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Boltzmann stochastic thermodynamics.

Mário J de Oliveira1

  • 1Universidade de São Paulo, Instituto de Física, Rua do Matão, 1371, 05508-090 São Paulo, SP, Brasil.

Physical Review. E
|June 20, 2019
PubMed
Summary

This study derives the Boltzmann kinetic equation from a master equation describing stochastic dynamics. It shows how this stochastic approach generates entropy, unlike Hamiltonian dynamics.

Area of Science:

  • Statistical Mechanics
  • Thermodynamics
  • Kinetic Theory

Background:

  • The Boltzmann kinetic equation is fundamental to understanding gas behavior.
  • Hamiltonian dynamics, described by the Liouville equation, conserves entropy.
  • Stochastic processes offer an alternative framework for microscopic dynamics.

Purpose of the Study:

  • To derive the Boltzmann kinetic equation from a stochastic master equation.
  • To investigate the role of stochastic dynamics in entropy generation.
  • To provide a stochastic interpretation of Maxwell-Boltzmann kinetic theory.

Main Methods:

  • Formulating an integrodifferential master equation for isolated thermodynamic systems.
  • Analyzing stochastic evolution and entropy generation.

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  • Considering transition rates for two-particle collisions.
  • Examining the reduction to Fokker-Planck and Liouville equations under specific conditions.

    • Main Results:

    • The Boltzmann kinetic equation is derived from the stochastic master equation.
    • Stochastic dynamics leads to entropy generation, increasing Gibbs entropy.
    • Hamiltonian dynamics, via the Liouville equation, results in constant entropy.
    • The master equation reduces to a Fokker-Planck type equation for small-angle scattering.

    Conclusions:

    • A stochastic interpretation of Maxwell and Boltzmann's kinetic theory is presented.
    • The master equation framework unifies descriptions of stochastic and Hamiltonian dynamics.
    • This approach clarifies the microscopic origins of entropy increase in isolated systems.