Thermodynamics of histories: some examples

Transcription

1 Thermodynamics of histories: some examples Vivien Lecomte 1,2, Cécile Appert-Rolland 1, Estelle Pitard 3, Frédéric van Wijland 1,2 1 Laboratoire de Physique Théorique, Université d Orsay 2 Laboratoire Matière et Systèmes Complexes, Université Paris 7 3 Laboratoire Colloïdes, Verres et Nanomatériaux, Université Montpellier 2 Nancy - 19th May 2006 V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

2 Introduction Thermodynamics of histories Histories vs configurations In equilibrium Boltzmann weight P(C) e βh(c) time average from configurational average V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

3 Introduction Thermodynamics of histories Histories vs configurations In equilibrium Boltzmann weight P(C) e βh(c) time average from configurational average V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

4 Introduction Thermodynamics of histories Histories vs configurations In equilibrium Boltzmann weight P(C) e βh(c) time average from configurational average V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

5 Introduction Thermodynamics of histories Histories vs configurations In equilibrium Boltzmann weight P(C) e βh(c) time average from configurational average V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

6 Introduction Histories vs configurations Thermodynamics of histories ÓÒ ÙÖ Ø ÓÒ Ì Ñ V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

7 Introduction Histories vs configurations Thermodynamics of histories ÓÒ ÙÖ Ø ÓÒ Ì Ñ V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

8 Introduction Histories vs configurations Thermodynamics of histories ÓÒ ÙÖ Ø ÓÒ Ì Ñ V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

9 Introduction Thermodynamics of histories Histories vs configurations ÓÒ ÙÖ Ø ÓÒ Ì Ñ Where dynamics matters out of equilibrium systems V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

10 Introduction Thermodynamics of histories Histories vs configurations ÓÒ ÙÖ Ø ÓÒ Ì Ñ Where dynamics matters out of equilibrium systems glassy dynamics evolution driven by rare events V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

11 Introduction Histories vs configurations Thermodynamics of histories ÓÒ ÙÖ Ø ÓÒ Historical background Ì Ñ Ruelle s thermodynamic formalism: deterministic dynamics V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

12 Introduction Thermodynamics of histories Histories vs configurations ÓÒ ÙÖ Ø ÓÒ Ì Ñ Historical background Ruelle s thermodynamic formalism: deterministic dynamics Pierre Gaspard: discrete time stochastic dynamics our contribution: continuous time stochastic dynamics V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

13 Contents Introduction Contents 1 Introduction histories versus configurations historical background 2 Thermodynamic formalism goals and concepts main results V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

14 Contents Introduction Contents 1 Introduction histories versus configurations historical background 2 Thermodynamic formalism goals and concepts main results 3 Examples example 1: the Ising ferromagnet example 2: the contact process example 3: a kinetically constrained model V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

15 Thermodynamic formalism Dynamical partition function Dynamical partition function Motivation Thermodynamics of configurations Z(β, N) = C e β H(C) = e N f(β) (large N) V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

16 Thermodynamic formalism Dynamical partition function Dynamical partition function Motivation Thermodynamics of configurations Z(β, N) = C e β H(C) = e N f(β) (large N) Thermodynamics of histories Z dyn (s, t) = Prob{history} 1 s = e t f dyn (s) (large t) histories from 0 to t V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

17 Thermodynamic formalism Dynamical partition function Dynamical partition function Motivation Thermodynamics of configurations Z(β, N) = C e β H(C) = e N f(β) (large N) Thermodynamics of histories Z dyn (s, t) = Prob{history} 1 s = e t f dyn (s) (large t) histories from 0 to t Conjugate variables µ conjugated to number of particles s conjugated to Kologorov-Sinai entropy V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

18 Thermodynamic formalism Dynamical partition function Dynamical partition function Motivation Thermodynamics of configurations Z(β, N) = C e β H(C) = e N f(β) (large N) Thermodynamics of histories Z dyn (s, t) = Prob{history} 1 s = e t f dyn (s) (large t) histories from 0 to t Breaking of analyticity f(β) not analytic in β c (configurational) phase transition f dyn (s) not analytic in s c phases in space of histories V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

19 Thermodynamic formalism Results System with Markovian dynamics: results t P(C, t) = [ W(C C)P(C, t) W(C C ] )P(C, t) }{{}}{{} C gain term loss term Advantages of the Markov approach f dyn (s) is the largest eigenvalue of W +, of eigenvector Π + exact results, numerical approach Dynamical state observable O + (s) mean value in the state Π + typical value of O in highly chaotic histories (s ) O + (s) = typical value of O in steady state (s 0) typical value of O in lowly chaotic histories (s ) V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

20 Thermodynamic formalism Results System with Markovian dynamics: results t P(C, t) = [ W(C C)P(C, t) W(C C ] )P(C, t) }{{}}{{} C gain term loss term Advantages of the Markov approach f dyn (s) is the largest eigenvalue of W +, of eigenvector Π + exact results, numerical approach Dynamical state observable O + (s) mean value in the state Π + typical value of O in highly chaotic histories (s ) O + (s) = typical value of O in steady state (s 0) typical value of O in lowly chaotic histories (s ) V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

