Out of Equilibrium Dynamics of Complex Systems

Transcription

1 Out of Equilibrium Dynamics of Complex Systems Leticia F. Cugliandolo Sorbonne Universités, Université Pierre et Marie Curie Laboratoire de Physique Théorique et Hautes Energies Institut Universitaire de France leticia/seminars Beg Rohu, France, 2017

2 Plan of Lecture 1. Langevin equation (derivation, time-scales) 2. Stochastic calculus (discretisation, chain-rule, Fokker-Planck, drift-force) 3. Generating functional formalism (Onsager-Machlup, Martin-Siggia-Rose) 4. Time-reversal symmetry (fluctuation-dissipation theorem, fluctuation theorems)

3 Formalism

4 Dissipative systems Aim Interest in describing the statics and dynamics of a classical or quantum physical system coupled to a classical or quantum environment. The Hamiltonian of the ensemble is Environment H = H syst + H env + H int Interaction System The dynamics of all variables are given by Newton or Heisenberg rules, depending on the variables being classical or quantum. The total energy is conserved, E = ct, but each contribution is not, in particular, E syst ct, and we ll take E syst E env.

5 Reduced system Model the environment and the interaction E.g., an ensemble of harmonic oscillators and a linear in q a and non-linear in x, via the function V(x), coupling : H env + H int = N α=1 [ p 2 α + m ] αωα 2 qα 2 2m α 2 + N α=1 c α q α V(x) Equilibrium. Imagine the whole system in contact with a bath at inverse temperature β. Compute the reduced classical partition function or quantum density matrix by tracing away the bath degrees of freedom. Dynamics. Classically (coupled Newton equations) and quantum (easier in a path-integral formalism) to get rid of the bath variables. In all cases one can integrate out the oscillator variables as they appear only quadratically.

6 Reduced system Statistics of a classical system Imagine the coupled system in canonical equilibrium with a megabath Z syst + env = env, syst e βh Integrating out the environmental (oscillator) variables Zsyst red = e β syst ( H syst 1 2 a c 2 a maω 2 a [V(x)] 2 ) Z syst = syst e βh syst One possibility : assume weak interactions and drop the new term. Trick : add H counter to the initial coupled Hamiltonian, and choose it in such a way to cancel the quadratic term in V(x) to recover Z red syst = Z syst i.e., the partition function of the system of interest.

7 Reduced system Dynamics of a classical system : general Langevin equations The system, p, x, coupled to an equilibrium environment evolves according to the multiplicative noise non-markov Langevin equation Inertia friction {{}}{ }}{ mẍ(t) +V (x(t)) dt γ(t t )ẋ(t ) V (x(t )) = t 0 δv (x) +V (x(t)) ξ(t) δx(t) }{{}}{{} deterministic force noise The friction kernel is γ(t t ) = Γ(t t )θ(t t ) The noise has zero mean and correlation ξ(t)ξ(t ) = k B T Γ(t t ) with T the temperature of the bath and k B the Boltzmann constant.

8 Reduced system Dynamics of a classical system : general Langevin equations The system, p, x, coupled to an equilibrium environment evolves according to the multiplicative noise non-markov Langevin equation Inertia friction {{}}{ }}{ mẍ(t) +V (x(t)) dt γ(t t )ẋ(t ) V (x(t )) = t 0 δv (x) +V (x(t)) ξ(t) δx(t) }{{}}{{} deterministic force noise Important Noise arises from lack of knowledge on bath ; noise can be multiplicative ; memory kernel generated ; equilibrium assumption on bath variables implies detailed balance between friction and noise

9 Additive white noise Separation of time-scales & special coupling In classical systems one usually takes a bath kernel with the smallest relaxation time, t env t all other time scales. The bath is approximated by the white form Γ(t t ) = 2γ 0 δ(t t ) Moreover, one assumes the coupling is bi-linear, H int = a c aq a x. The Langevin equation becomes δv (x) mẍ(t) + γ 0 ẋ(t) = δx(t) + ξ(t) with ξ(t) = 0 and ξ(t)ξ(t ) = 2k B T γ 0 δ(t t ).

10 Separation of time-scales Velocities and coordinates For t τ v = m/γ one expects the velocities to equilibrate to the Maxwell distribution P ({ v}) = i P ( v i ) i e βmv2 i /2 In this limit, one can drop m v a i overdamped equation and work with the γṙ a i = δv ({ r i}) δr a i + ξ a i. The positions can have highly non-trivial dynamics, see examples. Message : be very careful when trying to prove equilibration. Different variables could behave very differently.

