Emergence and Breaking of Duality Symmetry in Generalized Fundamental Thermodynamic Relations

TL;DR

This paper generalizes thermodynamics to arbitrary systems with a priori probabilities, revealing a duality symmetry between different entropies that emerges in repeated measurements and explaining its breaking in traditional thermodynamic limits.

Contribution

It introduces a generalized thermodynamic framework that replaces the infinite-size limit with a multiple-measurement limit, uncovering a duality symmetry and its conditions for breaking.

Findings

Duality symmetry between Massieu's and Gibbs' entropy in repeated measurements.

Breaking of symmetry when entropy is an Eulerian function satisfying Callen's postulate.

Provides a foundation for classical and nanothermodynamics based on measurement limits.

Abstract

Thermodynamics as limiting behaviors of statistics is generalized to arbitrary system with probability {\it a priori} where thermodynamic infinite-size limit is replaced by multiple-measurement limit. A duality symmetry between Massieu's and Gibbs' entropy arises in the limit of infinitely repeated observations, yielding the Gibbs equation and Hill-Gibbs-Duhem equation (HGDE) as dual pair. If a system has thermodynamic limit satisfying Callen's postulate, entropy being an Eulerian function, the symmetry is lost: the HGDE reduces to the Gibbs-Duhem equation. This theory provides a de-mechanized foundation for classical and nanothermodynamics and offers a framework for distilling emergence from large data, free from underlying details.