Stochastic action principle and maximum entropy

TL;DR

This paper revisits the stochastic action principle, demonstrating that path probabilities depend exponentially on average Lagrangian action, and unifies maximum entropy and least action principles within stochastic dynamics.

Contribution

It introduces a unified framework linking maximum entropy and classical mechanics principles through stochastic dynamics and path uncertainty measures.

Findings

Path probability depends exponentially on average Lagrangian action.

Path entropy can be measured by Shannon information.

Maximum entropy and least action principles are unified in stochastic dynamics.

Abstract

A stochastic action principle for stochastic dynamics is revisited. We present first numerical diffusion experiments showing that the diffusion path probability depend exponentially on average Lagrangian action. This result is then used to derive an uncertainty measure defined in a way mimicking the heat or entropy in the first law of thermodynamics. It is shown that the path uncertainty (or path entropy) can be measured by the Shannon information and that the maximum entropy principle and the least action principle of classical mechanics can be unified into a concise form. It is argued that this action principle, hence the maximum entropy principle, is simply a consequence of the mechanical equilibrium condition extended to the case of stochastic dynamics.