Stochastic motion in phase space on a surface of constant energy

TL;DR

This paper investigates stochastic dynamics of particles constrained to constant energy surfaces, demonstrating entropy increase and irreversibility despite energy conservation, contrasting with traditional Liouville dynamics.

Contribution

It introduces a novel stochastic framework that conserves energy exactly and reveals irreversible behavior in closed systems, unlike standard approaches.

Findings

Entropy increases over time, indicating irreversibility.

Stochastic dynamics on constant energy surfaces differ from Liouville equation predictions.

The approach maintains energy conservation while allowing entropy growth.

Abstract

We study closed systems of particles that are subject to stochastic forces in addition to the conservative forces. The stochastic equations of motion are set up in such a way that the energy is strictly conserved at all times. To ensure this conservation law, the evolution equation for the probability density is derived using an appropriate interpretation of the stochastic equation of motion that is not the It\^o nor the Stratonovic interpretation. The trajectories in phase space are restricted to the surface of constant energy. Despite this restriction, the entropy is shown to increase with time, expressing irreversible behavior and relaxation to equilibrium. This main result of the present approach contrasts with that given by the Liouville equation, which also describes closed systems, but does not show irreversibility.