π<π(
1πΏ3 )
π€ (Ξπ» )=min (1 ,πβπ½Ξπ» )
Fast drive
the coupling within the lane can be ignored. As a resultthe spins on the driven lane become uncorrelated and they arerandomly distributed (TASEP)
The driven lane applies a boundary field on the two otherparts
Due to the slow exchange rate with the bulk, the two bulk sub-systems reach the equilibrium distribution of an Ising modelwith a boundary field
Low temperature limit
In this limit the steady state of the bulk sub systems canbe expanded in T and the exchange rate with the driven line canbe computed.
with rate
with rate
performs a random walk with a rate which depends on
π (ππ )=β βπ=0
ππ πΏ2
β1
lnπ ( 2ππΏ )+ βπ=1
ππ πΏ2
π( 2ππΏ )
π (ππ=2ππΏ )=
π (0)β―β―π (2 (πβ1 )
πΏ )
π ( 2πΏ)β―β―π( 2ππΏ )
β‘πβπ(ππ)
π (π0)π (π0)
+ - ++ + + + +
- - - - -
+ + + + +
- - - - -
Calculate p at low temperature
+ - ++ + + + +
- - - - -
+ + + + +
- - - - -
contribution to p
is the exchange rate between the driven line and the adjacent lines
The magnetization of the driven lane changes in steps of
Expression for rate of increase, p
π (π )=ββ«0
π
lnπ (π) ππ+β«0
π
lnπ (π ) ππ
π (ππ )=π (βππ)
This form of the large deviation function demonstrates the spontaneous symmetry breaking. It also yield theexponential flipping time at finite
π
π (π0 )=πβπΏπ (π0)
ΒΏππ>1βπ (πβ 4 π½ π½)
Summary
Simple examples of the effect of long range correlations in drivenmodels have been presented.
A limit of slow exchange rate is discussed which enablesthe evaluation of some large deviation functions far fromequilibrium.