Path integrals for classical Markov processes

Transcription

1 Path integrals for classical Markov processes Hugo Touchette National Institute for Theoretical Physics (NITheP) Stellenbosch, South Africa Chris Engelbrecht Summer School on Non-Linear Phenomena in Field Theory Stellenbosch, South Africa -3 January Slides: Hugo Touchette (NITheP) Path integrals January / 8 Quantum path integral x i. x f t i t f Propagator: K(x f, t f x i, t i ) = x f, t f x i, t i Propagation: ψ(x f, t f ) = K(x f, t f x i, t i ) ψ(x i, t i ) dx i Path integral: K(x f, t f x i, t i ) = x(tf ) x(t i ) D[x] e is[x]/ = all paths e is[path]/ Hugo Touchette (NITheP) Path integrals January / 8

2 Classical path integral x i. x f t i t f Propagator: P(x f, t f x i, t i ) = Prob{x(t i ) x(t f )} Propagation (Kolmogorov-Chapmann equation): P(x f, t f ) = P(x f, t f x i, t i ) P(x i, t i ) dx i Path integral: P(x f, t f x i, t i ) = x(tf ) x(t i ) D[x] e S[x]/ɛ = all paths e S[path]/ɛ Hugo Touchette (NITheP) Path integrals January 3 / 8 Comparison Quantum path integral K quantum amplitude Interference possible Schrödinger equation: i ψ(x, t) = Hψ(x, t) t H hermitian : semi-classical limit Classical path integral P classical probability No interference Fokker-Planck equation: P(x, t) = LP(x, t) t L non-hermitian in general ɛ : low-noise limit Type of stochastic processes Continuous time Markov processes Stochastic differential equations Hugo Touchette (NITheP) Path integrals January / 8

3 Stochastic differential equations Deterministic dynamics (ODE): ẋ(t) = F (x) Perturbed dynamics (SDE): Ẋ ɛ (t) = F (X ɛ ) + ɛ ξ(t) }{{} noise x x(t) x X (t) ǫ τ τ x t+ t = x t + F (x t ) t X t+ t = X t +F (X t ) t+ ɛ W t Gaussian white noise: W t N (, t) Hugo Touchette (NITheP) Path integrals January 5 / 8 Path integral (a) (b) x x x(t) x x(t) x x x n t= Time discretization: Infinitesimal propagator: t=τ P[x] = lim n P(x, x,..., x n x ) = lim n P(x n x n ) P(x x )P(x x ) P(x, t + t x, t) = Path distribution: P[x] = c e S[x]/ɛ, S[x] = Δt ( exp t πɛ t ɛ τ L(ẋ, x) dt, [ x x t ] ) F (x) L = [ẋ F (x)] Hugo Touchette (NITheP) Path integrals January 6 / 8

4 Dominant path approximation Low-noise approximation: P(x f, t f x i, t i ) = D[x] e S[x]/ɛ e min S[x]/ɛ ɛ = e S[x ]/ɛ Final result: Dominant path: P(x f, t f x i, t i ) e V (x f,t f x i,t i )/ɛ V (x f, t f x i, t i ) = min x(t):x(t i )=x i,x(t f )=x f S[x] x (t) : min S[x] δs = Euler-Lagrange equation: d dt L ẋ L x =, x(t i) = x i, x(t f ) = x f Hugo Touchette (NITheP) Path integrals January 7 / 8 Applications Stationary distribution: V (x)/ɛ P(x) e V (x) = min S[x] x(t):x( )=,x()=x x(t) x Exit time: τ ɛ e V /ɛ V = min x D in probability min V (x, t x, ) t Exit path = dominant path = instanton Exit location = x(τ ɛ ) Semiclassical or WKB approximation D t D Hugo Touchette (NITheP) Path integrals January 8 / 8

5 Example: Ornstein-Uhlenbeck process SDE: Stationary distribution: ẋ(t) = γx(t) + ɛ ξ(t) P(x) e V (x)/ɛ, V (x) = min x(t):x()=x,x( )=x S[x] Euler-Lagrange equation: ẍ γ x =, x( ) =, x() = x Instanton solution: x (t) = xe γt, (ẋ = γx ) Solution: V (x) = S[x ] = γx P(x) Gaussian Instanton = time reverse of deterministic dynamics (ɛ = ) Hugo Touchette (NITheP) Path integrals January 9 / 8 Example: Noisy van der Pol oscillator SDE: ẋ = v Bifurcation: Stable fixed point: α < Stable limit cycle: α > v = x + v(α x v ) + ɛ ξ(t) v 3 t Stationary distribution: P(r, θ) e W (r)/ɛ W (r) given by Hamilton-Jacobi equation Hugo Touchette (NITheP) Path integrals January / 8 x

