Phase diagram and density large deviation of a nonconserving A B C model

Phase diagram and density large deviation of a nonconserving ABC model Or Cohen and David Mukamel International Workshop on Applied Probability, Jerusalem, 2012

Driven diffusive systems Bulk driven Boundary driven T2 T1 Studied via simplified

Motivation What is the effect of bulk nonconserving dynamics on bulk driven system ? q p w- w+ Can it be inferred from the conserving steady state properties ?

Outline ABC model Phase diagram under conserving dynamics Slow nonconserving dynamic Phase diagram and inequivalence of ensembles Conclusions

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

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

ABC model B C A x t Evans, Kafri , Koduvely & Mukamel - Phys. Rev. Lett. 1998

Equal densities For equal densities NA=NB=NC AAAAABBAABBBCBBCCCCCC Potential induced by other species BB BB BBB

Weak asymmetry Coarse graining Clincy, Derrida & Evans - Phys. Rev. E 2003

Weak asymmetry Coarse graining Weakly asymmetric thermodynamic limit Clincy, Derrida & Evans - Phys. Rev. E 2003

Phase transition For low β is minimum of F[ρα] Clincy, Derrida & Evans - Phys. Rev. E 2003

Phase transition For low β is minimum of F[ρα] 2nd order phase transition at Clincy, Derrida & Evans - Phys. Rev. E 2003

Nonequal densities ? AAAAABBAABBBCBBCCC • No detailed balance • (Kolmogorov criterion violated) • Steady state current • Stationary measure unknown

Nonequal densities ? AAAAABBAABBBCBBCCC • No detailed balance • (Kolmogorov criterion violated) • Steady state current • Stationary measure unknown Hydrodynamics equations : Drift Diffusion

Nonequal densities ? AAAAABBAABBBCBBCCC • No detailed balance • (Kolmogorov criterion violated) • Steady state current • Stationary measure unknown Hydrodynamics equations : Drift Diffusion Full steady-state solution or Expansion around homogenous

Nonconserving ABC model 1 2 q 1 ABBA 0X X0 X=A,B,C 1 1 q BCCB 1 q CAAC 1 Conserving model (canonical ensemble) + 1 2 A B C 0 Lederhendler & Mukamel - Phys. Rev. Lett. 2010

Nonconserving ABC model 1 2 q 1 ABBA 0X X0 X=A,B,C 1 1 q BCCB 3 pe-3βμ 1 ABC 000 q CAAC p 1 Conserving model (canonical ensemble) + 1 2 A B Nonconserving model (grand canonical ensemble) + + 1 2 3 C 0 Lederhendler & Mukamel - Phys. Rev. Lett. 2010

Nonequal densities Hydrodynamics equations : Drift Diffusion Deposition Evaporation

Nonequal densities Hydrodynamics equations : Drift Diffusion Deposition Evaporation e-β/L ABBA 1 1 pe-3βμ e-β/L ABC 000 BCCB 0X X0 1 1 p e-β/L X= A,B,C CA AC 1

Conserving steady-state Drift Diffusion Conserving model Steady-state profile Nonequal densities : Cohen & Mukamel - Phys. Rev. Lett.2012 Equal densities : Ayyer et al. - J. Stat. Phys. 2009

Nonconserving steady-state Drift Diffusion Deposition Evaporation

Nonconserving steady-state Drift + Diffusion Deposition + Evaporation Nonconserving modelwith slow nonconserving dynamics

Dynamics of particle density

Dynamics of particle density After time τ1 :

Large deviation function of r After time τ1 :

Large deviation function of r = 1D - Random walk in a potential

Large deviation function of r = 1D - Random walk in a potential Large deviation function pe-3βμ ABC 000 p

Large deviation function of r High µ

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

Inequivalence of ensembles For NA=NB≠NC : Conserving = Canonical Nonconserving = Grand canonical disordered disordered ordered ordered 1st order transition 2nd order transition tricritical point

Conclusions ABC model Slow nonconserving dynamics Inequivalence of ensemble, and links to long range interacting systems. Relevance to other driven diffusive systems. Thank you ! Any questions ?

Phase diagram and density large deviation of a nonconserving A B C model