Research Discussion: Path integrals and conserved quantities in stochastic dynamics

Dr Steven Fitzgerald, University of Leeds. Part of the Leeds Applied Nonlinear Dynamics seminar series.

Starting from a Langevin equation for a particle at x(t) moving in a potential V(x), with white noise \eta(t) 

\dot x(t) = - V’(x) + \eta(t) 

a path integral expression for the transition probability from x(0) = 0 to x(T) = X can be derived almost immediately (Onsager-Machlup): 

P(X,T | 0,0)  = \int Dx \exp -S[x]/4D 

where D is the noise strength, \int Dx is a functional (path) integral over functions x(t) with the transition boundary conditions, and S is an action 

S[x] = \int_0^T (\dot x + V’)^2 dt >= 0

This is analogous to the Feynman path integral in quantum mechanics with the noise strength playing the role of hbar, except there is no factor of i (meaning the integral is more likely to converge), and the “Lagrangian” looks a bit weird. I will derive a few results familiar from basic stat mech, and discuss a possible general expression for the transition probability (aka Green function) for a general potential V in the weak noise limit. It turns out that the most probable path from (0,0) to (X,T) through the potential V can be mapped on to classical trajectories in an effective potential -|V’|^2. Generalisations are possible. Time permitting I could also discuss a related approach to calculating escape rates from a potential well driven by coloured noise.