Effect of local violation of detailed balance Tridib Sadhu - - PowerPoint PPT Presentation

Effect of local violation of detailed balance

Tridib Sadhu

Department of physics of complex systems, Weizmann Institute of Science, Rehovot 76100, Israel.

Joint work with

Satya N. Majumdar and David Mukamel Ref:arXiv:1106.1838

Tridib Sadhu Long-range profile

The model

◮ N particles with symmetric

simple exclusion interactions

• n a square lattice.

◮ Jump rates = 1, across every

bond, in both directions.

◮ Detailed balance is satisfied,

and the equilibrium density is φ = N/L2 = ρ, everywhere.

Tridib Sadhu Long-range profile

Local perturbation

Locally perturb the system, but still maintaining detailed balance.

◮ For example, introduce a local

potential V ( r) =

• −h

at r = elsewhere

◮ Equilibrium density.

φ ( r) =     

1 ρ−1(1−ρ)e−h+1

at r = ρ − O(1/L2) elsewhere Effect is local

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What if the perturbation locally breaks detailed balance?

◮ For example, change the

jump rate across a single bond.

◮ Effect is non-local.

|φ ( r)−ρ| ∼ 1/r2 y direction 1/r elsewhere

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Main results

• 1. For arbitrary local configuration of driving bonds, in

dimensions d ≥ 2

local violation of detailed balance = ⇒ Algebraically decaying density profiles.

• 2. Decay exponent depends on local arrangement of driving

bonds.

• 3. A correspondence with electrostatic is established where φ is

the potential due to electric dipoles at the driving bonds.

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Outline

◮ Locally driven non-interacting particles

◮ Analogy to electrostatic potential due to charges ◮ Exact solution

◮ Local drive with exclusion interaction

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Non-interacting particles

Let φ ( r, t) = the density of particles at site r, at time t. and 0 ≡ (0, 0), e1 ≡ (1, 0) The master equation for the density profile φ( r, t): ∂tφ( r, t) = ∇2φ( r, t) + ǫφ( 0)

• δ

r, 0 − δ r, e1

• ,

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

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◮ Steady state equation

∇2φ( r) = −ǫφ( 0)

• δ

r, 0 − δ r, e1

• ◮ Equation for potential due to a dipole.

◮ Strength of the dipole is not known a priori, but can be

determined self-consistently.

Tridib Sadhu Long-range profile

◮ Steady state equation

∇2φ( r) = −ǫφ( 0)

• δ

r, 0 − δ r, e1

• ◮ Solution:

φ( r) = ρ + ǫφ( 0)

• G(

r, 0) − G( r, e1)

• ,

where ρ = global average density, and ∇2G( r, ro) = −δ

r, ro

Table: Exact values of G( r, 0) − G( 0, 0) q \ p 1 2 . . . − 1

4 2 π − 1

. . . 1 − 1

4

− 1

π 1 4 − 2 π

. . . 2

2 π − 1 1 4 − 2 π

− 4

3π

. . . . . . . . . . . . . . .

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◮ Solution:

φ( r) = ρ + ǫφ( 0)

• G(

r, 0) − G( r, e1)

• ,

◮ Self-consistency equation:

φ( 0) = ρ + ǫφ( 0)

• G(

0, 0) − G( 0, e1)

• ,

⇒ φ( 0) = ρ/(1 − ǫ/4)

Tridib Sadhu Long-range profile

◮ Solution:

φ( r) = ρ + ǫφ( 0)

• G(

r, 0) − G( r, e1)

• ,

◮ Self-consistency equation:

φ( 0) = ρ + ǫφ( 0)

• G(

0, 0) − G( 0, e1)

• ,

⇒ φ( 0) = ρ/(1 − ǫ/4)

◮ At large

r, φ( r) − ρ = −ǫφ( 0) 2π

• e1 ·

r r 2 + O( 1 r 2 ) and current

• j(

r) = −∇φ( r) = ǫφ( 0) 2π 1 r 2

• e1 − 2(

e1 · r) r r 2

• + O( 1

r 3 ).

