A probabilistic mechanics theory for random dynamics

TL;DR

This paper develops a probabilistic mechanics framework extending classical laws to systems with random motion, enabling new insights into thermodynamics and entropy without relying on local equilibrium assumptions.

Contribution

It introduces a stochastic extension of classical mechanics principles, such as virtual work and least action, applicable to random and thermodynamic systems.

Findings

Derived entropy change in free gas expansion without local equilibrium

Linked entropy production to work by random forces in nonequilibrium

Showed violation of Liouville and Poincaré recurrence theorems in this framework

Abstract

This is a general description of a probabilistic formalism of mechanics, i.e., an extension of the Newtonian mechanics principles to the systems undergoing random motion. From an analysis of the induction procedure from experimental data to the Newtonian laws, it is shown that the experimental verification of Newton law in a random motion implies a stochastic extension of the virtual work principle and the least action principle, i.e., <dW>=0 and <dA>=0 averaged over all the random paths instead of dW=0 and dA=0 for single path in regular dynamics. A probabilistic mechanics is formulated and applied to thermodynamic system. Several known results, rules and principles can be reproduced and justified from this new point of view. To mention some, we have obtain the entropy variation of the free expansion of gas and heat conduction without considering local equilibrium, and a violation of the Liouville theorem and Poincar\'e recurrence theorem, which allows to relate the entropy production to the work performed by random forces in nonequilibrium process.