11 Stochastic calculus Two ways of writing the multiplicative noise equation The physical eq. that comes from integrating away the bath (oscillators) (V [x(t)]) 2 d t x(t) = F [x(t)] + V [x(t)]ξ(t) and the equation usually found in the mathematics literature are equivalent after identification d t x(t) = f[x(t)] + g[x(t)]ξ(t) g[x(t)] = f[x(t)] = 1 V [x(t)] 1 (V [x(t)]) F (x(t)) = 2 (g[x(t)])2 F [x(t)]

12 Stochastic calculus Discretization prescriptions d t x(t) = f[x(t)] + g[x(t)] ξ(t) with ξ(t) = 0 and ξ(t)ξ(t ) = 2D δ(t t ) means x(t k+1 ) = x(t k ) + f[x(t k )] dt + g[x(t k )] ξ(t k )dt with x(t k ) = αx(t k+1 ) + (1 α)x(t k ) and 0 α 1. Particular cases are α = 0 Itō ; α = 1/2 Stratonovich. Stratonovich 67, Gardiner 96, Øksendal 00, van Kampen 07

13 Stochastic calculus Orders of magnitude & different stochastic processes ξ k ξ(t k ) = O(dt 1/2 ) because of the Dirac-delta correlations of a dx x(t k+1 ) x(t k ) = O(dt 1/2 ) Variable increment white bath What is the difference between the two terms in the right-hand-side of the Langevin eq. when they are evaluated using different discretisation schemes, that is to say different α? f[x α (t k )] f[x α (t k )] = O(dt 1/2 ) vanishes for dt 0 g[x α (t k )]ξ(t k ) g[x α (t k )]ξ(t k ) = O(dt 0 ) remains finite for dt 0 For multiplicative noise processes the discretisation matters: different α yields different stochastic processes.

14 Stochastic calculus Discretization prescriptions d t x(t) = f[x(t)] + g[x(t)] ξ(t) with ξ(t) = 0 and ξ(t)ξ(t ) = 2D δ(t t ) means x(t k+1 ) = x(t k ) + f[x(t k )] dt + g[x(t k )] ξ(t k )dt with x(t k ) = αx(t k+1 ) + (1 α)x(t k ) The chain rule for the time-derivative is (just from Taylor expansion) d t Y (x) = d t x d x Y (x) + D(1 2α) g 2 (x) d 2 xy (x) Only for α = 1/2 (Stratonovich) one recovers the usual expression. Not even for additive noise the chain rule is the usual one if α 1/2

15 Stochastic calculus Fokker-Planck equation The probability of x at time t k P (y, t k ) = dx k 1 T (y, t k x k 1, t k 1 ) P (x k 1, t k 1 ) with the condition (or transition) probability T (y, t k x k 1, t k 1 ) δ(y x k 1 dx) ξ(tk 1 ) = δ(y x k 1 y {δ(y x k 1 ) dx ξ(tk 1 )} y{δ(y x k 1 ) (dx) 2 ξ(tk 1 )} + O(dx 3 ) From the Langevin equation, dx ξ(tk 1 ) (dx) 2 ξ(tk 1 ) = f[x(t k 1 )] dt + 2Dαg[x(t k 1 )]g [x(t k 1 )] dt = 2D g 2 [x(t k 1 )] dt

16 Stochastic calculus Fokker-Planck equations for different α Call y x, perform the integral over x k 1 and rearrange terms. The Fokker-Planck equation t P (x, t) = x [(f(x) + 2Dαg(x)d x g(x))p (x, t)] +D x[g 2 2 (x)p (x, t)] depends on α and g Two processes will be statistically the same if f + 2D α gd x g = f drifted + 2D α gd x g Correspondence between (f, α) and (f drifted, α)

17 Stochastic calculus Fokker-Planck & stationary measure The Fokker-Planck equation t P (x, t) = x [(f(x) + 2Dαg(x)d x g(x))p (x, t)] +D 2 x[g 2 (x)p (x, t)] has the stationary measure P st (x) = Z 1 [g(x)] 2(α 1) e 1 D x f(x ) g 2 (x ) = Z 1 e 1 D U eff(x) with U eff (x) = x f(x ) g 2 (x ) + 2D(1 α) ln g(x) Remark : the potential U eff (x) depends upon α and g(x) Noise induced phase transitions Stratonovich 67, Sagués, Sancho & García-Ojalvo 07

