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.

arXiv comment5 pages

References in corpus (3)

• The large deviation approach to statistical mechanics
• Emergence and Breaking of Duality Symmetry in Generalized Fundamental Thermodynamic Relations
• Representations and Divergences in the Space of Probability Measures and Stochastic Thermodynamics

Cited by in corpus (2)

• Discontinuous phase transition of feature detection in lateral predictive coding
• Matrix approach to generalized ensemble theory for nonequilibrium discrete systems