21 Contents Thermodynamic formalism Contents 1 Introduction histories versus configurations historical background 2 Thermodynamic formalism goals and concepts main results 3 Examples example 1: the Ising ferromagnet example 2: the contact process example 3: a kinetically constrained model V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

22 Examples The Ising ferromagnet (1) Example: the Ising ferromagnet Equilibrium N spins {σ i } of total magnetisation M = i σ i Hamiltonian (infinite range interaction) H ( {σ i } ) = 1 2N σ i σ j i,j Phase transition high temperature (disordered) phase: β < 1 m = 0 low temperature (ordered) phase: β > 1 m = ±m 0 m 0 solution of: tanh m 0 = β tanh m 0 V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

23 Examples The Ising ferromagnet (1) Example: the Ising ferromagnet Equilibrium N spins {σ i } of total magnetisation M = i σ i Hamiltonian (infinite range interaction) H ( {σ i } ) = 1 2N mean magnetisation: m = M/N Phase transition i,j σ i σ j = M2 2N high temperature (disordered) phase: β < 1 m = 0 low temperature (ordered) phase: β > 1 m = ±m 0 m 0 solution of: tanh m 0 = β tanh m 0 V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

24 Examples The Ising ferromagnet (1) Example: the Ising ferromagnet Equilibrium N spins {σ i } of total magnetisation M = i σ i Hamiltonian (infinite range interaction) H ( {σ i } ) = 1 2N mean magnetisation: m = M/N Phase transition i,j σ i σ j = M2 2N high temperature (disordered) phase: β < 1 m = 0 low temperature (ordered) phase: β > 1 m = ±m 0 m 0 solution of: tanh m 0 = β tanh m 0 V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

25 Examples Example: the Ising ferromagnet The Ising ferromagnet (2) Dynamics transition rates W(σ i σ i ) = e βσ i M/N f ÝÒ(s) ÑÓÖ ÓØ s Ð ÓØ s Dynamical phase transition at s = s. V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

26 Magnetization Examples Example: the Ising ferromagnet m(s) Ð ÓØ 0.2 ÑÓÖ ÓØ s V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

27 Examples Example II: the contact process Contact process Dynamics (infinite range interaction) N sites {n i } with n i = 1 or 0 Total number of particles n = i n i Transition rates: W(n i 1 n i ) = { λ n N if n i = 0 1 if n i = 1 Collective variable: n Transition rates: W(n n ± 1) = { (N n)λ n N n V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

28 Examples Example II: the contact process Contact process Dynamics (infinite range interaction) N sites {n i } with n i = 1 or 0 Total number of particles n = i n i Transition rates: W(n i 1 n i ) = { λ n N if n i = 0 1 if n i = 1 Collective variable: n Transition rates: W(n n ± 1) = { (N n)λ n N n V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

29 Density Examples Contact process ρ(s) Ð Ø Ú ÑÓÖ Ø Ú s V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

30 Examples FA 1-1 model Example III: a kinetically constrained model Dynamics (Fredrickson Andersen model) L sites {n i } with n i = 1 or 0 in one dimension Constraint: at least one neighbor is alive to allow a move Transition rates: annihilation with rate k creation with rate k annihilation: creation: k k k k k k V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

31 Examples FA 1-1 model Example III: a kinetically constrained model Dynamics (Fredrickson Andersen model) L sites {n i } with n i = 1 or 0 in one dimension Constraint: at least one neighbor is alive to allow a move Transition rates: annihilation with rate k creation with rate k diffusion with rate D (not constrained) annihilation: creation: diffusion: k k k k k k D D D D V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

32 Examples FA 1-1 model Example III: a kinetically constrained model Comparison with the unconstrained model Same equilibrium distribution Slow decay in the constrained model Numerical simulation Generalization of [Giardinà, Kurchan, Peliti] s algorithm Biased rates to explore fixed-s dynamics Direct measurement of ρ(s) V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

33 Examples Dynamical phase transition FA 1-1 model ρ K (s) ÓÒ ØÖ Ò ÙÒÓÒ ØÖ Ò s ÑÓÖ Ø Ú Ð Ø Ú Comparison between constrained and unconstrained model V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

34 Examples FA 1-1 model Dynamical phase transition ρ(s) rho_f1_s0_l50 rho_f1_s0_l100 rho_f1_s0_l200 L = 50 L = 100 L = s Large size scaling V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17

35 Examples Conclusion Summary Thermodynamic formalism large time limit Dynamical free energy f dyn dynamical phase transition Dynamical order parameter m(s) Perspective Other kinds of phase transition Caracterization of glassy dynamics References: Chaotic properties of systems with Markov dynamics, PRL 05 cond-mat/next-week V. Lecomte (LPT-Orsay, MSC-Paris 7) Thermodynamics of histories 05/19/ / 17