Parametric invariance

TL;DR

This paper explores the concept of parametric invariance across various physical theories, linking it to entropy and adiabatic processes, and extends classical invariants to systems with variable particle numbers.

Contribution

It provides a unified analysis of parametric invariance in classical, quantum, and statistical mechanics, and extends invariants to variable particle systems, connecting them to entropy.

Findings

Parametric invariance relates to entropy in thermodynamics.

Classical and quantum invariants are explicitly calculated.

Extension of Hertz invariant to variable particle systems.

Abstract

We examine the development of the concept of parametric invariance in classical mechanics, quantum mechanics, statistical mechanics, and thermodynamics, and particularly its relation to entropy. The parametric invariance was used by Ehrenfest as a principle related to the quantization rules of the old quantum mechanics. It was also considered by Rayleigh in the determination of pressure caused by vibration, and the general approach we follow here is based on his. Specific calculation of invariants in classical and quantum mechanics are determined. The Hertz invariant, which is a volume in phase space, is extended to the case of a variable number of particles. We show that the slow parametric change leads to the adiabatic process, allowing the definition of entropy as a parametric invariance.