STATIONARY NON EQULIBRIUM STATES FROM A MICROSCOPIC AND A - PowerPoint PPT Presentation

STATIONARY NON EQULIBRIUM STATES FROM A MICROSCOPIC AND A MACROSCOPIC POINT OF VIEW Davide Gabrielli University of L’Aquila 1 July 2014 GGI Firenze Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

References • L. Bertini; A. De Sole; D. G. ; G. Jona-Lasinio; C. Landim Stochastic interacting particle systems out of equilibrium J. Stat. Mech. (2007) • D. G. From combinatorics to large deviations for the invariant measures of some multiclass particle systems Markov Processes Relat. (2008) • L. Bertini; D. G.; G. Jona-Lasinio; C. Landim Thermodynamic transformations of nonequilibrium states J. Stat. Phys. (2012) Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

SNS: Microscopic description Lattice: Λ N Configuration of particles: η ∈ { 0 , 1 } Λ N or η ∈ N Λ N η t ( x ) = number of particles at x ∈ Λ N at time t Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

SNS: Microscopic description • Stochastic Markovian dynamics • r ( η, η ′ ) = rate of jump from configuration η to configuration η ′ • η ′ = local perturbation of η • µ N ( η ) = invariant measure of the process, probability measure on the state space η ′ r ( η, η ′ ) = � η ′ µ N ( η ′ ) r ( η ′ , η ) µ N ( η ) � µ N = ⇒ MICROSCOPIC description of the SNS Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

SNS: Macroscopic description • η = ⇒ π N ( η ) Empirical measure (positive measure on [0 , 1]) π N ( η ) = 1 � x ∈ Λ N η ( x ) δ x N δ x = delta measure (Dirac) at x ∈ [0 , 1]; since x ∈ Λ N we have i x = N , i ∈ N . Given f : [0 , 1] → R � fdπ N = 1 � η ( x ) f ( x ) N [0 , 1] x ∈ Λ N Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

SNS: Macroscopic description When η is distributed according to µ N and N is large LAW OF LARGE NUMBERS π N → ¯ ρ ( x ) dx This means � � fdπ N → f ( x )¯ ρ ( x ) dx [0 , 1] [0 , 1] ρ ( x ) = typical density profile of the SNS ¯ Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

SNS: Macroscopic description When η is distributed according to µ N and N is large, a refinement of the law of large numbers LARGE DEVIATIONS � � ≃ e − NV ( ρ ) P π N ( η ) ∼ ρ ( x ) dx V = Large deviations rate function V = ⇒ MACROSCOPIC DESCRIPTION OF THE SNS V contains less information than µ N but is easier to compute and is independent from microscopic details of the dynamics Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Example: Equilibrium SEP • Equilibrium: C L = C R = C ; A L = A R = A • Microscopic state: product of Bernoulli measures of C parameter p = A + C x ∈ Λ N p η ( x ) (1 − p ) 1 − η ( x ) µ N ( η ) = � Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Example: Equilibrium SEP MACROSCOPIC DESCRIPTION � � � π N ( η ) ∼ ρ ( x ) dx = µ N ( η ) P { η, : π N ( η ) ∼ ρ ( x ) dx } � � � [0 , 1] dπ N ( η ) log 1 − p − N p − log(1 − p ) � = e { η, : π N ( η ) ∼ ρ ( x ) dx } Using the combinatorial estimate � 1 � � � ≃ e − N 0 ρ ( x ) log ρ ( x )+(1 − ρ ( x )) log(1 − ρ ( x )) dx � { η, : π N ( η ) ∼ ρ ( x ) dx } � � � 1 0 ρ ( x ) log ρ ( x ) + (1 − ρ ( x )) log (1 − ρ ( x )) V ( ρ ) = dx p (1 − p ) Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Contraction Average number of particles 1 � � η ( i ) = dπ N ( η ) N [0 . 1] i satisfies LDP � � 1 � ≃ e − NJ ( y ) η ( i ) ∼ y P N i BY CONTRACTION J ( y ) = inf { ρ : 0 ρ ( x ) d x = y } V ( ρ ) � 1 Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Relative entropy Relative entropy of the probability measure µ 2 N with respect to µ 1 N N ( η ) log µ 2 � � � N ( η ) µ 2 � µ 1 η µ 2 = � H � N N µ 1 N ( η ) H ≥ 0, not symmetric!! Density of relative entropy � � � h = lim N → + ∞ 1 µ 2 � µ 1 N H � N N Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

An example x ∈ Λ N p η ( x ) (1 − p ) 1 − η ( x ) , product of Bernoulli µ 1 N ( η ) = � measures of parameter p x ∈ Λ N ρ ( x ) η ( x ) (1 − ρ ( x )) 1 − η ( x ) , slowly varying µ 2 N ( η ) = � product of Bernoulli measures associated to the density profile ρ ( x ) 1 � � � µ 2 � µ 1 N H � N N    1 η ( x ) log ρ ( x ) + (1 − η ( x )) log (1 − ρ ( x )) � µ 2 � = N ( η )  N p (1 − p ) η x ∈ Λ N = 1 ρ ( x ) log ρ ( x ) + (1 − ρ ( x )) log (1 − ρ ( x )) � N p (1 − p ) x ∈ Λ N Riemann sums, convergence when N → + ∞ to V ( ρ ) Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

