Geometrical derivation of the Boltzmann factor

TL;DR

This paper presents a geometric derivation of the Boltzmann factor from the microcanonical ensemble, offering new insights into its fundamental origin and applications like the Maxwell-Boltzmann distribution and wealth distribution.

Contribution

It introduces a geometric perspective to derive the Boltzmann factor, connecting microcanonical ensembles to practical distributions in physics and economics.

Findings

Boltzmann factor has a geometric origin

Derivation from the microcanonical ensemble

Applications to Maxwell-Boltzmann and wealth distributions

Abstract

We show that the Boltzmann factor has a geometrical origin. Its derivation follows from the microcanonical picture. The Maxwell-Boltzmann distribution or the wealth distribution in human society are some direct applications of this new interpretation.