Driven ABC model under particle-nonconserving dynamics Or Cohen and David Mukamel International Seminar on Large Fluctuations in Non-Equilibrium Systems, Dresden, July 2011
Motivation Equilibrium systems with System driven out of Long-range interactions equilibrium GMm v ( r ) T 2 r T 1 • • Much is known Much remains unknown • • Exhibit long-range correlations Exhibit long-range correlations • • Exhibit unique phenomena : Exhibit similar phenomena ? inequivalence of ensembles, negative specific heat in MC ensemble, slow relaxation, quasi-stationary states
Outline 1. Long-range interactions Equilibrium Long-range 2. Inequivalence of ensembles 3. ABC model Driven Models 4. Inequivalence of ensembles 5. Conclusions
Long-range interactions 1 r v ( r ) d r Long range Short range σ >0 Energy 1 E / d V E V scaling YES NO Additive S S Micro- C V <0 C V ≥ 0 C V <0 canonical E E E E E E 1 1 2 2 C V ≥ 0 C V ≥ 0 Canonical
Inequivalence of ensembles Microcanonical Canonical T T disordered disordered ordered ordered C V <0 K K T K = interaction strength disordered 1 st order transition 2 nd order transition ordered inequivalence K
Outline 1. Long-range interactions Equilibrium Long-range 2. Inequivalence of ensembles 3. ABC model Driven Models 4. Inequivalence of ensembles 5. Conclusions
ABC model Ring of size L Dynamics : q AB BA 1 q BC CB 1 q CA AC 1 B C A q=1 ABBCACCBACABACB L q<1 AAAAABBBBBCCCCC Evans, Kafri , Koduvely & Mukamel - Phys. Rev. Lett. 1998
Currents & Detailed Balance 1. Equal densities N A =N B =N C Although q ≠ 1 detailed balance obeyed with respect with to L L 1 k 2 H ({ X }) A B B C C A ~ L i i i k i i k i i k L i 1 k 1 H ({ X }) P ({ X }) q i i 2. Nonequal densities, e.g. N B ≠ N C No effective Hamiltonian
Weak asymmetry 2 E ~ L q exp( ) L S ~ L E S E L / L f ( ) P E ( ) ( ) E q e e Clincy, Derrida & Evans - Phys. Rev. E 2003
Weak asymmetry 2 E ~ L q exp( ) L S ~ L E S E L / L f ( ) P E ( ) ( ) E q e e 2 nd order phase transiton c 2 3 10 . 9 at the critical temp. Clincy, Derrida & Evans - Phys. Rev. E 2003
Outline 1. Long-range interactions Equilibrium Long-range 2. Inequivalence of ensembles 3. ABC model Driven Models 4. Inequivalence of ensembles 5. Conclusions
ABC model + vacancies ‘Canonical’ ensemble : 1 0X X0 X=A,B,C 1 C A B 0 Lederhendler & Mukamel - Phys. Rev. Lett. 2010
Nonconserving dynamics ‘Grand Canonical’ ensemble : 1 0X X0 X=A,B,C 1 pe -3 βμ ABC 000 p A B C 0 Fluctuating parameter : Conjugate field : N N N A B C r L Lederhendler & Mukamel - Phys. Rev. Lett. 2010
Inequivalence of ensembles For N A =N B =N C : Conserving = Nonconserving = Canonical Grand canonical disordered disordered T= T= ordered ordered 1 st order transition 2 nd order transition tricritical point Lederhendler, Cohen & Mukamel - J. Stat. Mech: Theory Exp. 2010
Nonequal densities Hydrodynamics equations : i ( ) A A B A B A i i i 1 i i 1 L Drift Diffusion Deposition Evaporation d 1 d d 3 3 A A p e A B C 0 A B C 2 dt L dx dx
