Thermodynamical equivalence of physical systems

TL;DR

This paper introduces a concept of thermodynamic equivalence between different physical systems, classifying them into classes based on their thermodynamic potentials and transformations, revealing deep connections among seemingly different systems.

Contribution

It formalizes the notion of thermodynamic equivalence, demonstrating its transitive property and classifying systems into equivalence classes based on their thermodynamic potentials and scaling behaviors.

Findings

Ideal classical and quantum Fermi gases form one class.

Systems with scaled interactions form another class.

Ideal Bose gases in any dimension form a third class.

Abstract

Two different physical systems are said to be thermodynamically equiv- alent if one of the thermodynamic potentials of the first system is pro- portional to the corresponding potential of the second system after expressing the state variables of the first system in terms of those of the second by a transformation reversible throughout the state pa- rameter domain. The thermodynamic equivalence has a transitive nature so that physical systems divide into classes of thermodynam- ically equivalent systems that have similar phase diagrams. A first class of thermodynamically equivalent systems is formed by the ideal classical and quantum Fermi gases, whatever the dimensions of the confining spaces, and the one dimensional hard rod gas. A second class is formed by the physical systems characterized by interactions that coincide by a scaling of the distance and the coupling constant. A third class is formed by the ideal Boses gases in arbitrary spatial dimensions. The thermodynamic equivalence can also be defined in a more general way. By so doing the first and the third class combine into a single class of thermodynamically equivalent systems.