18 Stochastic calculus Fokker-Planck & stationary measure e.g. f = g 2 U and U eff = U + 2D(1 α) ln g x 2 + 2D(1 α) ln x x 2 + 2D(1 α) ln (1 x 2 ) g(x) = x g(x) = (1 x 2 ) U(x) = x 2 U(x) = x 2

19 Multiplicative noise Noise induced coarsening for t φ = 2 φ µφ gφ 3 φξ NCOP COP R(t, T ) t 1/2 R(t, T ) t 1/3 Ibañes, García-Ojalvo, Toral & Sancho 00

20 Stochastic calculus Drift The Gibbs-Boltzmann equilibrium P GB (x) = Z 1 e βu(x) is approached if (recall the physical writing of the equation) f(x) g 2 (x)d x U(x) }{{} + } 2D(1 α)g(x)d {{ xg(x) } Potential drift The drift is also needed for the Stratonovich mid-point scheme Important choice: if one wants the dynamics to approach thermal equilibrium independently of α and g the drift term has to be added.

21 Stochastic calculus Fokker-Planck & stationary measure The Fokker-Planck equation t P (x, t) = x [(f(x) + 2Dαg(x)d x g(x))p (x, t)] +D x[g 2 2 (x)p (x, t)] for the drifted force f(x) g 2 (x)d x U(x) + 2D(1 α)g(x)d x g(x) becomes t P (x, t) = x [( g 2 (x)d x U(x) + 2Dg(x)d x g(x))p (x, t)] +D x[g 2 2 (x)p (x, t)] with the expected Gibbs-Boltzmann measure stationary measure P st (x) = Z 1 e 1 D U(x) independently of g(x) and α

22 Methods Dynamic generating functional Glassy models with and without disorder: The "order parameter" is a composite object depending on two-times. It s handy to use functional methods to write a dynamic generating functional as a path-integral Onsager-Machlup & Martin-Siggia-Rose-Janssen-deDominicis formalisms Similar to Feynman path-integral The construction will follow LFC & Lecomte, Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager-Machlup approach, arxiv : , J. Phys. A 50, (2017) where special care of discretisation effects was taken.

23 Generating functional Onsager-Machlup representation Langevin equation in generic discretization scheme x k = R(x k, x k 1, ξ k 1 ) Definition of the infinitesimal transition probability T (x k, t k x k 1, t k 1 ) = dξ k 1 P noise (ξ k 1 )δ(x k R(x k, x k 1, ξ k 1 )) In order to integrate over dξ k 1 we have to transform the δ in which ξ k 1 appears within R to J 1 δ(ξ k 1... ). The Jacobian J is needed Generalisation of f (f 1 (a)) δ(f(z) a) = δ(z f 1 (a)) J = det kk [ ] δ{[xk R[x k, x k 1, ξ k 1 ; α]} δξ k T (x k, t k x k 1, t k 1 ) = dξ k 1 P noise (ξ k 1 )J 1 δ(ξ k R(xk, x k 1 )) One can forget the modulus since only one solution is expected

24 Generating functional Onsager-Machlup representation The transition probability now reads T (x k, t k x k 1, t k 1 ) = 1 4πk B T dt 1 g(x k 1 ) es OM[x k,x k 1 ;α] For d t x(t) = f(x(t)) + g(x(t))ξ(t), the Onsager-Machlup action is S OM [x k, x k 1 ; α] ln P i (x T ) dt [ ( 1 (xk x k 1 ) f(x 4k B T g 2 k 1 ) (x k 1 ) dt ) ] 2 +2Dαg(x k 1 )g (x k 1 ) αf (x }{{}}{{ k 1) } From the integration over the noise Jacobian

25 Generating functional Onsager-Machlup representation, continuous time notation The transition probability now reads T (x k, t k x k 1, t k 1 ) = 1 4πk B T dt 1 g(x k 1 ) es OM[{x};α] For d t x(t) = f(x(t)) + g(x(t))ξ(t), the Onsager-Machlup action is S OM [{x}; α] ln P i (x T ) dt [ ] 1 (d t x t f t + 2Dαg t d x g t 4k B T }{{} )2 αd x f }{{} t g 2 t From the integration over the noise Jacobian

