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 $\hat{\mathbf{\nu }}$ w.r.t. a probability prior $\mathbf{p}$. Following Callen's postulate and through Legendre-Fenchel transform, without help from mechanics, we show an internal energy $\mathbf{\mu }$ 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. $\mathbf{\mu }$ is to $\hat{\mathbf{\nu }}$ what $\omega$ is to $t$ in Fourier analysis.

5 pages

References in corpus (2)

• The large deviation approach to statistical mechanics
• Emergence and Breaking of Duality Symmetry in Generalized Fundamental Thermodynamic Relations

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• Matrix approach to generalized ensemble theory for nonequilibrium discrete systems