Derivation of Boltzmann Principle

TL;DR

This paper provides a modern, self-contained derivation of Boltzmann's principle from classical mechanical models, highlighting its historical roots and its significance as a bridge between classical and statistical mechanics.

Contribution

It offers a novel, accessible derivation of Boltzmann's principle based on classical mechanics and the heat theorem, connecting historical methods with modern understanding.

Findings

Derivation based on classical mechanical models and heat theorem

Historical analysis linking Helmholtz and Boltzmann's work

Clarification of Boltzmann principle's foundational role

Abstract

We present a derivation of Boltzmann principle $S_{B}=k_{B}\ln \mathcal{W}$ based on classical mechanical models of thermodynamics. The argument is based on the heat theorem and can be traced back to the second half of the nineteenth century with the works of Helmholtz and Boltzmann. Despite its simplicity, this argument has remained almost unknown. We present it in a modern, self-contained and accessible form. The approach constitutes an important link between classical mechanics and statistical mechanics.