Origin of the Zeroth Law of Thermodynamics and its Role in Statistical Mechanics

Kim Sharp

Published 2025-02-07Version 1

In statistical mechanics the zeroth law of thermodynamics is taken as a postulate which, as its name indicates, logically precedes the first and second laws. Treating it as a postulate has consequences for how temperature is introduced into statistical mechanics and for the molecular interpretation of temperature. One can, however, derive the zeroth law from first principles starting from a classical Hamiltonian using basic mechanics and a geometric representation of the phase space of kinetic energy configurations - the velocity hypersphere. In this approach there is no difficulty in providing a molecular interpretation of temperature, nor in deriving equality of temperature as the condition of thermal equilibrium. The approach to the macroscopic limit as a function of the number of atoms is easily determined. One also obtains with little difficulty the Boltzmann probability distribution, the statistical mechanical definition of entropy and the configuration partition function. These relations, along with the zeroth law, emerge as straightforward consequences of atoms in random motion.