Eur. Phys. J. H 40, 489–526 (2015)
https://doi.org/10.1140/epjh/e2015-60034-2

The development of ensemble theory

A new glimpse at the history of statistical mechanics

The University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo, 153-8902, Japan

^a e-mail: Hajime.Inaba@gmail.com

Received: 21 July 2015
Received in final form: 28 September 2015
Published online: 13 November 2015

Abstract

This paper investigates the history of statistical mechanics from the viewpoint of the development of the ensemble theory from 1871 to 1902. In 1871, Ludwig Boltzmann introduced a prototype model of an ensemble that represents a polyatomic gas. In 1879, James Clerk Maxwell defined an ensemble as copies of systems of the same energy. Inspired by H.W. Watson, he called his approach “statistical”. Boltzmann and Maxwell regarded the ensemble theory as a much more general approach than the kinetic theory. In the 1880s, influenced by Hermann von Helmholtz, Boltzmann made use of ensembles to establish thermodynamic relations. In Elementary Principles in Statistical Mechanics of 1902, Josiah Willard Gibbs tried to get his ensemble theory to mirror thermodynamics, including thermodynamic operations in its scope. Thermodynamics played the role of a “blind guide”. His theory of ensembles can be characterized as more mathematically oriented than Einstein’s theory proposed in the same year. Mechanical, empirical, and statistical approaches to foundations of statistical mechanics are presented. Although it was formulated in classical terms, the ensemble theory provided an infrastructure still valuable in quantum statistics because of its generality.