Thermodynamics with generalized ensembles: The class of dual orthodes

TL;DR

This paper develops a theoretical framework for generalized ensembles in statistical physics based on Boltzmann's orthodes, introducing the class of dual orthodes that unify various ensemble types through a generalized equipartition theorem.

Contribution

It extends the concept of orthodes to a broader class called dual orthodes, unifying diverse ensembles and providing a mechanical foundation alternative to information-theoretic approaches.

Findings

Tsallis ensembles are orthodes and interpolate between canonical and microcanonical ensembles.

Dual orthodes admit both microcanonical-like and canonical-like parametrizations.

The theory demonstrates the equivalence of all dual orthodes.

Abstract

We address the problem of the foundation of generalized ensembles in statistical physics. The approach is based on Boltzmann's concept of orthodes. These are the statistical ensembles that satisfy the heat theorem, according to which the heat exchanged divided by the temperature is an exact differential. This approach can be seen as a mechanical approach alternative to the well established information-theoretic one based on the maximization of generalized information entropy. Our starting point are the Tsallis ensembles which have been previously proved to be orthodes, and have been proved to interpolate between canonical and microcanonical ensembles. Here we shall see that the Tsallis ensembles belong to a wider class of orthodes that include the most diverse types of ensembles. All such ensembles admit both a microcanonical-like parametrization (via the energy), and a canonical-like one (via the parameter $\beta$). For this reason we name them ``dual''. One central result used to build the theory is a generalized equipartition theorem. The theory is illustrated with a few examples and the equivalence of all the dual orthodes is discussed.