A Probabilistic Foundation of Elementary Particle Statistics. Part II.

Abstract

The long history of ergodic and quasi-ergodic hypotheses provides the best example of the attempt to supply non-probabilistic justifications for the use of statistical mechanics in describing mechanical systems. In this paper we reverse the terms of the problem. We aim to show that accepting a probabilistic foundation of elementary particle statistics dispenses with the need to resort to ambiguous non-probabilistic notions like that of (in)distinguishability. In the quantum case, starting from suitable probability conditions, it is possible to deduce elementary particle statistics in a unified way. Following our approach Maxwell–Boltzmann statistics can also be deduced, and this deduction clarifies its status. Thus our primary aim in this paper is to give a mathematically rigorous deduction of the probability of a state with given energy for a perfect gas in statistical equilibrium; that is, a deduction of the equilibrium distributions for a perfect gas. A crucial step in this deduction is the statement of a unified statistical theory based on clearly formulated probability conditions from which the particle statistics follows. We believe that such a deduction represents an important improvement in elementary particle statistics, and a step towards a probabilistic foundation of statistical mechanics. The present Part II is devoted to this deduction. Part I presented the necessary tools. After the deduction of the probability of a state with given energy for a system in statistical equilibrium, we will propose in the last section a simple model giving an ergodic interpretation of the equilibrium distributions.