Simulating rare events in dynamical processes. Cristian Giardina, Jorge Kurchan, Vivien Lecomte, Julien Tailleur

Transcription

1 Simulating rare events in dynamical processes Cristian Giardina, Jorge Kurchan, Vivien Lecomte, Julien Tailleur Laboratoire MSC CNRS - Université Paris Diderot Computation of transition trajectories and rare events in non-equilibrium systems J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

2 Outline Fluctuations of chaoticity in dynamical systems Large deviation functions in interacting particle systems J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

3 Outline Fluctuations of chaoticity in dynamical systems Large deviation functions in interacting particle systems J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

4 Fluctuations in dynamical systems Phase space is non uniform KAM Theorem, Arnol d Web Laminar vs turbulent flows Solitons and Breathers in a non-linear crystals Regular orbits in planetary systems J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

5 Fluctuations in dynamical systems Phase space is non uniform KAM Theorem, Arnol d Web Laminar vs turbulent flows Solitons and Breathers in a non-linear crystals Regular orbits in planetary systems Fluctuations of chaoticity Sensitivity to initial conditions Fluct. of Lyapunov exponents [Ruelle 78, Benzi 84, Grassberger 88] Lots of theory, few applications in physics J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

6 Lyapunov exponents Divergence of nearby trajectories u(t) u(0) exp[λ(t)t] λ(t) λ : Lyapunov exponent t ẋ = f(x) u = f(x) x u Fluctuations P (λ, t) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

7 A large deviation problem P (λ, t) = exp[s(λ, t)] Total time t δt correlation time τ; u(t) u(0) eλt = k u(kδt) u((k 1)δt) = e(λ 1+ +λ t/δt )δt δt δt δt J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

8 A large deviation problem P (λ, t) = exp[s(λ, t)] Total time t δt correlation time τ; u(t) u(0) eλt = k u(kδt) u((k 1)δt) = e(λ 1+ +λ t/δt )δt P (λ, t) = Π i dλ i P (λ 1, δt)... P (λ t/δt, δt) (λ 1 + +λ t/δt )δt=λt δt δt δt J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

9 A large deviation problem P (λ, t) = exp[s(λ, t)] Total time t δt correlation time τ; u(t) u(0) eλt = k u(kδt) u((k 1)δt) = e(λ 1+ +λ t/δt )δt P (λ, t) = Π i dλ i P (λ 1, δt)... P (λ t/δt, δt) (λ 1 + +λ t/δt )δt=λt δt = π i dλ i e S(λ 1,δt)+ +S(λ t/δt,δt) δt δt (λ 1 + +λ t/δt )δt=λt J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

10 A large deviation problem P (λ, t) = exp[s(λ, t)] Total time t δt correlation time τ; u(t) u(0) eλt = k u(kδt) u((k 1)δt) = e(λ 1+ +λ t/δt )δt P (λ, t) = Π i dλ i P (λ 1, δt)... P (λ t/δt, δt) (λ 1 + +λ t/δt )δt=λt δt = π i dλ i e S(λ 1,δt)+ +S(λ t/δt,δt) δt δt (λ 1 + +λ t/δt )δt=λt P (λ, t) e ts(λ) The larger the time, the more peaked P (λ, t) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

11 Numerical methods Sample P (λ, t) e ts(λ) Frequency map analysis (Laskar 93) Spectral analysis (Sepulveda, Badi, Pollak 95) Fast Lyapunov indicator (Lega 96) Correlation functions (Pollner, Vattay 96) Fast Lyapunov indicator (Froeschlé, Lega, Gongzi 97) SALI (Skokos 01) Random sampling λ s.t. s (λ ) = 0 Grid Low dimension only J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

12 The thermodynamics of trajectories Give a weight exp(αλt) to each trajectory P α (λ, t) = 1 Z α P (λ, t)e αλt Z α (t) = e αλt J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

13 The thermodynamics of trajectories Give a weight exp(αλt) to each trajectory P α (λ, t) = 1 P (λ, t)e αλt e t[s(λ)+αλ] Z α (t) = e αλt Z α t J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

14 The thermodynamics of trajectories Give a weight exp(αλt) to each trajectory P α (λ, t) = 1 P (λ, t)e αλt e t[s(λ)+αλ] Z α (t) = e αλt Z α t Trajectories with λ (α) dominate s (λ ) = α differs from s (λ ) = 0 J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

