  We examine the fundamental aspects of statistical mechanics, dividing the
problem into a discussion purely about probability, which we analyse from a
Bayesian standpoint. We argue that the existence of a unique maximising
probability distribution $\{p(j\vert K)\}$ for states labelled by $j$ given
data $K$ implies that the corresponding maximal value of the information
entropy $\sigma(\{(p_j\vert K)\}) = -\sum_j (p_j \vert K)\ln{(p_j\vert K)}$
depends explicitly on the data at equilibrium and on the Hamiltonian of the
system. As such, it is a direct measure of our uncertainty about the exact
state of the body and can be identified with the traditional thermodynamic
entropy. We consider the well known microcanonical, canonical and grand
canonical methods and ensure that the fluctuations about mean values are
generally minuscule for macroscopic systems before identifying these mean
values with experimental observables and thereby connecting to many standard
results from thermodynamics. Unexpectedly, we find that it is not generally
possible for a quantum process to be both isentropic and reversibly adiabatic.
This is in sharp contrast to traditional thermodynamics where it is assumed
that isentropic, reversible adiabatic processes can be summoned up on demand
and easily realised. By contrast, we find that linear relations between
pressures $P_j$ and energies $E_j$ are necessary and sufficient conditions for
a quasi-static and adiabatic change to be isentropic, but, of course, this
relationship only holds for a few especially simple systems, such as the
perfect gas, and is not generally true for more complicated systems. By
considering the associated entropy increases up to second order in small volume
changes we argue that the consequences are in practice negligible.