26 Generating functional From Onsager-Machlup to Martin-Siggia-Rose Use the Hubbard-Stratonovich (Gaussian integral) trick to go from the exponential of a square to the one of a linear term e 1 2 y2 = dˆx e ±iˆxy (iˆx)2 that with the parameters in the action and a convenient choice of sign 1 e 2(2k B T ) y2 = dˆx e iˆxy (2k BT )(iˆx) 2 In our case y t = (d t x t f t + 2Dαg t d x g t )/g t Further change of variables ˆx t g tˆx t

27 Generating functional MSR path-integral representation The joint probability distribution of the trajectory" reads P ({x}; α) = D{ˆx} (4πk B T dt) 1/2 e S MSR[{x},{iˆx};α] N 1 k=0 and the Martin-Siggia-Rose-Janssen 79 action is S MSR [{x}, {iˆx}; α] ln P i (x T ) [ ] + iˆx t (d t x }{{} t f t + 2Dαg t d x g t ) + D(iˆx t ) 2 gt 2 }{{} αd }{{ xf } t proportional to γ 0 (not written) Jacobian Observable averages can now be calculated as A(x t, iˆx t ) = D{x}D{ˆx} P MSR ({x}, {iˆx}; α) A(x t, iˆx t )

28 Generating functional Path-integral representation For the drifted force f t = g 2 d x V t + 2D(1 α)g t d x g t S MSR ({x}, {iˆx}; λ, α) ln P i (x T, λ T ) + [ iˆxt (d t x t + g 2 t d x V t 2D(1 2α)g t d x g t ) +D(iˆx t ) 2 g 2 t αd x f t ] Remark: The action depends on α and g. Observable averages can now be calculated as A(x t, iˆx t ) = D{x}D{ˆx} P MSR ({x}, {iˆx}; α, λ) A(x t, iˆx t ) and do not depend on α

29 Stochastic calculus Path-integral representation for additive noise For g = 1 and the force f t = d x V t the action is S MSR [{x}, {iˆx}; α] ln P i (x T ) ] + [ iˆxt (d t x t + d x V t ) + D(iˆx t ) 2 + αd 2 xv t Observable averages can now be calculated as A(x t, iˆx t ) = D{x}D{ˆx} P MSR ({x}, {iˆx}; λ, α) A(x t, iˆx t ) and do not depend on α

30 Methods Dynamical symmetry & exact results The functional path-integral formalism allows one to obtain exact identities (fluctuation-dissipation theorem, fluctuation theorems) as consequences of a dynamic symmetry and its symmetry breaking. Details in : Symmetries of generating functionals of Langevin processes with colored multiplicative noise Aron, Biroli & LFC, J. Stat. Mech. P11018 (2010) ; Dynamical symmetries of Markov processes with multiplicative white noise, Aron, Barci, LFC, González Arenas & Lozano, J. Stat. Mech (2016) Possible (though not easy) to extend to quantum system. (Non) equilibrium dynamics : a (broken) symmetry of the Keldysh generating functional Aron, Biroli & LFC, arxiv:

31 Symmetry Transformations in the path-integral representation Let us define d (α) t x t d t x t 2D(1 2α)g t d x g t and group two terms in the action due to the coupling to the bath S diss [x, iˆx] = iˆx t [d (α) t x t Diˆx t g 2 t ] This expression suggests to use the generalized transformation on the time-dependent variables {x t, iˆx t } T c = x t x t, iˆx t iˆx t + D 1 g 2 t d(α) t x t, and α 1 α Remember D = β 1 = k B T

32 Symmetry Transformations in the path-integral representation For initial conditions drawn from P i (x) = Z 1 e βv (x) and f(x) = g 2 (x)d x V (x) + 2D(1 α)g(x)d x g(x) one proves S det+jac [T c iˆx, T c x; T c α] = S det+jac [iˆx, x; α] that implies P [T c iˆx, T c x; T c α] = P [iˆx, x; α] Note that we have to use the non-trivial chain rule. Moreover, the transformation leaves the integral measure invariant (no Jacobian) and the interval of integration as well.

