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

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

1

1

1

1

Effect of a local drive: a single driving bond

In d ≥ 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

• 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]

𝛻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( �⃗�1𝑟

𝑟2 )2]+𝑂 (

1𝑟 4

)

dimensions

𝜕𝑡 𝜙 (𝑟 , 𝑡 )=𝛻2𝜙 (𝑟 ,𝑡 )+𝜖 ⟨𝜏 ( 0⃗ ) {1−𝜏 ( �⃗�1 ) }⟩ [𝛿𝑟 ,0⃗−𝛿�⃗� ,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 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:

( Δ𝑟 +Δ𝑠 )𝐶 (𝑟 ,𝑠)=𝜎❑ (𝑟 , 𝑠)

𝐶𝜖 (r , s )=C−𝜖(−𝑟 ,−𝑠)

𝐶𝜖 ,𝜌 (r , s )=C−𝜖 ,1−𝜌(r , s )inversionparticle-hole

∫𝑑𝑑𝑟 𝐶 (𝑟 ,𝑠 )=0at 𝐶𝜖 (𝑟 ,𝑠 )=𝐶−𝜖 (𝑟 ,𝑠)

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).

𝐶 (𝑟 ,𝑠) 1 / (𝑟2+𝑠2 )𝑑

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

𝐶 (𝑟 ,𝑠) 1 / (𝑟2+𝑠2 )𝑑

𝐶 (𝑟 ,𝑠 )=𝜖 𝛼1 (𝑟 ,𝑠 )+𝜖2𝛼2 (𝑟 ,𝑠 )+…

Since (no dipole) and the net charge is zero the leading contribution is quadrupolar

( Δ𝑟 +Δ𝑠 )𝐶 (𝑟 ,𝑠 )=𝜎 1 (𝑟 ,𝑠 )+𝜎2 (𝑟 ,𝑠)+𝜎3(𝑟 ,𝑠)

+

𝑄=𝑛 (0 ) (1−𝑛 (−𝑒1 ))+𝑛(−𝑒1)(1−𝑛 (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

𝐶 (𝑟 ,𝑠) 1 / (𝑟2+𝑠2 )𝑑

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

𝐻=− 𝐽 ∑¿ 𝑖 𝑗>¿ 𝑆𝑖𝑆 𝑗¿

¿

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.

𝑃 (𝑚 ( 𝑥 , 𝑦 )) 𝑒−𝐿2𝐹 (𝑚 ( 𝑥 , 𝑦 ) )

𝑃 (𝑚0 )=𝑒−𝐿𝑈 (𝑚0)

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

𝑈

The large deviations function

𝑃 (𝑚0 )=𝑒−𝐿𝑈 (𝑚0)

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

𝑚𝑜 𝐿2

−1

ln𝑝 ( 2𝑘𝐿 )+ ∑𝑘=1

𝑚𝑜 𝐿2

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

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.