On the classical approximation in the quantum statistics of equivalent particles

It is shown here that the microcanonical ensemble for a system of noninteracting bosons and fermions contains a subensemble of state vectors for which all particles of the system are distinguishable. This IQC (inner quantum-classical) subensemble is therefore fully classical, except for a rather extreme quantization of particle momentum and position, which appears as the natural price that must be paid for distinguishability. The contribution of the IQC subensemble to the entropy is readily calculated, and the criterion for this to be a good approximation to the exact entropy is a logarithmically strengthened form of the usual criterion for the validity of classical statistics in terms of the thermal de Broglie wavelength and the average volume per particle. Thus, it becomes possible to derive the Maxwell-Boltzmann distribution directly from the ensemble in the classical limit, using fully classical reasoning about the distinguishability of particles. The entropy is additive—theN! factor of the Boltzmann count cancels out in the course of the calculation, and the N! paradox is thereby resolved. The method of correct Boltzmann counting and the lowest term of the Wigner-Kirkwood series for the partition function are seen to be partly based on the IQC subensemble, and their partly nonclassical nature is clarified. The clear separation in the full ensemble of classical and nonclassical components makes it possible to derive the classical statistics of indistinguishable particles from their quantum statistics in a controlled, explicit way. This is particularly important for nonequilibrium theory. The treatment of molecular collisions along too-literally classical lines turns out to require exorbitantly high temperatures, although there are suggestions of indirect ways in which classical nonequilibrium theory might be justified at ordinary temperatures. The applicability of exact classical ergodic and mixing theory to systems at ordinary temperatures is called into question, although the general idea of coarse-graining is confirmed. The concepts on which the IQC idea is based are shown to give rise to a series development of thermostatistical quantities, starting with the distinguishable-particle approximation.This work was supported in part by the Air Force Office of Scientific Research, through Grants No. AF-AFSOR 557-64 and 557-67.