From microscopic to MACROSCOPIC • Driving parameters ( λ, E ) • λ = ⇒ rates of injection and annihilation at the boundary • E = ⇒ external field driving the particles on the bulk • µ λ,E = ⇒ corresponding invariant measure N • ¯ ρ λ,E = ⇒ corresponding typical density profile • V λ,E ( ρ ) = ⇒ corresponding LD rate function � � � ρ λ 2 ,E 2 ) = lim N → + ∞ 1 µ λ 2 ,E 2 � µ λ 1 ,E 1 V λ 1 ,E 1 (¯ N H � N N Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

From microscopic to MACROSCOPIC • This relation between relative entropy and LD rate function can be easily verified for the boundary driven Zero Range Process • It is true also for boundary driven SEP; proof based on matrix representation of µ N • In general the computation of V through relative entropy is difficult • An alternative powerful approach to compute V is the dynamic variational one of the Macroscopic Fluctuation Theory Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Boundary driven TASEP: a microscopic view Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Boundary driven TASEP: a microscopic view • Duchi E., Schaeffer G A combinatorial approach to jumping particles, J. Comb. Theory A (2005) Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Boundary driven TASEP: a microscopic view • η = ⇒ configuration of particles above • ξ = ⇒ configuration of particles below • ( η, ξ ) = ⇒ full configuration of particles • Stochastic Markov dynamics for ( η, ξ ) • Observing just η = ⇒ still Markov and boundary driven TASEP • ν N ( η, ξ ) = ⇒ invariant measure for the joint dynamics, it has a combinatorial representation µ N ( η ) = � ξ ν N ( η, ξ ) Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Boundary driven TASEP: a microscopic view Complete configurations � E ( x ) = ( η ( y ) + ξ ( y )) − Nx − 1 y ≤ x ( η, ξ ) is a complete configuration if � E ( x ) ≥ 0 E (1) = 0 ν N is concentrated on complete configurations ( η, ξ ) complete = ⇒ N 1 ( η, ξ ) , N 2 ( η, ξ ) Z N A N 1 ( η,ξ ) C N 2 ( η,ξ ) 1 ν N ( η, ξ ) = Special case A = C = 1 = ⇒ ν N uniform measure on complete configurations Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Boundary driven TASEP: a macroscopic view Joint Large deviations � � ≃ e − N G ( ρ,f ) ( π N ( η ) , π N ( ξ )) ∼ ( ρ ( x ) , f ( x )) P Contraction principle � � ≃ e − NV ( ρ ) P π N ( η ) ∼ ρ ( x ) V ( ρ ) = inf f G ( ρ, f ) Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Boundary driven TASEP: a macroscopic view Complete density profiles � x E ( x ) = ( ρ ( y ) + f ( y )) dy − x 0 The pair ( ρ, f ) is a complete density profile if � E ( x ) ≥ 0 E (1) = 0 When C = A = 1 since ν N is uniform on complete configurations a classic simple computation gives � 1 � � G ( ρ, f ) = h 1 2 ( ρ ( x )) + h 1 2 ( f ( x )) dx 0 if ( ρ, f ) is complete; here p + (1 − α ) log (1 − α ) h p ( α ) = α log α 1 − p Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Boundary driven TASEP: a macroscopic view � 1 � � V ( ρ ) = inf h 1 2 ( ρ ( x )) + h 1 2 ( f ( x )) dx f : ( ρ,f ) ∈C 0 To be compared with B. Derrida, J.L. Lebowitz, E.R. Speer Exact large deviation functional of a stationary open driven diffusive system: the asymmetric exclusion process J. Stat. Phys. (2003) � 1 � V ( ρ ) = sup ρ ( x ) log [ ρ ( x )(1 − f ( x ))] f 0 � + (1 − ρ ( x )) log [(1 − ρ ( x )) f ( x )] dx + log 4 where f (0) = 1, f (1) = 0 and f is monotone Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Boundary driven TASEP: a macroscopic view Both variational problems have the same minimizer � � x � f ρ ( x ) = CE (1 − ρ ( y )) dy 0 V ( ρ ) = G ( ρ, f ρ ) See Bahadoran C. A quasi-potential for conservation laws with boundary conditions arXiv:1010.3624 for a dynamic variational approach, using MFT Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

2-class TASEP Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

The invariant measure Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR

Collapsing particles � � � � (˜ η 1 , ˜ η T ) : ˜ η 1 ( x ) ≤ ˜ η T ( x ) = ⇒ ( η 1 , η T ) = C (˜ η 1 , ˜ η T )) x x Flux across bond ( x, x + 1) � � � J ( x ) = sup η 1 ( z ) − ˜ ˜ η T ( z ) + y z ∈ [ y,x ] Davide Gabrielli STATIONARY NON EQULIBRIUM STATES FR