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

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Phase diagram and density large deviation of a nonconserving A B C model Or Cohen and David Mukamel. International Workshop on Applied Probability, Jerusalem, 2012. Driven diffusive systems. Bulk driven . Boundary driven . T 2. T 1. Studied via simplified. Motivation.
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Phase diagram and density large deviation of a nonconservingABC modelOr Cohen and David MukamelInternational Workshop on Applied Probability, Jerusalem, 2012Driven diffusive systemsBulk driven Boundary driven T2T1Studied via simplifiedMotivationWhat is the effect of bulk nonconserving dynamics on bulk driven system ?qpw-w+Can it be inferred from the conserving steady state properties ?OutlineABC modelPhase diagram under conserving dynamicsSlow nonconserving dynamicPhase diagram and inequivalence of ensemblesConclusionsABC modelBCARing of size LDynamics : qAB BA1qBC CB1qCA AC1 Evans, Kafri , Koduvely & Mukamel - Phys. Rev. Lett. 1998 ABC modelBCARing of size LDynamics : qAB BA1qBC CB1qCA AC1ABBCACCBACABACBq=1q<1AAAAABBBBBCCCCC Evans, Kafri , Koduvely & Mukamel - Phys. Rev. Lett. 1998 ABC modelBCAxt Evans, Kafri , Koduvely & Mukamel - Phys. Rev. Lett. 1998 Equal densitiesFor equal densities NA=NB=NCAAAAABBAABBBCBBCCCCCCPotential induced by other speciesBBBBBBBWeak asymmetryCoarse grainingClincy, Derrida & Evans - Phys. Rev. E 2003Weak asymmetryCoarse grainingWeakly asymmetric thermodynamic limitClincy, Derrida & Evans - Phys. Rev. E 2003Phase transitionFor low βis minimum of F[ρα]Clincy, Derrida & Evans - Phys. Rev. E 2003Phase transitionFor low βis minimum of F[ρα]2nd order phase transition at Clincy, Derrida & Evans - Phys. Rev. E 2003Nonequal 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 DiffusionNonequal densities ?AAAAABBAABBBCBBCCC
  • No detailed balance
  • (Kolmogorov criterion violated)
  • Steady state current
  • Stationary measure unknown
  • Hydrodynamics equations :Drift DiffusionFull steady-state solution orExpansion around homogenousNonconserving ABC model12q1ABBA0X X0 X=A,B,C11qBCCB1qCAAC1Conserving model(canonical ensemble)+12ABC0Lederhendler & Mukamel - Phys. Rev. Lett. 2010Nonconserving ABC model12q1ABBA0X X0 X=A,B,C11qBCCB3pe-3βμ1ABC 000qCAACp1Conserving model(canonical ensemble)+12ABNonconserving model(grand canonical ensemble)++123C0Lederhendler & Mukamel - Phys. Rev. Lett. 2010Nonequal densitiesHydrodynamics equations :Drift Diffusion Deposition Evaporation Nonequal densitiesHydrodynamics equations :Drift Diffusion Deposition Evaporation e-β/LABBA11pe-3βμe-β/LABC 000BCCB0X X011pe-β/LX= A,B,CCA AC1Conserving steady-stateDrift DiffusionConserving model Steady-state profile Nonequal densities : Cohen & Mukamel - Phys. Rev. Lett.2012 Equal densities : Ayyer et al. - J. Stat. Phys. 2009Nonconserving steady-stateDrift Diffusion Deposition Evaporation Nonconserving steady-state Drift + Diffusion Deposition + Evaporation Nonconserving modelwith slow nonconserving dynamics Dynamics of particle densityDynamics of particle densityAfter time τ1 :Dynamics of particle densityAfter time τ2 :Dynamics of particle densityAfter time τ1 :Dynamics of particle densityAfter time τ2 :Dynamics of particle densityAfter time τ1 :Large deviation function of rAfter time τ1 :Large deviation function of r = 1D - Random walk in a potentialLarge deviation function of r = 1D - Random walk in a potentialLarge deviation functionpe-3βμABC 000pLarge deviation function of rHigh µ Large deviation function of rHigh µ Low µ First order phase transition (only in the nonconserving model)Inequivalence of ensemblesFor NA=NB≠NC :Conserving = CanonicalNonconserving = Grand canonicaldisordereddisorderedorderedordered1st order transition2nd order transitiontricritical pointConclusionsABC modelSlow nonconserving dynamicsInequivalence of ensemble, and links to long range interacting systems.Relevance to other driven diffusive systems.Thank you ! Any questions ?
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