Canonical Thermodynamics

Abstract

The paper demonstrates that the canonical probability distribution of the occupancy numbers of a bosonic system is multinomial, and shows how the thermodynamics of the canonical system descends from this distribution. The categorical distribution (i.e. the one-particle probability distribution of occupancy of the quantum eigenstates allowed to a particle of the system) of the multinomial distribution should be derived from constrained maximization of the Shannon entropy of the multinomial distribution. However, since the multinomial distribution intractable, one must renounce to a closed-form solution to the constrained maximization problem. The analysis is then focused on the thermal state, that is characterized by the constraint on system's expected energy. In this case, the paper proposes to consider a suboptimal tractable categorical distribution, which is likely to be close to the actual categorical maximizer, and shows that the one-particle Boltzmann distribution is a good approximation to the actual categorical maximizer only in certain cases, including the classical regime. The unexpected result is that, in the general case, the approximation that we find to the categorical maximizer is not of exponential type, or, in other words, is not the one-particle Boltzmann distribution. As a consequence, the probability distribution of microstates is not of exponential type. As in the standard analysis, it is always equal to the product of factors but, in the general case, these factors are not the Boltzmann factors, therefore the probability of a microstate can be different from the probability of another microstate even when the two have the same energy.