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In d-dimensions

The analogy to electrostatics holds in higher dimensions.

◮ Then, in d ≥ 2

φ( r) ∼ 1/rd−1

◮ In d = 1, Green’s function G(x, xo) = −|x − xo|/2, then

φ(x) = ρ − (ǫ/2) φ(0) sgn(x),

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Arbitrary driving configuration

φ( r) = ρ + ǫφ(

• i1)
• G(

r, i1) − G( r, i1 + e1)

• +

ǫφ(

• i2)
• G(

r, i2) − G( r, i2 + e1)

• +

· · · k self-consistency equations obtained by putting r = i1, i2 · · · . These are a set of linear equations, and can be solved using known solutions of G.

Tridib Sadhu Long-range profile

Quadrupolar charge configuration

The steady state equation ∇2φ( r) = −ǫφ( 0)

• 2δ

r, 0 − δ r, e1 − δ r,− e1

• .

Solution φ( r) − ρ = −ǫφ( 0) 2π

• 1

r2 − 2

• e1 ·

r r2 2 + O( 1 r4 ), with φ( 0) = ρ/(1 − ǫ/2) .

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A side note

◮ Collection of biased bonds does

not necessarily imply breakdown

• f detailed balance.

◮ Detailed balance with respect

to a Gibbs distribution φ( r) ∝ exp[−V ( r)], where V ( r) = − ln(1 − ǫ) δ

r,

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Analogy to magnetic fields

◮ In 2-d, magnetic field by

(i → j) link H = ln[eij]

◮ Then for a bond

H = ln[eij] − ln[eji] = ln[eij eji ]

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◮ Kolmogorov criterion: Detailed balance if

and only if α1α2α3α4 = β4β3β2β1

• n all loops

◮ In terms of magnetic field:

H = ln[α1α2α3α4 β4β3β2β1 ] =

• zero ⇐

⇒ Detailed balance non-zero ⇐ ⇒ No detailed balance

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Exclusion interaction

◮ The steady state equation for density

∇2φ( r) = −ǫτ( 0)(1 − τ( e1)

• δ

r, 0 − δ r, e1

• ,

where τ ( r) = 1 If there is a particle No particle

• and φ (

r) = τ ( r)

◮

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Exclusion interaction

◮ The steady state equation for density

∇2φ( r) = −ǫτ( 0)(1 − τ( e1)

• δ

r, 0 − δ r, e1

• ,

where τ ( r) = 1 If there is a particle No particle

• and φ (

r) = τ ( r)

◮ Unlike the non-interacting case, the pre-factor has to be

determined separately. However, the exponent of the power-law decay remains

same.

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Exclusion interaction

◮ The d = 1 result is very similar to

the profile obtained in SSEP with a battery by [Bodineau, Derrida and

Lebowitz].

Tridib Sadhu Long-range profile

Exclusion interaction

◮ In d = 2

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Numerical results

On a 200 × 200 lattice with ρ = N/L2 = 0.6 Non-Interacting: φ( 0) =

ρ 1−ǫ/4

Exclusion interaction: τ( 0)(1 − τ( e1) = 0.3209 measured separately

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Global bias

Steady state equation for Non-interacting case ∇2φ( r) = −ǫφ( 0)

• δ

r, 0 − δ r, e1

• + µ [φ(

r) − φ( r − e1)]

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Take home message

System in Equilibrium Local changes local perturbation with detailed balance

Tridib Sadhu Long-range profile

Take home message

System in Equilibrium Local changes local perturbation with detailed balance Non-equilibrium localy break detailed balance No changes Non-local Changes

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Summary

◮ An electrostatic correspondence, where density φ is the

potential due to an electric dipole at the driving bonds. For the non-interacting case the current is analogous to electric field.

◮ Analogous quantity of magnetic field to check detailed

balance. Open problem What would happen, if other kind of local interparticle interactions (Ising like) are switched on?

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Thank you

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