Second Law in Classical Non-Extensive Systems

TL;DR

This paper introduces a geometric approach to Thermo-Statistics that accurately describes equilibrium in complex, inhomogeneous systems where traditional canonical ensembles fail, extending the Second Law's applicability.

Contribution

It presents a new derivation of the Second Law based on a geometric foundation, enabling Thermo-Statistics to be applied to non-diluted and non-homogeneous systems.

Findings

Provides a geometric framework for Thermo-Statistics

Addresses limitations of canonical ensembles in complex systems

Resolves an open problem discussed by Uffink

Abstract

Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble, whereas canonical ones fail in the most interesting, mostly inhomogeneous, situations like phase separations or away from the thermodynamic ``limit'' (e.g. self-gravitating systems and small quantum systems). A new derivation of the Second Law is presented that respects these fundamental complications. Our ``geometric foundation of Thermo-Statistics'' opens the fundamental (axiomatic) application of Thermo-Statistics to non-diluted systems or to ``non-simple'' systems which are not similar to (homogeneous) fluids. Supprisingly, but also understandably, a so far open problem c.f. Uffink: cond-mat/0005327, page 50 and page 72.