Internal Energy, Fundamental Thermodynamic Relation, and Gibbs' Ensemble Theory as Laws of Statistical Counting

Abstract: Counting ad infinitum is the holographic observable to a statistical dynamics with finite states under independent repeated sampling. Entropy provides the infinitesimal probability for an observed frequency $\hat{\boldsymbol{\nu}}$ w.r.t. a probability prior ${\bf p}$. Following Callen's postulate and through Legendre-Fenchel transform, without help from mechanics, we show an internal energy $\boldsymbol{\mu}$ emerges; it provides a linear representation of real-valued observables with full or partial information. Gibbs' fundamental thermodynamic relation and theory of ensembles follow mathematically. $\boldsymbol{\mu}$ is to $\hat{\boldsymbol{\nu}}$ what $\omega$ is to $t$ in Fourier analysis.