Rare Events, Extremely Rare Events and Fluctuations in a Thermodynamic System

TL;DR

This paper investigates the paths of particles in thermodynamic systems using a Metropolis algorithm to derive an Onsager-Machlup-like functional, revealing issues with continuous-time limits and thermodynamic consistency.

Contribution

It introduces a Metropolis-based derivation of the Onsager-Machlup functional and analyzes the thermodynamic inconsistencies in the continuous-time limit.

Findings

Sampling the Ito-Girsanov limit produces unphysical paths.

Unphysical effects stem from correlations violating thermodynamics.

The study highlights limitations of the continuous-time approach in thermodynamic path sampling.

Abstract

In this paper, we follow in the footsteps of Onsager and Machlup (OM) and consider diffusion-like paths that are explored by a particle moving via a conservative force while being in thermal equilibrium with its surroundings. Instead of considering diffusion (Brownian dynamics), we use a Metropolis algorithm to derive an OM-like functional. Through the lens of the Metropolis algorithm, we are able to elucidate the errors made when using a nonzero time increment. Of particular interest are transition paths that transverse an energy barrier that is large (but not too large) compared to the typical thermal energy. These transitions have probabilities that are only small and yet not so small as to be considered a violation of thermodynamics. As such, we turn our attention to the "double-ended" problem, where the OM functional can be interpreted as a "thermodynamic" action and employed to sample paths that start and end at predetermined points. We find that sampling the continuous-time limit of the OM functional, i.e., the Ito-Girsanov form, produces unphysical paths due to thermodynamic inconsistency arising from the singular nature of the limit. These unphysical effects are due to the correlation between the particles position and the random fluctuating noise that is introduced in the limiting process. Such a correlation is a direct consequence of the form of the Ito-Girsanov functional, does not originate from numerical inaccuracies, and is a violation of thermodynamics as embodied by Seifert's Integral Fluctuation Theorem.