Phenomenological thermodynamics in a nutshell

TL;DR

This paper provides a rigorous, mathematically grounded overview of phenomenological equilibrium thermodynamics for single-phase systems, emphasizing stability and foundational principles derived from statistical mechanics.

Contribution

It offers a modified, more fundamental approach based on statistical mechanics, improving the mathematical rigor of thermodynamic stability analysis over Callen's traditional framework.

Findings

Derives thermodynamic stability from kinematical properties of states.

Provides a rigorous mathematical foundation for equilibrium thermodynamics.

Facilitates application to standard textbook examples.

Abstract

This paper gives a concise, mathematically rigorous description of phenomenological equilibrium thermodynamics for single-phase systems in the absence of chemical reactions and external forces. The present approach is similar to that of Callen, who introduces in his well-known thermodynamics book the basic concepts by means of a few postulates from which everything else follows. His setting is modified to match the more fundamental approach based on statistical mechanics. Thermodynamic stability is derived from kinematical properties of states outside equilibrium by rigorous mathematical arguments, superseding Callen's informal arguments that depend on a dynamical assumption close to equilibrium. From the formulas provided, it is an easy step to go to various examples and applications discussed in standard textbooks such as Callen or Reichl. A full discussion of global equilibrium would also involve the equilibrium treatment of multiple phases and chemical reactions. Since their discussion offers no new aspects compared with traditional textbook treatments, they are not treated here.