Canonical Thermodynamics

TL;DR

The paper addresses deriving the canonical occupancy-number distribution for a bosonic system and connects its Shannon entropy to thermodynamic entropy, challenging the standard grand-canonical Boltzmann picture in the canonical ensemble. It derives a multinomial canonical occupancy distribution $P_n = W_n \prod_c P_c^{n_c}$ by tracing over a permutation-invariant, maximally mixed universe and justifies typical behavior via quantum typicality (Levy's lemma). A key contribution is the realization that the entropy $H_n$ contains the Gibbs term $-\log(N!)$ plus a nontrivial term $\sum_c E\{\log(n_c!)\}$, which prevents a closed-form Boltzmann maximizer in general; a tractable approximation partitions categories into well-populated ${\cal C}_w$ and sparsely populated ${\cal C}_s$, yielding a non-exponential maximizing distribution except in special limits. In the classical limit, the framework yields Sackur–Tetrode-like expressions; comparisons with Bose–Einstein grand-canonical results show BE entropies can be higher, but enforcing $\beta=1/T$ in a multinomial-Boltzmann form is not generally compatible with constrained maximum-entropy; collectively, the work clarifies how thermodynamics can emerge from occupancy statistics without relying on microstate entropy as fundamental.

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.

Figures (4)

• Figure 1: Various entropies in $k$ units versus expected energy per particle in $k$ units. The circles mark the entropy of the product of multinomial distributions obtained from Einstein's constrained entropy maximization. The squares mark the entropy of the multinomial distribution with optimized categorical distribution. The asterisks mark the entropy of the multinomial-Boltzmann distribution. The dashed line is the entropy of the Boltzmann distribution $-\log(N!)$.
• Figure 2: Expected energy per particle in $k$ units vs. temperature in various cases. The dashed line is the energy of the Boltzmann distribution with $\beta=1/T$. The circles mark the energy of the Bose-Einstein statistics.The squares mark the energy of our proposed categorical distribution and temperature obtained from (\ref{['clausiusmult']}). The symbols $+$ mark the energy of the Boltzmann distribution and temperature obtained from (\ref{['clausiusmult']}).
• Figure 3: Various entropies in $k$ units versus temperature. The dashed line is the entropy of the Boltzmann distribution $-\log(N!)$, with $\beta=1/T$. The circles mark the Bose-Einstein entropy (\ref{['einsteinentropy']}). The asterisks mark the entropy of the Multinomial-Boltzmann distribution with $\beta=1/T$. The squares mark the entropy of the Multinomial distribution with our proposed categorical distribution and temperature obtained by imposing (\ref{['clausiusmult']}). The $+$ symbols mark the entropy of the Multinomial-Boltzmann distribution and temperature obtained by imposing (\ref{['clausiusmult']}).
• Figure 4: A detail of Fig. \ref{['fig:thermoentropy']}, see the caption to Fig. \ref{['fig:thermoentropy']}.