15 Temperature and entropy P α (λ, t) t e t[s(λ)+αλ] Z α (t) = e αλt λ Macrostate J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

16 Temperature and entropy P α (λ, t) t e t[s(λ)+αλ] Z α (t) = e αλt λ α Macrostate Temperature α > 0 favors chaos α < 0 favors order J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

17 Temperature and entropy P α (λ, t) t e t[s(λ)+αλ] Z α (t) = e αλt λ α Macrostate Temperature α > 0 favors chaos α < 0 favors order Z α is a dynamical partition function µ(α) = 1 t log[z α] is a dynamical free energy Language of dynamical transitions (e.g. transition to turbulence) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

18 Lyapunov Weighted Dynamics [Nat. Phys., 3, 203 (2007)] N copies/clones of the system (x, u). At each dt: J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

19 Lyapunov Weighted Dynamics [Nat. Phys., 3, 203 (2007)] N copies/clones of the system (x, u). At each dt: Each copy evolve with ẋ = f(x); u = f x u For each copy, we compute ( ) α u(t+dt) u(t) exp(αλdt) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

20 Lyapunov Weighted Dynamics [Nat. Phys., 3, 203 (2007)] N copies/clones of the system (x, u). At each dt: Each copy evolve with ẋ = f(x); u = f x u For each copy, we compute ( ) α u(t+dt) u(t) exp(αλdt) Each copy is replaced by [on average] exp(αλdt) copies At time t, 1 clone exp(αλt) copies N (t)/n (0) exp(αλt) exp[µ(α)t] J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

21 Two important tricks 1 To prevent the number of clones to diverge or vanish overall cloning rate R(t + dt) = N (t)/n (t + dt) exp(αλt) exp[µ(α)t] = t R(t) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

22 Two important tricks 1 To prevent the number of clones to diverge or vanish overall cloning rate R(t + dt) = N (t)/n (t + dt) exp(αλt) exp[µ(α)t] = t R(t) 2 To prevent degeneracy of clones small noise: when we replicate the clones to the dynamics J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

23 An integrable case : The double well potential Localizing the separatrix H(q, p) = p2 2 + (1 q2 ) 2 LWD with α = 1 J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

24 An integrable case : The double well potential Localizing the separatrix H(q, p) = p2 2 + (1 q2 ) 2 LWD with α = 1 J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

25 A toy model to study the transition to Chaos The Standard Map p n+1 = p n kδ 2π sin(2πq n) q n+1 = q n + δp n+1 Chaoticity increases with k, δ J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

26 A toy model to study the transition to Chaos The Standard Map p n+1 = p n kδ 2π sin(2πq n) q n+1 = q n + δp n+1 Chaoticity increases with k, δ Almost Integrable Case (δ = 0.45, k = 1); LWD with α = 1 J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

27 A mixed case k = 1, δ = 1, α = 1 LWD α = 1 J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

28 Integrable islands in a chaotic sea k = 7.8, δ = 1, α = 1 LWD α = 1 J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

29 Dynamical free energy µ(α) = 1 t log(z α) λ(α) = µ (α) = Z 1 α dλ λ P (λ) e αλt µ,, λ µ(α) λ(α) α 1 J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

30 Dynamical free energy µ(α) = 1 t log(z α) λ(α) = µ (α) = Z 1 α dλ λ P (λ) e αλt µ,, λ µ(α) λ(α) α 1 J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

31 Dynamical free energy µ(α) = 1 t log(z α) λ(α) = µ (α) = Z 1 α dλ λ P (λ) e αλt µ,, λ µ(α) λ(α) α 1 First order transition point J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

32 The Fermi Pasta Tsingou Ulam chain Chain of non-linear oscillators H = N i=1 ( 1 2 p2 i (x i x i+1 ) 2 + β ) 4 (x i x i+1 ) 4 β = 0 uncoupled fourier modes ω k = 2 sin ( ) πk N J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

