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

Statistical counting is the holographic observable to a statistical dynamics with finite states under independent and identically distributed sampling. Entropy provides the infinitesimal probability for an observed empirical frequency ν^ with respect to a probability prior p, when ν^≠p as N→∞. Following Callen's postulate and through Legendre-Fenchel transform, without help from mechanics, we show that an internal energy u 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. u is to ν^ what chemical potential μ is to particle number in Gibbs' chemical thermodynamics, what β=T-1 is to internal energy in classical thermodynamics, and what ω is to in Fourier analysis.