Translation of Ludwig Boltzmann’s Paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium” Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI 1877, pp 373-435 (Wien. Ber. 1877, 76:373-435). Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, p. 164-223, Barth, Leipzig, 1909

Kim Sharp,Franz Matschinsky

doi:10.3390/e17041971

Abstract

Translation of the seminal 1877 paper by Ludwig Boltzmann which for the first time established the probabilistic basis of entropy. Includes a scientific commentary.

Highlights

Any of Boltzmann’s original scientific work is available in translation. This is remarkable given his central role in the development of both equilibrium and non-equilibrium statistical mechanics, his statistical mechanical explanation of entropy, and our understanding of the Second Law of thermodynamics
Liouville’s theorem, that dV remains constant in time (See Equation (41)), and so it cannot describe the entropy increase upon approach to equilibrium that Boltzmann was so concerned with
In the mechanical theory of heat we are always dealing with extremely large numbers of molecules, so such small differences disappear, and our approximate formula provides an exact solution to the problem

Summary

Introduction

Any of Boltzmann’s original scientific work is available in translation. This is remarkable given his central role in the development of both equilibrium and non-equilibrium statistical mechanics, his statistical mechanical explanation of entropy, and our understanding of the Second Law of thermodynamics. Liouville’s theorem, that dV remains constant in time (See Equation (41)), and so it cannot describe the entropy increase upon approach to equilibrium that Boltzmann was so concerned with He avoids at the outset the considerable difficulty Gibbs had accounting for changes in entropy with time (see p144 in [14]) Boltzmann gave us, for the first time, a definition of entropy applicable to every state (distribution), at equilibrium or not: “ the entropy of the initial and final states is not defined, (Tr: by Clausius’ equation) but one can still calculate the quantity which we have called the permutability measure”, LB1877, Section V. The calculations are very difficult; to facilitate understanding, I will, as in earlier work, consider a limiting case

Kinetic Energy Has Discrete Values

Kinetic Energies Exchange in a Continuous Manner

Consideration of Polyatomic Gas Molecules and External Forces