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 on a square lattice. ◮ Jump rates = 1, across every bond, in both directions. ◮ Detailed balance is satisfied, and the equilibrium density is φ = N / L 2 = ρ , everywhere. Tridib Sadhu Long-range profile

Local perturbation Locally perturb the system, but still maintaining detailed balance. ◮ For example, introduce a local potential r = � � − h at � 0 V ( � r ) = 0 elsewhere ◮ Equilibrium density.  1 r = � at � 0  ρ − 1 (1 − ρ ) e − h +1  φ ( � r ) = ρ − O (1 / L 2 )  elsewhere  Effect is local Tridib Sadhu Long-range profile

What if the perturbation locally breaks detailed balance? ◮ For example, change the jump rate across a single bond. ◮ Effect is non-local . � 1 / r 2 y direction | φ ( � r ) − ρ | ∼ 1 / r elsewhere Tridib Sadhu Long-range profile

Main results 1. For arbitrary local configuration of driving bonds, in dimensions d ≥ 2 Algebraically decaying local violation of ⇒ = density profiles. detailed balance 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. Tridib Sadhu Long-range profile

Outline ◮ Locally driven non-interacting particles ◮ Analogy to electrostatic potential due to charges ◮ Exact solution ◮ Local drive with exclusion interaction Tridib Sadhu Long-range profile

Non-interacting particles Let φ ( � r , t ) = the density of particles at site � r , at time t . and � 0 ≡ (0 , 0), � e 1 ≡ (1 , 0) The master equation for the density profile φ ( � r , t ): � � r , t ) + ǫφ ( � r , t ) = ∇ 2 φ ( � ∂ t φ ( � 0) δ � 0 − δ � , r ,� r ,� e 1 where discrete Laplacian ∇ 2 φ ( m , n ) = φ ( m +1 , n )+ φ ( m − 1 , n )+ φ ( m , n +1)+ φ ( m , n − 1) − 4 φ ( m , n ) Tridib Sadhu Long-range profile

◮ Steady state equation � � ∇ 2 φ ( � r ) = − ǫφ ( � 0) δ � 0 − δ � r ,� r ,� e 1 ◮ 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 � � ∇ 2 φ ( � r ) = − ǫφ ( � 0) δ � 0 − δ � r ,� r ,� e 1 � � r ) = ρ + ǫφ ( � r ,� ◮ Solution: φ ( � 0) G ( � 0) − G ( � r ,� e 1 ) , where ρ = global average density, and ∇ 2 G ( � r ,� r o ) = − δ � r ,� r o r ,� 0) − G ( � 0 ,� Table: Exact values of G ( � 0) q \ p 0 1 2 . . . − 1 2 0 0 π − 1 . . . 4 − 1 − 1 4 − 2 1 1 . . . 4 π π 2 4 − 2 1 − 4 2 π − 1 . . . π 3 π . . . . . . . . . . . . Tridib Sadhu Long-range profile

� � r ) = ρ + ǫφ ( � r ,� ◮ Solution: φ ( � G ( � 0) − G ( � r ,� 0) e 1 ) , ◮ Self-consistency equation: � � φ ( � 0) = ρ + ǫφ ( � G ( � 0 ,� 0) − G ( � 0) 0 ,� e 1 ) , ⇒ φ ( � 0) = ρ/ (1 − ǫ/ 4) Tridib Sadhu Long-range profile

� � r ) = ρ + ǫφ ( � r ,� ◮ Solution: φ ( � G ( � 0) − G ( � r ,� 0) e 1 ) , ◮ Self-consistency equation: � � φ ( � 0) = ρ + ǫφ ( � G ( � 0 ,� 0) − G ( � 0) 0 ,� e 1 ) , ⇒ φ ( � 0) = ρ/ (1 − ǫ/ 4) ◮ At large � r , r ) − ρ = − ǫφ ( � 0) � e 1 · � r + O ( 1 φ ( � r 2 ) 2 π r 2 and current r ) = ǫφ ( � � � 0) 1 e 1 − 2( � e 1 · � r ) � r + O ( 1 � j ( � r ) = −∇ φ ( � � r 3 ) . 2 π r 2 r 2 Tridib Sadhu Long-range profile

