Path integral solutions for Fokker–Planck conditional propagators in nonequilibrium systems: Catastrophic divergences of the Onsager–Machlup–Laplace approximation

Abstract

When the conditional propagator for a single-variable Fokker–Planck equation is represented in functional path integral form, it separates into a path-independent thermodynamic factor and a reversible, time-dependent kinetic factor. The validity of the Onsager–Machlup–Laplace (OML) approximation to the conditional propagator is determined solely by the mechanical potential associated with the kinetic factor. The OML approximation is exact if the underlying mechanical force is linear; Hongler’s model exemplifies a class of nonlinear Fokker–Planck equations that are solved exactly by the OML approximation, because the nonlinearity is confined to the thermodynamic factor. Contrary to earlier suggestions, neither the existence of turning points in the mean paths nor limited information content of the path of least thermodynamic action indicates the breakdown of the OML approximation. When classical paths on the mechanical potential surface coalesce, however, the OML approximation breaks down catastrophically. Divergences analogous to the classical catastrophes of scattering theory are found for extremal path durations; these extremal times must exist whenever the underlying mechanical potential surface possesses two or more nonidentical maxima. Path coalescence and the consequent divergence of the OML approximation are illustrated for a system with the deterministic kinetics of the Schlögl model and constant probability diffusion coefficient. Catastrophic divergences of the OML approximation may be predicted and classified by generalization of similar problems found in the stationary phase approximation of semiclassical scattering theory; additionally, methods of semiclassical collision theory suggest approaches for eliminating erroneous divergences of the conditional propagator for nonequilibrium thermodynamic systems.