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

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Statistical counting ad infinitum is the holographic observable to a statistical dynamics with finite states under independent and identically distributed N 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 N in Gibbs' chemical thermodynamics, what β=T-1 is to internal energy U in classical thermodynamics, and what ω is to t in Fourier analysis.