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We examine the fundamental aspects of statistical mechanics, dividing theproblem into a discussion purely about probability, which we analyse from aBayesian standpoint. We argue that the existence of a unique maximisingprobability distribution $\{p(j\vert K)\}$ for states labelled by $j$ givendata $K$ implies that the corresponding maximal value of the informationentropy $\sigma(\{(p_j\vert K)\}) = -\sum_j (p_j \vert K)\ln{(p_j\vert K)}$depends explicitly on the data at equilibrium and on the Hamiltonian of thesystem. As such, it is a direct measure of our uncertainty about the exactstate of the body and can be identified with the traditional thermodynamicentropy. We consider the well known microcanonical, canonical and grandcanonical methods and ensure that the fluctuations about mean values aregenerally minuscule for macroscopic systems before identifying these meanvalues with experimental observables and thereby connecting to many standardresults from thermodynamics. Unexpectedly, we find that it is not generallypossible for a quantum process to be both isentropic and reversibly adiabatic.This is in sharp contrast to traditional thermodynamics where it is assumedthat isentropic, reversible adiabatic processes can be summoned up on demandand easily realised. By contrast, we find that linear relations betweenpressures $P_j$ and energies $E_j$ are necessary and sufficient conditions fora quasi-static and adiabatic change to be isentropic, but, of course, thisrelationship only holds for a few especially simple systems, such as theperfect gas, and is not generally true for more complicated systems. Byconsidering the associated entropy increases up to second order in small volumechanges we argue that the consequences are in practice negligible.

Probabilistic Foundations of Statistical Mechanics: A Bayesian Approach