# Boltzmann stochastic thermodynamics

## TL;DR

This paper derives the Boltzmann kinetic equation from a stochastic master equation framework, showing how entropy production arises in stochastic thermodynamics and connecting it to classical kinetic theory.

## Contribution

It presents a stochastic interpretation of Boltzmann's kinetic theory, deriving the Boltzmann and Fokker-Planck equations from a master equation approach.

## Key findings

- Derivation of Boltzmann equation from stochastic master equation.
- Reduction to Fokker-Planck equation for small-angle scattering.
- Contrast between stochastic and Hamiltonian dynamics in entropy evolution.

## Abstract

The Boltzmann kinetic equation is obtained from an integro-differential master equation that describes a stochastic dynamics in phase space of an isolated thermodynamic system. The stochastic evolution yields a generation of entropy, leading to an increase of Gibbs entropy, in contrast to a Hamiltonian dynamics, described by the Liouville equation, for which the entropy is constant in time. By considering transition rates corresponding to collisions of two particles, the Boltzmann equation is attained. When the angle of the scattering produced by collisions is small, the master equation is shown to be reduced to a differential equation of the Fokker-Planck type. When the dynamics is of the Hamiltonian type, the master equation reduces to the Liouville equation. The present approach is understood as a stochastic interpretation of the reasonings employed by Maxwell and Boltzmann in the kinetic theory of gases regarding the microscopic time evolution.