From Random Motion of Hamiltonian Systems to Boltzmann H Theorem and Second Law of Thermodynamics -- a Pathway by Path Probability

TL;DR

This paper proposes a new formalism for stochastic Hamiltonian systems where path probability depends exponentially on action, extending classical mechanics and providing insights into the Second Law of Thermodynamics and the Boltzmann H theorem.

Contribution

It introduces a generalized mechanics framework for random dynamics, modifying Liouville's theorem and connecting path probability with action to explain thermodynamic irreversibility.

Findings

Path probability depends exponentially on action.

Modified Liouville theorem allows phase density evolution.

Provides a pathway from Hamiltonian mechanics to thermodynamics.

Abstract

A numerical experiment of ideal stochastic motion of a particle subject to conservative forces and Gaussian noise reveals that the path probability depends exponentially on action. This distribution implies a fundamental principle generalizing the least action principle of the Hamiltonian-Lagrangian mechanics and yields an extended formalism of mechanics for random dynamics. Within this theory, Liouville theorem of conservation of phase density distribution must be modified to allow time evolution of phase density and consequently the Boltzmann H theorem. We argue that the gap between the regular Newtonian dynamics and the random dynamics was not considered in the criticisms of the H theorem.