Discrete calculus of variations and Boltzmann distribution without Stirling's approximation

TL;DR

This paper introduces a discrete calculus of variations framework that accurately derives the Boltzmann distribution without relying on Stirling's approximation, addressing theoretical limitations.

Contribution

It presents a novel double extrema form of discrete calculus of variations that directly yields the Boltzmann distribution without Stirling's approximation.

Findings

Derivation of the Boltzmann distribution without Stirling's approximation

Introduction of a double extrema form of discrete calculus of variations

Elimination of common theoretical issues in statistical mechanics

Abstract

A \emph{double extrema form} of the calculus of variations is put forward in which only the smallest one of the finite differences is physically meaningful to represent the variational derivatives defined on the discrete points. The most probable distribution for the Boltzmann system is then reproduced without the Stirling's approximation, and free from other theoretical problems.