33 Equilibrium J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

34 LWD with α = 5N (N=128) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

35 LWD with α = 5N (N=1024) 1024 Particles 0 0 time J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

36 LWD with α 1 J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

37 Phase transition? Scaling with N... How do the fluctuations of λ scale with N? P (λ, t) e tn ξ s(λ,t,n) ; s(λ, t, N) N,t O(1) Z α = dλe αλt+tn ξ s(λ) Select λ : s (λ ) = α N ξ 0 N Need to find the good scaling (hard!) see [Kuptsov& Politi PRL 2011] J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

38 Conclusion Part I Numerical method to sample the fluctuations of λ Detect atypical trajectories Study dynamical phase transition (turbulence!) [J. Tailleur, J. Kurchan, Nature Physics, 3, p , (2007)] J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

39 Dynamical free energies t 0 = 0 C 0 t 1 C 1 t 2 C 2 C K 1... t K 1 t K C K t C = C K Statistical mechanics in trajectories space J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

40 Dynamical free energies t 0 = 0 C 0 t 1 C 1 t 2 C 2 C K 1... t K 1 t K C K t C = C K Statistical mechanics in trajectories space Observable Q = k Q C k,c k+1 q t J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

41 Dynamical free energies t 0 = 0 C 0 t 1 C 1 t 2 C 2 C K 1... t K 1 t K C K t C = C K Statistical mechanics in trajectories space Observable Q = k Q C k,c k+1 q t Microcanonical ensemble P (Q 0 ) = δ(q Q 0 ) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

42 Dynamical free energies t 0 = 0 C 0 t 1 C 1 t 2 C 2 C K 1... t K 1 t K C K t C = C K Statistical mechanics in trajectories space Observable Q = k Q C k,c k+1 q t Microcanonical ensemble P (Q 0 ) = δ(q Q 0 ) Canonical ensemble Z(β) = e βq e tψ(β) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

43 Dynamical free energies t 0 = 0 C 0 t 1 C 1 t 2 C 2 C K 1... t K 1 t K C K t C = C K Statistical mechanics in trajectories space Observable Q = k Q C k,c k+1 q t Microcanonical ensemble P (Q 0 ) = δ(q Q 0 ) Canonical ensemble Z(β) = e βq e tψ(β) ψ(β) is a dynamical free energy How can one compute it? J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

44 Computation of Z(β) = e βq Transition rates W (C C ) t P (C) = C C W (C C)P (C ) r(c)p (C) Escape rate r(c) = C C W (C C ) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

45 Computation of Z(β) = e βq Transition rates W (C C ) t P (C) = C C W (C C)P (C ) r(c)p (C) Escape rate r(c) = C C W (C C ) Joint probability P (Q, C, t) ˆPβ (C, t) = Q e βq P (Q, C, t) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

46 Computation of Z(β) = e βq Transition rates W (C C ) t P (C) = C C W (C C)P (C ) r(c)p (C) Escape rate r(c) = C C W (C C ) Joint probability P (Q, C, t) ˆPβ (C, t) = Q e βq P (Q, C, t) Z(β) = C ˆP β (C, t) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

47 Computation of Z(β) = e βq W β (C C ) = e βq C,C W (C C ) t ˆPβ (C) = C C W β (C C) ˆP β (C ) r β (C) ˆP β (C) + (r β (C) r(c)) ˆP β (C) Escape rate r β (C) = C C W β(c C ) Joint probability P (Q, C, t) ˆPβ (C, t) = Q e βq P (Q, C, t) Z(β) = C ˆP β (C, t) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

48 Computation of Z(β) = e βq W β (C C ) = e βq C,C W (C C ) t ˆPβ (C) = C C W β (C C) ˆP β (C ) r β (C) ˆP β (C) + (r β (C) r(c)) ˆP β (C) Escape rate r β (C) = C C W β(c C ) (r β (C) r(c)) ˆP β (C) C ˆP β not conserved Z(β) = C ˆP β (C, t) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

49 Computation of Z(β) = e βq W β (C C ) = e βq C,C W (C C ) t ˆPβ (C) = C C W β (C C) ˆP β (C ) r β (C) ˆP β (C) + (r β (C) r(c)) ˆP β (C) Escape rate r β (C) = C C W β(c C ) (r β (C) r(c)) ˆP β (C) C ˆP β not conserved Population dynamics: Z(β, t) = C ˆP β (C, t) = N(t)/N(0) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

50 Population dynamics N copies of the system evolve with rates W β t 0 = 0 C 0 t 1 C 1 t 2 t K C 2... t K 1 C K 1 C K t C = C K J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