6 Noisy van der Pol oscillator (cont d) Solution: W (r) = αr + r Hugo Touchette (NITheP) Path integrals January / 8 Gradient vs non-gradient Gradient system ẋ = F (x) + ɛ ξ(t), F (x) = U(x) V (x) = U(x) Instanton = time-reversed deterministic dynamics Instanton equation: ẋ = U(x ) Non-gradient system F U Instanton time-reversed deterministic dynamics Instanton dynamics: ẋ = F (x) + V (x) Equilibrium vs nonequilibrium systems Detailed balance vs non-detailed balance Hugo Touchette (NITheP) Path integrals January / 8

7 Example: Linear, non-gradient system Non-gradient: ẋ U Stationary distribution: ẋ = Bx + ɛ ξ, B = V (x,y)/ɛ P(x) e. ( ) Quasi-potential: Instanton: V (x, y) = x + y ẋ = ( B s + B a )x = B x ( ) B = = B T Hugo Touchette (NITheP) Path integrals January 3 / 8 Summary P(x f, t f x i, t i ) = x(tf ) x(t i ) D[x] P[x], P[x] = e S[x]/ɛ Action: S[x] = tf Dominant path approximation: t i V (x f, t f x i, t i ) = [ẋ t F (x t )] T A(x t )[ẋ t F (x t )] dt min x(t):x(t i )=x i,x(t f )=x f S[x] Calculate all sorts of transition, escape probabilities Similar to semiclassical or WKB approximation Nonlinear equations difficult to solve in general Other noises: colored, Poisson, etc. Hugo Touchette (NITheP) Path integrals January / 8

8 Applications Physics Brownian motion Diffusion, transport Irreversible processes Noisy systems Biophysics Engineering Control, reliability Signal analysis, filtering Queueing Statistics, sampling Finance Hugo Touchette (NITheP) Path integrals January 5 / 8 Further reading H. Kleinert Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets World Scientific, 5 [Chaps., and ] M. I. Freidlin, A. D. Wentzell Random Perturbations of Dynamical Systems Springer, New York, 98, Chap. [for the braves!] R. Graham Macroscopic potentials, bifurcations and noise in dissipative systems F. Moss, P. V. E. McClintock (eds), Noise in Nonlinear Dynamical Systems, Cambridge University Press, 989 [ask me a copy] H. Touchette The large deviation approach to statistical mechanics Physics Reports 78, -69, 9 [Sec. 6.] Hugo Touchette (NITheP) Path integrals January 6 / 8

9 Reading on stochastic processes N. G. van Kampen Stochastic Processes in Physics and Chemistry North-Holland, 99 B. Øksendal Stochastic Differential Equations Springer, [recommended] K. Jacobs Stochastic processes for Physicists: Understanding Noisy Systems Cambridge, [recommended] D. J. Higham An algorithmic introduction to numerical simulation of stochastic differential equations SIAM Review 3, 55-56, Hugo Touchette (NITheP) Path integrals January 7 / 8 Exercises Re-do the calculation of page 6 to obtain the path distribution and the corresponding action. Derive the expression of the propagator s quasi-potential V (x f, t f ; x i, t i ) for the Ornstein-Ulhenbeck process using the dominant path approximation. [Hint: Follow the stationary solution on page 9.] Check that the propagator verifies the time-dependent Fokker-Planck equation exactly. 3 Derive the stationary quasi-potential W (r) for the noisy van der Pol oscillator (page ). Can you write down another dynamical system having the same quasi-potential? Show for a gradient system that the instanton is the time-reverse of the deterministic path. 5 Derive the general result for the stationary quasi-potential shown on page. 6 What is the expression of the classical action S[x] using the Stratonovitch calculus convention? Is there any choice of calculus convention in quantum mechanics? Hugo Touchette (NITheP) Path integrals January 8 / 8