Density large-deviations of nonconserving driven models STATPHYS 25 Conference, SNU, Seoul, Korea, July 2013 Or Cohen and David Mukamel

T, µ Equilibrium Grand canonical ensemble out of equilibrium ?

T, µ Equilibrium Grand canonical ensemble out of equilibrium ? conserving steady state Helmholtz free energy

T, µ Equilibrium Grand canonical ensemble out of equilibrium ? conserving steady state Helmholtz free energy Driven system pq conserving steady state

T, µ Equilibrium Grand canonical ensemble out of equilibrium ? conserving steady state Helmholtz free energy Driven system pq e -βμ 1 conserving steady state

T, µ Equilibrium Grand canonical ensemble out of equilibrium ? conserving steady state Helmholtz free energy Driven system pq e -βμ 1 conserving steady state Dynamics-dependent chemical potential of conserving system

General particle-nonconserving driven model wRCwRC wLCwLC w - NC w + NC L sites

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

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

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

Slow nonconserving dynamics To leading order in L we obtain

Slow nonconserving dynamics = 1D - Random walk in a potential

Slow nonconserving dynamics = 1D - Random walk in a potential

Slow nonconserving dynamics w - NC w + NC

Outline 1.Limit of slow nonconserving 2.Example of the ABC model 3.Corrections to the rate function using MFT 4.Conclusions

ABC model A BC AB BA BC CB CA AC Dynamics : q 1 q 1 q 1 Ring of size L M R Evans, Y Kafri, H M Koduvely, D 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 M R Evans, Y Kafri, H M Koduvely, D Mukamel - Phys. Rev. Lett (1998 ) ABBCACCBACABACB AAAAABBBBBCCCCC

ABC model time site index A BC

Conserving ABC model 0X X0 X=A,B,C 1 1 AB C0 A Lederhendler, D Mukamel - Phys. Rev. Lett (2010) A Lederhendler, OC, D Mukamel - J. Stat. Mech (2010) A Lederhendler, D Mukamel - Phys. Rev. Lett (2010) A Lederhendler, OC, D Mukamel - J. Stat. Mech (2010) AB BA BC CB CA AC q 1 q 1 q Conserving model (canonical ensemble) + fixed

Conserving ABC model 1.M Clincy, B Derrida, M R Evans - Phys. Rev. E (2003) Weakly asymmetric thermodynamic limit 1 Density profile

Conserving ABC model 1.M Clincy, B Derrida, M R Evans - Phys. Rev. E (2003) 2.OC, D Mukamel - J. Phys. A (2011) 1.M Clincy, B Derrida, M R Evans - Phys. Rev. E (2003) 2.OC, D Mukamel - J. Phys. A (2011) Weakly asymmetric thermodynamic limit 1 Density profile known 2 2 nd order For low β’s

Conserving ABC model 1.M Clincy, B Derrida, M R Evans - Phys. Rev. E (2003) 2.OC, D Mukamel - J. Phys. A (2011) 3.T Bodineau, B Derrida - Comptes Rendus Physique (2007) 1.M Clincy, B Derrida, M R Evans - Phys. Rev. E (2003) 2.OC, D Mukamel - J. Phys. A (2011) 3.T Bodineau, B Derrida - Comptes Rendus Physique (2007) Weakly asymmetric thermodynamic limit 1 Density profile known 2 2 nd order For low β’s Stationary measure 3 :

Nonconserving ABC model 0X X0 X=A,B,C 1 1 AB C0 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) + ++ A Lederhendler, D Mukamel - Phys. Rev. Lett (2010) A Lederhendler, OC, D Mukamel - J. Stat. Mech (2010) A Lederhendler, D Mukamel - Phys. Rev. Lett (2010) A Lederhendler, OC, D Mukamel - J. Stat. Mech (2010)

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

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

Slow nonconserving ABC model ABC 000 pe -3βμ p

Slow nonconserving ABC model ABC 000 pe -3βμ p This is similar to equilibrium : f = Helmholtz free energy density

Rate function of r, G(r) High µ Low µ First order phase transition (only in the nonconserving model) OC, D Mukamel - Phys. Rev. Let. 108, (2012)

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 : OC, D Mukamel - Phys. Rev. Let. 108, (2012) Different nonconserving ABC model: J Barton, J L Lebowitz, E R Speer - J. Phys. A (2011) Discussion about ensemble inequivalence: OC, D Mukamel - J. Stat. Mech (2012) OC, D Mukamel - Phys. Rev. Let. 108, (2012) Different nonconserving ABC model: J Barton, J L Lebowitz, E R Speer - J. Phys. A (2011) Discussion about ensemble inequivalence: OC, D Mukamel - J. Stat. Mech (2012) Stability line

Corrections to G(r) using MFT ABC 000 pe -3βμ p - conserving action 1 - nonconserving action 2,3 - conserving current - nonconserving current 1.T Bodineau, B Derrida - Comptes Rendus Physique (2007) 2.G Jona-Lasinio, C Landim and M E Vares - Probability theory and related fields (1993) 3.T Bodineau, M Lagouge - J. Stat. Phys (2010) 1.T Bodineau, B Derrida - Comptes Rendus Physique (2007) 2.G Jona-Lasinio, C Landim and M E Vares - Probability theory and related fields (1993) 3.T Bodineau, M Lagouge - J. Stat. Phys (2010)

Corrections to G(r) using MFT ABC 000 pe -3βμ p τ r rμrμ 0 T Instanton path:

Conclusions 1.Nonequlibrium ‘grand canonical ensemble’ - Slow nonconserving dynamics 2.Example to ABC model 3.1 st order phase transition for nonmonotoneous µ s (r) and inequivalence of ensembles. ( µ s (r) is dynamics dependent ! ) 4.Corrections to rate function of r using MFT

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

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