TABLE OF CONTENTS Consider a system composed of N identical particles that interact with each other very weakly. That is, they can exchange energy, but the energy of each individual particle is otherwise independent of the other particles. This situation is completely analogous to that for a canonical ensemble in the sense that a physical system of nearly independent particles can be thought of as an "ensemble" with each particle being a member "system" of the ensemble. This is the same as simply renaming the terms used in the canonical ensemble theory. Then, the ensemble canonical distribution function becomes the particle distribution function. That is,
where f i is now the probability that a particle is in a quantum state i with energy ? i, N i is the number of such particles, and N is the total number of particles. Equation (3.1.1) is the Maxwell-Boltzmann distribution law for nearly independent particles.
But there is a basic flaw in the Maxwell-Boltzmann distribution function. Remember that in ensemble theory, the distribution function is derived from the number of complexions of the system, and since the systems are macroscopic, they are distinguishable. This fact is reflected in the combinatorial expression for the number of complexions, so (3.1.1) can be valid only if the nearly independent particles are distinguishable from each other. But we know from quantum theory that this is not the case. At the micro level, similar particles are indistinguishable. That is,...
