  We develop the argument that the Gibbs-von Neumann entropy is the appropriate
statistical mechanical generalisation of the thermodynamic entropy, for
macroscopic and microscopic systems, whether in thermal equilibrium or not, as
a consequence of Hamiltonian dynamics. The mathematical treatment utilises well
known results [Gib02, Tol38, Weh78, Par89], but most importantly, incorporates
a variety of arguments on the phenomenological properties of thermal states
[Szi25, TQ63, HK65, GB91] and of statistical distributions[HG76, PW78, Len78].
This enables the identification of the canonical distribution as the unique
representation of thermal states without approximation or presupposing the
existence of an entropy function. The Gibbs-von Neumann entropy is then
derived, from arguments based solely on the addition of probabilities to
Hamiltonian dynamics.
