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

𝑝 (𝑘)=𝑁𝑉Prob. 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 1𝑒𝛽𝑢

1

(𝑉 −1 )𝑃 (𝑘 )+𝑃 (0 )=𝑁 (𝑘≠0)

𝑃 (𝑘 )=𝑁−𝑃 (0)𝑉 −1 ≈ 𝑁𝑉 +𝑂( 1𝑉 )

1

1

1

1

How does a local drive affect the steady-state of a system?

A single driving bond

Main results:

In d ≥ 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

• Time evolution of density:

𝛻2=𝜙 (𝑚+1 ,𝑛 )+𝜙 (𝑚−1 ,𝑛)+𝜙 (𝑚 ,𝑛+1 )+𝜙 (𝑚 ,𝑛−1 )−4𝜙 (𝑚 ,𝑛)

𝛻2𝜙 (𝑟 )=−𝜖𝜙( 0⃗)[𝛿𝑟 ,0⃗−𝛿�⃗� ,𝑒1]

Non-interacting particles

The steady state equation

particle density electrostatic potential of an electric dipole

𝛻2𝜙 (𝑟 )=−𝜖𝜙( 0⃗)[𝛿𝑟 ,0⃗−𝛿�⃗� ,𝑒1]

The dipole strength has to be determined self consistently.

𝛻2𝐺 (𝑟 , �⃗�𝑜 )=−𝛿�⃗� ,𝑟 𝑜Green’s function

solution

Unlike electrostatic configuration here the strength ofthe dipole should be determined self consistently.

p\q 0 1 2

0 0

1

2

Green’s function of the discrete Laplace equation

𝐺 (𝑟 ,𝑟 𝑜 )≈− 12𝜋 ln ∨ �⃗�− 𝑟0∨¿

To find one uses the values

𝜙 (0⃗ )= 𝜌

1− 𝜖4

determining

𝜙 ( �⃗� )=𝜌− 𝜖𝜙 (0⃗ )2𝜋

�⃗�1𝑟𝑟 2

+𝑂 ( 1𝑟2

)

𝑗 (𝑟 )=𝜖𝜙 (0⃗ )2𝜋

1𝑟2

¿

density:

current:

at large

Multiple driven bonds

Using the Green’s function one can solve for , …by solving the set of linear equations for

The steady state equation:

Two oppositely directed driven bonds – quadrupole field

𝜙 ( �⃗� )=𝜌− 𝜖𝜙 (0⃗ )2𝜋 [ 1𝑟 2 −2( �⃗�1𝑟𝑟2 )

2]+𝑂 (1𝑟 4

)

dimensions

𝜕𝑡𝜙 (𝑟 , 𝑡 )=𝛻2 ( �⃗� ,𝑡 )+𝜖 ⟨𝜏 ( 0⃗ ) {1−𝜏 ( �⃗�1 ) } ⟩[𝛿𝑟 , 0⃗−𝛿𝑟 ,𝑒1]

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

𝜙 ( �⃗� )=𝜌−𝜖 ⟨𝜏 (0⃗ ) {1−𝜏 (�⃗�1 )}⟩

2𝜋�⃗�1𝑟𝑟2

+𝑂(1𝑟 2

)

𝜙 ( �⃗� )=𝜌−𝜖 ⟨𝜏 (0⃗ ) {1−𝜏 (�⃗�1 )}⟩

2𝜋�⃗�1𝑟𝑟2

+𝑂(1𝑟 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)

𝐻=− 𝐽 ∑¿ 𝑖 𝑗>¿𝑆𝑖𝑆 𝑗 ¿

¿ 𝑆 𝑖=±1

+ particle - vacancy

particle exchange (Kawasaki) dynamics

+ - - + with rate

+

-

-

at

2d Ising model with a row of weak bonds (equilibrium)

𝐽 1

𝐽

𝐽 1< 𝐽

+

-

The weak-bonds row localizes the interface at any temperature

𝐽 1< 𝐽𝐽 1𝐽

+

-

T=0

+

-

T>0

Interface energy at low temperature:

𝐻= 𝐽∑𝑖=1

𝐿

|h𝑖−h𝑖+1|−( 𝐽− 𝐽 1)∑𝑖=1

𝐿

𝛿(h𝑖)

- 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

𝐻=− 𝐽 ∑¿ 𝑖 𝑗>¿𝑆𝑖𝑆 𝑗¿

¿

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)

𝑃 (𝑚0 )=𝑒−𝐿𝑈 (𝑚0)

𝑈

Slow exchange between the line and the rest of the system

In between exchange processes the systems iscomposed of 3 sub-systems evolving independently

+−⇆−+¿𝑟𝑤 (Δ𝐻 )

𝑟𝑤 (− Δ𝐻 )𝑟<𝑂(

1𝐿3

)

𝑤 (Δ𝐻 )=min (1 ,𝑒−𝛽Δ𝐻 )

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

𝑈 (𝑚𝑜 )=− ∑𝑘=0

𝑚𝑜𝐿2

−1

ln𝑝 ( 2𝑘𝐿 )+ ∑𝑘=1

𝑚𝑜𝐿2

𝑞( 2𝑘𝐿 )

𝑃 (𝑚𝑜=2𝑘𝐿 )=

𝑝 (0)⋯⋯𝑝 (2 (𝑘−1 )𝐿 )

𝑞 ( 2𝐿)⋯⋯𝑞( 2𝑘𝐿 )≡𝑒−𝑈(𝑚𝑜)

𝑝 (𝑚0)𝑞 (𝑚0)

+ - ++ + + + +

- - - - -

+ + + + +

- - - - -

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

𝑈 (𝑚 )=−∫0

𝑚

ln𝑝 (𝑘) 𝑑𝑘+∫0

𝑚

ln𝑞 (𝑘 )𝑑𝑘

𝑞 (𝑚𝑜 )=𝑝 (−𝑚𝑜)

This form of the large deviation function demonstrates the spontaneous symmetry breaking. It also yield theexponential flipping time at finite

𝑈

𝑃 (𝑚0 )=𝑒−𝐿𝑈 (𝑚0)

¿𝑚𝑜>1−𝑂 (𝑒− 4 𝛽 𝐽)

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.