33 Symmetry Consequences of the transformation: FDT From this result we can prove exact equilibrium relations such as the fluctuation-dissipation theorem linking the (causal) linear response to a field that changes the force as f t f t + h t R(t, t ) = δ x(t) δh(t ) θ(t t ) h=0 and the correlation function in a model independent way : R(t, t ) R( t, t ) = β t x( t)x( t ) that for a stationary problem (in equilibrium) becomes R(t t ) R(t t) = β t C(t t) = β t C(t t )

34 Broken symmetry Relation under "any" transformation P (T c x, T cˆx; T c α, λ) P (x, ˆx; α, λ) = e S(x,ˆx;α,λ) with S the variation of the full action (and measure) T c x and T cˆx are transformed trajectories, λ the transformed parameter in the potential, T c α a different discretisation parameter ; and from here obtain relations between observables by averaging this relation : equilibrium fluctuation dissipation ( S = 0), or out of equilibrium theorems ( S 0). e.g., Jarzinsky 97, Crooks 00 & many others

35 Broken Symmetry Consequences of the transformation: Fluctuation-theorems For initial conditions drawn from P i (x) = Z 1 e βv (x) and f(x, λ t ) = g 2 (x) x V (x, λ t ) + 2D(1 α)g(x)d x g(x) proves with P [T c iˆx, T c x; T c α, λ t = λ t ] βw β F = e P [iˆx, x; α, λ t ] W = dt d t λ t λ V (x, λ) one F = ln Z(λ T ) ln Z(λ T ) the work, and free-energy difference between initial and fictitious final states. Exact out of equilibrium relations such as the Jarzinsky relation follow e βw = e β F

36 Coloured noise Langevin equation & generating functional The generic Langevin equation for a particle in 1d is mẍ(t) + V [x(t)] t T dt Γ(t t )V [x(t )]ẋ(t ) = F (t) + ξ(t)m [x(t)] with the coloured noise ξ(t)ξ(t ) = k B T Γ(t t ) The dynamic generating functional is a path-integral Z dyn [η] = dx T dẋ T DxDˆx e S[x,iˆx;η] with iˆx(t) the response variable. x T and ẋ T are the initial conditions at time T. Martin-Siggia-Rose-Jenssen-deDominicis formalism

37 Why care about multiplicative noise?

38 Magnetisation precession Bloch equation Evolution of the time-dependent 3d magnetisation density per unit volume, M = (M x, M y, M z ), with constant modulus M s = M d t M = µ M H eff µ γµ 0 is the product of γ = µ B g/, the gyromagnetic ratio, and µ 0, the vacuum permeability constant (µ B Bohr s magneton and g Lande s g-factor) For the initial condition M(t i ) = M i the magnetisation precesses around H eff with 2M d t M = d t M 2 = 0 and d t (M H eff ) = 0 (if H eff = ct) Bloch 32

39 Dissipative effects Landau-Lifshitz & Gilbert equations µ d t M = 1 + γ0 2 M µ2 [ H eff + γ ] 0µ (M H eff ) M s Landau & Lifshitz 35 d t M = µ M ( H eff γ ) 0 d t M M s Gilbert 55 2nd terms in RHS: dissipative mechanisms slow down the precession and push M towards H eff with 2M d t M = d t M 2 = 0 and d t (M H eff ) > 0

40 Thermal fluctuations À la Langevin in Gilbert s formulation d t M = µm ( H eff + H γ ) 0 d t M M s H is a white random noise, with zero mean H i (t) = 0 and correlations H i (t)h j (t ) = 2Dδ ij δ(t t ) The (diffusion) parameter D is proportional to k B T Brown 63 The noise H multiplies the magnetic moment M and one cannot always write 2M d t M = d t M 2 (only if the Stratonovich calculus is used) This is the Markov stochastic Landau-Lifshitz-Gilbert-Brown (sllgb) multiplicative white noise stochastic differential equation. Subtleties of Markov multiplicative noise processes are now posed.

41 Thermal fluctuations À la Langevin in Gilbert s formulation d t M = µm ( H eff + H γ ) 0 d t M M s 2D(1 2α)µ2 1 + µ 2 γ 2 0 M H is a white random noise, with correlations H i (t)h j (t ) = 2Dδ ij δ(t t ) The (diffusion) parameter D is proportional to k B T Brown 63 The modulus of the magnetic moment is now conserved d t M 2 = 0 for all α One also proves that the dynamics approaches the asymptotic Gibbs-Boltzmann distribution P GB (M) e 1 D M H eff