Rare Events, the Thermodynamic Action and the Continuous-Time Limit

TL;DR

This paper critically examines the use of the Onsager-Machlup functional in continuous-time limits for diffusion paths, revealing fundamental issues with its physical interpretation and the concept of most probable paths.

Contribution

It demonstrates that the Ito-Girsanov regularization leads to unphysical path ensembles and shows the incompatibility of thermodynamic action with continuous-time diffusion measures.

Findings

Ito-Girsanov measure produces unphysical path ensembles

Thermodynamic action is incompatible with continuous-time diffusion paths

Most Probable Path concept does not hold under these measures

Abstract

We consider diffusion-like paths that are explored by a particle moving via a conservative force while being in thermal equilibrium with its surroundings. To probe rare transitions, we use the Onsager-Machlup (OM) functional as a path probability distribution function for double-ended paths that are constrained to start and stop at predesignated points after a fixed time. We explore the continuous-time limit where the OM functional has been commonly regularized by using the Ito-Girsanov change of measure. When used as a path measure, the Ito-Girsanov expression generates an ensemble of double-ended paths that are unphysical. We expose the underlying reasons why this continuous-time limit does not, and cannot, generate a thermodynamic ensemble of paths. Furthermore, we show that the concept of the Most Probable Path and the Thermodynamic action are incompatible with such measures for discrete or continuous time diffusion processes.