Long-range correlations in driven systems
David Mukamel
Driven, non-equilibrium systems
''
)()'()'()'()(CC
CPCCWCPCCWtCP
the probability to be at a microscopic configuration C at time t
)()'()'()'( CPCCWCPCCW
systems with currents
do not obey detailed balance
T1 T2
T1>T2
E
heat current
charge current
What are the steady state properties of such systems?
Typical simple examples
It is well known that such systems exhibit long-range correlations when the dynamics involves some conserved parameter.
Outline
Will discuss a few 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
(π β1 )π (π )+π (0 )=π (πβ 0)
π (π )=πβπ (0)π β1 β ππ +π( 1π )
1
1
1
1
How does a local drive affect the steady-state of a system?
A single driving bond
Main results:
In d β₯ 2 dimensions both the density and the local currentdecay algebraically with the distance from the driven bond.
The same is true for local arrangements of driven bond.The power law of the decay depends on the specificconfiguration.
In d=2 dimensions a close correspondence to electrostatics is found, with analogous variables to electric and magneticfields E, H.
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]
The dipole strength has to be determined self consistently.
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( οΏ½βοΏ½1ππ2 )
2]+π (1π 4
)
dimensions
ππ‘π (π , π‘ )=π»2 ( οΏ½βοΏ½ ,π‘ )+π β¨π ( 0β ) {1βπ ( οΏ½βοΏ½1 ) } β©[πΏπ , 0ββπΏπ ,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 non-interacting case strength of the dipole was measured separately .
Simulation results
Magnetic field analog
for process
for the bond
Zero-charge configuration
The density is flat however there are currents
Zero magnetic field configuration
no currents but inhomogeneous density (equilibrium)
In general
non-zero electric field inhomogeneous density
non-zero magnetic field currents
zero magnetic field equilibrium configuration
Example II: a two dimensional model with a driven line
T. Sadhu, Z. Shapira, DM
Two dimensional lattice gas (Ising) model (equilibrium)
π»=β π½ βΒΏ π π>ΒΏπππ π ΒΏ
ΒΏ π π=Β±1
+ particle - vacancy
particle exchange (Kawasaki) dynamics
+ - - + with rate
+
-
-
at
2d Ising model with a row of weak bonds (equilibrium)
π½ 1
π½
π½ 1< π½
+
-
The weak-bonds row localizes the interface at any temperature
π½ 1< π½π½ 1π½
+
-
T=0
+
-
T>0
Interface energy at low temperature:
π»= π½βπ=1
πΏ
|hπβhπ+1|β( π½β π½ 1)βπ=1
πΏ
πΏ(hπ)
- the probability of finding the interface at height
1d Quantum mechanical particle (discrete space) with alocal attractive potential. The wave function is localized.
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
+
-
-
Consider now a line of driven bonds
+ - - + with rate
- + + - with rate
π»=β π½ βΒΏ π π>ΒΏπππ πΒΏ
ΒΏ
Main results
The driven line attracts the interface
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 notantisymmetric, (mesoscopic transition).
The fluctuation of the interface are not symmetric aroundthe driven line.
These results can be demonstrated analytically in certain limit.
Time= 6e9Time= 5e9 Time= 3e9
The is attracted by the driven line.
Results of numerical studies
A configuration of the periodic 500X501 lattice at temperature 0.85Tc.
Periodic 30X31 lattice at temperature 0.6Tc. Driven lane at y=0,there are around 15 macro-switches on a 10^9 MC steps.
Temporal evolution of the interface position
Periodic 30X31 lattice at temperature 0.57Tc.
Zoom in
Mesoscopic switches
Macroscopic switch
Example of configurations in the two mesoscopic states for a 100X101 with fixed boundary at T=0.85Tc
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
Asymmetric magnetization profile for a periodic 500X501 lattice at temperature T=0.85Tc.
A snapshot of the magnetization profile near the two interfaces on a 500X501 square lattice with periodic boundary condition at T=0.85Tc.
Non-symmetric fluctuations of the interface
Closed boundary conditions
In order to study the mesoscopic switches in more detailand to establish the existence of spontaneous symmetry breaking of the driven line we consider the case of closed boundary conditions
+ + + + + + + +
- - - - - - - -
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.
L=100 T=0.85Tc
Averaged magnetization profile in the two states
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 deviation function) of the magnetization of the driven line can be calculated.
In this limit the probability distribution of is where the potential (large deviation function) can be computed.
Large driving field
Slow exchange rate between the driven line and the rest of the system
Low temperature
Schematic potential (large deviation function)
π (π0 )=πβπΏπ (π0)
π
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
πππΏ2
β1
lnπ ( 2ππΏ )+ βπ=1
πππΏ2
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
Driven systems exhibit long range correlations under generic conditions.
Such correlations sometimes lead to long-range orderand spontaneous symmetry breaking which are absentunder equilibrium conditions.
Simple examples of these phenomena have been presented.
A limit of slow exchange rate is discussed which enablesthe evaluation of some large deviation functions far fromequilibrium.