On the (Boltzmann) Entropy of Nonequilibrium Systems

TL;DR

This paper explores Boltzmann's entropy for nonequilibrium systems, generalizing formulas for dilute gases and fluids, and demonstrates an ${ m H}$-theorem for macro-variables with autonomous evolution.

Contribution

It extends Boltzmann's entropy framework to nonequilibrium systems and establishes an ${ m H}$-theorem under deterministic macro-variable evolution.

Findings

Generalized Boltzmann entropy formulas for dilute gases and fluids.

Proved an ${ m H}$-theorem for macro-variables with autonomous dynamics.

Connected nonequilibrium entropy to deterministic evolution equations.

Abstract

Boltzmann defined the entropy of a macroscopic system in a macrostate $M$ as the $\log$ of the volume of phase space (number of microstates) corresponding to $M$. This agrees with the thermodynamic entropy of Clausius when $M$ specifies the locally conserved quantities of a system in local thermal equilibrium (LTE). Here we discuss Boltzmann's entropy, involving an appropriate choice of macro-variables, for systems not in LTE. We generalize the formulas of Boltzmann for dilute gases and of Resibois for hard sphere fluids and show that for macro-variables satisfying any deterministic autonomous evolution equation arising from the microscopic dynamics the corresponding Boltzmann entropy must satisfy an ${\cal H}$-theorem.