In d -dimensions The analogy to electrostatics holds in higher dimensions. ◮ Then, in d ≥ 2 r ) ∼ 1 / r d − 1 φ ( � ◮ In d = 1, Green’s function G ( x , x o ) = −| x − x o | / 2, then φ ( x ) = ρ − ( ǫ/ 2) φ (0) sgn ( x ) , Tridib Sadhu Long-range profile

Arbitrary driving configuration � � � r ,� r ,� φ ( � r ) = ρ + ǫφ ( i 1 ) G ( � i 1 ) − G ( � i 1 + � e 1 ) � � � r ,� r ,� + ǫφ ( i 2 ) G ( � i 2 ) − G ( � i 2 + � e 1 ) + · · · k self-consistency equations obtained by r = � i 1 ,� putting � i 2 · · · . 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 � � ∇ 2 φ ( � r ) = − ǫφ ( � 0) 2 δ � 0 − δ � e 1 − δ � . r ,� r ,� r , − � e 1 Solution � � 2 � r ) − ρ = − ǫφ ( � 0) 1 � � e 1 · � r + O ( 1 φ ( � r 2 − 2 r 4 ) , r 2 2 π with φ ( � 0) = ρ/ (1 − ǫ/ 2) . Tridib Sadhu Long-range profile

A side note ◮ Collection of biased bonds does not necessarily imply breakdown of detailed balance. ◮ Detailed balance with respect to a Gibbs distribution φ ( � r ) ∝ exp[ − V ( � r )] , where V ( � r ) = − ln(1 − ǫ ) δ � r ,� 0 Tridib Sadhu Long-range profile

Analogy to magnetic fields ◮ In 2- d , magnetic field by ( i → j ) link H = ln[ e ij ] ◮ Then for a bond = ln[ e ij ] − ln[ e ji ] H ln[ e ij = ] e ji Tridib Sadhu Long-range profile

◮ Kolmogorov criterion: Detailed balance if and only if α 1 α 2 α 3 α 4 = β 4 β 3 β 2 β 1 on all loops ◮ In terms of magnetic field: H = ln[ α 1 α 2 α 3 α 4 � zero ⇐ ⇒ Detailed balance ] = non-zero ⇐ ⇒ No detailed balance β 4 β 3 β 2 β 1 Tridib Sadhu Long-range profile

Exclusion interaction ◮ The steady state equation for density � � ∇ 2 φ ( � r ) = − ǫ � τ ( � 0)(1 − τ ( � e 1 ) � δ � 0 − δ � , r ,� r ,� e 1 where � 1 If there is a particle � τ ( � r ) = and φ ( � r ) = � τ ( � r ) � � 0 No particle � ◮ Tridib Sadhu Long-range profile

Exclusion interaction ◮ The steady state equation for density � � ∇ 2 φ ( � r ) = − ǫ � τ ( � 0)(1 − τ ( � e 1 ) � δ � 0 − δ � , r ,� r ,� e 1 where � 1 If there is a particle � τ ( � r ) = and φ ( � r ) = � τ ( � r ) � � 0 No particle � ◮ Unlike the non-interacting case, the pre-factor has to be determined separately. However, the exponent of the power-law decay remains same . Tridib Sadhu Long-range profile

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 Tridib Sadhu Long-range profile

Numerical results On a 200 × 200 lattice with ρ = N / L 2 = 0 . 6 ρ Non-Interacting: φ ( � 0) = 1 − ǫ/ 4 Exclusion interaction: � τ ( � 0)(1 − τ ( � e 1 ) � = 0 . 3209 measured separately Tridib Sadhu Long-range profile

Global bias Steady state equation for Non-interacting case � � ∇ 2 φ ( � r ) = − ǫφ ( � 0) δ � 0 − δ � + µ [ φ ( � r ) − φ ( � r − � e 1 )] r ,� r ,� e 1 Tridib Sadhu Long-range profile

Take home message System in Equilibrium local perturbation with detailed balance Local changes Tridib Sadhu Long-range profile

Take home message System in Equilibrium localy break local perturbation detailed balance with detailed balance Non-equilibrium Local changes No changes Non-local Changes Tridib Sadhu Long-range profile

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? Tridib Sadhu Long-range profile

Thank you Tridib Sadhu Long-range profile