Internal Energy, Fundamental Thermodynamic Relation, and Gibbs’ Ensemble Theory as Emergent Laws of Statistical Counting - Hong Qian (Entropy)
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 <semantics><mover accent="true">ν^</mover></semantics> with respect to a probability prior <semantics>p</semantics>, when <semantics><mover accent="true">ν^</mover>≠p</semantics> as <semantics>N→∞</semantics>. Following Callen’s postulate and through Legendre–Fenchel transform, without help from mechanics, we show that an internal energy <semantics>u</semantics> 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. <semantics>u</semantics> is to <semantics><mover accent="true">ν^</mover></semantics> what chemical potential <semantics>μ</semantics> is to particle number N in Gibbs’ chemical thermodynamics, what <semantics>β=T−1</semantics> is to internal energy U in classical thermodynamics, and what <semantics>ω</semantics> is to t in Fourier analysis.