Adiabatic invariance with first integrals of motion

TL;DR

This paper discusses the extension of adiabatic invariance concepts to systems with additional first integrals of motion, linking classical thermodynamics, dynamical methods, and integrals of motion to better understand entropy calculations.

Contribution

It proposes an extension of adiabatic invariance formalism to systems with extra first integrals, explaining its relevance to entropy computation in classical fluids.

Findings

Extension of adiabatic invariance to systems with first integrals.

Connection between adiabatic invariance and entropy calculation methods.

Insight into the success of dynamical entropy methods in classical fluids.

Abstract

The construction of a microthermodynamic formalism for isolated systems based on the concept of adiabatic invariance is an old but seldom appreciated effort in the literature, dating back at least to P. Hertz [Ann. Phys. (Leipzig) 33, 225 (1910)]. An apparently independent extension of such formalism for systems bearing additional first integrals of motion was recently proposed by Hans H. Rugh [Phys. Rev. E 64, 055101 (2001)], establishing the concept of adiabatic invariance even in such singular cases. After some remarks in connection with the formalism pioneered by Hertz, it will be suggested that such an extension can incidentally explain the success of a dynamical method for computing the entropy of classical interacting fluids, at least in some potential applications where the presence of additional first integrals cannot be ignored.