Density large-deviations of nonconserving driven models 5 th KIAS Conference on Statistical Physics, Seoul, Korea, July 2012 Or Cohen and David Mukamel

Ensemble theory out of equilibrium ? T, µ Equilibrium

Ensemble theory out of equilibrium ? T, µ Equilibrium Driven diffusive systems pq w-w- w+w+ conserving steady state

Ensemble theory out of equilibrium ? T, µ Equilibrium Driven diffusive systems pq w-w- w+w+ Can we infer about the nonconserving system from the steady state properties of the conserving system ? conserving steady state

Ensemble theory out of equilibrium ? T, µ Equilibrium Driven diffusive systems pq w-w- w+w+

Generic driven diffusive model conserving (sum over η’ with same N) nonconserving (sum over η’ with N’≠N) wRCwRC wLCwLC w - NC w + NC L sites

Generic driven diffusive model conserving (sum over η’ with same N) nonconserving (sum over η’ with N’≠N) wRCwRC wLCwLC w - NC w + NC Guess a steady state of the form : L sites

Generic driven diffusive model conserving (sum over η’ with same N) nonconserving (sum over η’ with N’≠N) It is consistent if : wRCwRC wLCwLC w - NC w + NC In many cases : Guess a steady state of the form : L sites

Slow nonconserving dynamics To leading order in ε we obtain

Slow nonconserving dynamics = 1D - Random walk in a potential To leading order in ε we obtain

Slow nonconserving dynamics = 1D - Random walk in a potential Steady state solution : To leading order in ε we obtain

Outline 1.Limit of slow nonconserving 2.Example of the ABC model 3.Nonequilibrium chemical potential (dynamics dependent !) 4.Conclusions

ABC model A BC AB BA BC CB CA AC Dynamics : q 1 q 1 q 1 Ring of size L Evans, Kafri, Koduvely & Mukamel - Phys. Rev. Lett. 1998

ABC model A BC AB BA BC CB CA AC Dynamics : q 1 q 1 q 1 Ring of size L q=1 q<1 Evans, Kafri, Koduvely & Mukamel - Phys. Rev. Lett ABBCACCBACABACB AAAAABBBBBCCCCC

ABC model t x A BC

Nonconserving ABC model 0X X0 X=A,B,C 1 1 AB C0 Lederhendler & Mukamel - Phys. Rev. Lett AB BA BC CB CA AC q 1 q 1 q Conserving model (canonical ensemble) + fixed

Conserving model Clincy, Derrida & Evans - Phys. Rev. E 2003 Weakly asymmetric thermodynamic limit

Conserving model Clincy, Derrida & Evans - Phys. Rev. E 2003 Weakly asymmetric thermodynamic limit known For low β’s Density profile 2 nd order

Conserving model Clincy, Derrida & Evans - Phys. Rev. E 2003 Weakly asymmetric thermodynamic limit Density profile known 2 nd order For low β’s

Nonconserving ABC model 0X X0 X=A,B,C 1 1 AB C0 Lederhendler & Mukamel - Phys. Rev. Lett AB BA BC CB CA AC q 1 q 1 q 1 ABC 000 pe -3βμ p Conserving model (canonical ensemble) Nonconserving model (grand canonical ensemble) + ++

Slow nonconserving model Slow nonconserving limit ABC 000 pe -3βμ p

Slow nonconserving model Slow nonconserving limit ABC 000 pe -3βμ p saddle point approx.

Slow nonconserving model ABC 000 pe -3βμ p

Slow nonconserving model ABC 000 pe -3βμ p This is similar to equilibrium :

Large deviation function of r High µ Low µ First order phase transition (only in the nonconserving model)

Inequivalence of ensembles Conserving = Canonical Nonconserving = Grand canonical 2 nd order transition ordered 1 st order transitiontricritical point disordered ordered disordered For N A =N B ≠N C :

Why is µ S (N) the chemical potential ? N1N1 N2N2 SLOW

Why is µ S (N) the chemical potential ? N1N1 N2N2 Gauge measures SLOW

Conclusions 1.Nonequlibrium ‘grand canonical ensemble’ - Slow nonconserving dynamics 2.Example to ABC model 3.1 st order phase transition for nonmonotoneous µs(r) and inequivalence of ensembles. 4.Nonequilibrium chemical potential ( dynamics dependent ! ) Cohen & Mukamel - PRL 108, (2012)