Geometric Microcanonical Thermodynamics for Systems with First Integrals

TL;DR

This paper develops a geometric microcanonical thermodynamics framework for many-body Hamiltonian systems with multiple conserved quantities, providing explicit formulas for thermodynamic quantities as microcanonical averages, applicable even for non-separable Hamiltonians.

Contribution

It introduces a differential geometric approach to derive microcanonical entropy and thermodynamic derivatives in systems with multiple conserved quantities, extending applicability to non-separable Hamiltonians.

Findings

Derived explicit formulas for entropy and its derivatives.

Showed thermodynamic quantities as microcanonical averages.

Applicable to non-separable Hamiltonian systems.

Abstract

In the general case of a many-body Hamiltonian system, described by an autonomous Hamiltonian $H$, and with $K\geq 0$ independent conserved quantities, we derive the microcanonical thermodynamics. By a simple approach, based on the differential geometry, we derive the microcanonical entropy and the derivatives of the entropy with respect to the conserved quantities. In such a way, we show that all the thermodynamical quantities, as the temperature, the chemical potential or the specific heat, are measured as a microcanonical average of the appropriate microscopic dynamical functions that we have explicitly derived. Our method applies also in the case of non-separable Hamiltonians, where the usual definition of kinetic temperature, derived by the virial theorem, does not apply.