Nonequal densities Hydrodynamics equations : i ( ) A A B A B A i i i 1 i i 1 L Drift Diffusion Deposition Evaporation d 1 d d 3 3 A A p e A B C 0 A B C 2 dt L dx dx e - β /L AB BA 1 pe -3 βμ e - β /L 1 ABC 000 BC CB 0X X0 1 1 p e - β /L X= A,B,C CA AC 1
Conserving steady-state Drift Diffusion d 1 d d 3 3 A A p e A B C 0 A B C 2 dt L dx dx p 0 Conserving model Steady-state profile 1 sn c x , d * ( x , r ) r a b sn c x , d N N N A B C r L Nonequal densities : Cohen & Mukamel - Preprint Equal densities : Ayyer et al. - J. Stat. Phys. 2009
Nonconserving steady-state Drift Diffusion Deposition Evaporation d 1 d d 3 3 A A p e A B C 0 A B C 2 dt L dx dx
Nonconserving steady-state d 1 d d 3 3 A A p e A B C 0 A B C 2 dt L dx dx Drift + Diffusion Deposition + Evaporation Nonconserving model p ~ L , 2 with slow nonconserving dynamics Steady-state density Steady-state profile r ? * ( x , r )
Dynamics of particle density 2 1 ~ L ~ L 2 1 2 ( x ) ( x ) ( x ) A B C r r 1
Dynamics of particle density 2 1 ~ L ~ L 2 1 2 After time τ 1 : r r 1 * ( x , r ) 1
Dynamics of particle density 2 1 ~ L ~ L 2 1 2 After time τ 2 : r r 2
Dynamics of particle density 2 1 ~ L ~ L 2 1 2 After time τ 1 : r r 2 * ( x , r ) 2
Dynamics of particle density 2 1 ~ L ~ L 2 1 2 After time τ 2 : r r 3
Dynamics of particle density 2 1 ~ L ~ L 2 1 2 After time τ 1 : r r 3 * ( x , r ) 3
Large deviation function of r 2 1 ~ L ~ L 2 1 2 After time τ 1 : r r 3 * ( x , r ) 3 3 3 * ( x , r ) R ( r r ) , R ( r r ) 3 4 3 4 3 L L
Large deviation function of r R ( r ) V ( r ) R ( r ) r r r r min max = 1D - Random walk in a potential
Large deviation function of r R ( r ) V ( r ) R ( r ) r r r r min max = 1D - Random walk in a potential r R ( r ' ) P ( r ) exp[ LF ( r )] R ( r ' ) r ' r 0 1 3 * 3 e dx ( ) 0 r Large pe -3 βμ 0 F ( r ) dr ' log ABC 000 deviation 1 function p * * * r dx 0 A B C 0
Inequivalence of ensembles r r For N A =N B ≠ N C : r r , r 2 0 . 01 A B C 3 3 Conserving = Nonconserving = Canonical Grand canonical disordered disordered ordered ordered 1 st order transition 2 nd order transition tricritical point
Locating 1 st order transition Large deviation function ‘Chemical potential’ in conserving system r 1 1 * * * * 3 F ( r ) dr ' ( ( r ' )) ( r ' ) log dx ( ( ) ) A B C 0 3 r 0 0 Conserving Nonconserving 2 nd order trans. 1 st order trans. Maxwell’s μ μ Critical point construction Ordered phase F ( r ) F ( r ) 1 2 Homogenous phase r r r r 1 2
Fast evaporation & deposition p ~ L 2 Conserving Nonconserving 1 x Flat vacancies ( x ) r Oscillatory vacancies ( ) const profile 0 0 profile d No moving ( x ) 0 Moving ( x , ) ( x v ) solutions d solutions 2 3 2 3 c c 1 2 2 r 36 2 2 2 2 r ( 1 k ) 36 ( 1 k ) 1 1 r 2 NESS is sensitive to the dynamics
Results & Conclusions 1. Inequivalence of ensembles in the ABC model Open questions : Other similarities to system with LRI ? (dynamical features etc.) In other driven models ? 2. Dynamical definition of ensembles in driven models ? Conserving ABC model + slow nonconserving dynamics Obtain LDF of particle density Applies to other driven models
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