Least action principle and stochastic motion : a generic derivation of path probability

TL;DR

This paper analytically derives the probability of stochastic paths in mechanical systems governed by Langevin dynamics, showing it depends exponentially on an action functional, with the most probable paths being those of least action.

Contribution

It provides a unified analytical derivation linking path probability to action for both dissipative and non-dissipative systems, extending previous numerical results.

Findings

Path probability depends exponentially on the action.

Most probable paths are those minimizing the action.

Results unify dissipative and non-dissipative stochastic dynamics.

Abstract

This work is an analytical calculation of the path probability for random dynamics of mechanical system described by Langevin equation with Gaussian noise. The result shows an exponential dependence of the probability on the action. In the case of non dissipative limit, the action is the usual one in mechanics in accordance with the previous result of numerical simulation of random motion. In the case of dissipative motion, the action in the exponent of the exponential probability is just the one proposed in a previous work (Q.A. Wang, R. Wang, arXiv:1201.6309), an action defined for the total system including the moving system and its environment receiving the dissipated energy. In both cases, the result implies that the most probable paths are the paths of least action which, in the limit of vanishing randomness, become the regular paths minimizing the action.