51 Population dynamics N copies of the system evolve with rates W β t 0 = 0 C 0 t 1 C 1 t 2 t K C 2... t K 1 C K 1 C K t C = C K Cloning term e (t k+1 t k )[r β (C k ) r(c k )] J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

52 Population dynamics N copies of the system evolve with rates W β t 0 = 0 C 0 t 1 C 1 t 2 t K C 2... t K 1 C K 1 C K t C = C K Cloning term e (t k+1 t k )[r β (C k ) r(c k )] Z(β) = N(t) N J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

53 Population dynamics N copies of the system evolve with rates W β t 0 = 0 C 0 t 1 C 1 t 2 t K C 2... t K 1 C K 1 C K t C = C K Cloning term e (t k+1 t k )[r β (C k ) r(c k )] N k copies The population is rescaled by R k = N k N J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

54 Population dynamics N copies of the system evolve with rates W β t 0 = 0 C 0 t 1 C 1 t 2 t K C 2... t K 1 C K 1 C K t C = C K Cloning term e (t k+1 t k )[r β (C k ) r(c k )] N k copies The population is rescaled by R k = N k N Z(β) = k R k 1 ψ(β) = lim t t log R k k J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

55 p p SSEP p p p p 1 L 1 LDF of the particle current J: ψ J (β) L (N = 200 L = 400) β J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

56 p p SSEP p p p p 1 L 1 LDF of the particle current J: ψ J (β) L (N = 200 L = 400) β Fluctuation theorem: ψ(β) = ψ(log q p β) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

57 p p SSEP p p p p 1 L 1 LDF of the particle current J: ψ J (β) L (N = 200 L = 400) β Fluctuation theorem: ψ(β) = ψ(log q p β) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

58 p p SSEP p p p p 1 L 1 LDF of the particle current J: ψ J (β) L (N = 200 L = 400) β Fluctuation theorem: ψ(β) = ψ(log q p β) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

59 p q ASEP p q p q 1 L 1 LDF of the particle current J (N = 200 L = 400 p = 1.2 q = 0.8) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

60 p q ASEP p q p q 1 L 1 LDF of the particle current J (N = 200 L = 400 p = 1.2 q = 0.8) ψ J (β) L β 6 10 Fluctuation theorem: ψ(β) = ψ(log q p β) J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

61 p q ASEP p q p q 1 L 1 Typical density profiles for β 0 Small current J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

62 p q ASEP p q p q 1 L 1 Typical density profiles for β 0 Small current Large current J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

63 Conclusion Part II A population dynamics with modified rates allows us to compute large deviation functions [C. Giardina, J. Kurchan, L. Peliti, PRL 96, (2006)] [V. Lecomte, J. Tailleur, J. Stat. Mech, P03004 (2007)] [J. Tailleur, V. Lecomte, arxiv: ] [C. Giardina, J. Kurchan, V. Lecomte, J. Tailleur, J. Stat. Phys. 145, 787 (2011)] Several methods available (DMRG, cloning, TPS) Clarify what works best in which case?! J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

64 Conclusion Part II A population dynamics with modified rates allows us to compute large deviation functions [C. Giardina, J. Kurchan, L. Peliti, PRL 96, (2006)] [V. Lecomte, J. Tailleur, J. Stat. Mech, P03004 (2007)] [J. Tailleur, V. Lecomte, arxiv: ] [C. Giardina, J. Kurchan, V. Lecomte, J. Tailleur, J. Stat. Phys. 145, 787 (2011)] Several methods available (DMRG, cloning, TPS) Clarify what works best in which case?! J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25

65 Conclusion Part II A population dynamics with modified rates allows us to compute large deviation functions [C. Giardina, J. Kurchan, L. Peliti, PRL 96, (2006)] [V. Lecomte, J. Tailleur, J. Stat. Mech, P03004 (2007)] [J. Tailleur, V. Lecomte, arxiv: ] [C. Giardina, J. Kurchan, V. Lecomte, J. Tailleur, J. Stat. Phys. 145, 787 (2011)] Several methods available (DMRG, cloning, TPS) Clarify what works best in which case?! J. Tailleur (CNRS-Univ Paris Diderot) 13 June / 25