Thermodynamic Entropy from Sadi Carnot's Cycle using Gauss' and Doll's-Tensor Molecular Dynamics

TL;DR

This paper connects macroscopic thermodynamic entropy with microscopic phase-space volume using novel Hamiltonian mechanics approaches applied to Carnot's cycle, enhancing understanding of thermodynamics from a microscopic perspective.

Contribution

It introduces two new Hamiltonian mechanics methods, Gauss' isokinetic and Doll's tensor, to analyze Carnot's cycle microscopically, bridging microscopic and macroscopic thermodynamics.

Findings

Microscopic and macroscopic views of Carnot's cycle are shown to be equivalent.

The methods are applied to ideal gases and simple fluids, demonstrating broad applicability.

The analysis deepens understanding of entropy as phase-space volume in thermodynamics.

Abstract

Carnot's four-part ideal-gas cycle includes both isothermal and adiabatic expansions and compressions. Analyzing this cycle provides the fundamental basis for statistical thermodynamics. We explore the cycle here from a pedagogical view in order to promote understanding of the macroscopic thermodynamic entropy, the state function associated with thermal energy changes. From the alternative microscopic viewpoint the Hamiltonian ${\cal H}(q,p)$ is the energy and entropy is the (logarithm of the) phase-space volume $\Omega$ associated with a macroscopic state. We apply two novel forms of Hamiltonian mechanics to Carnot's Cycle: [1] Gauss' isokinetic mechanics for the isothermal segments and [2] Doll's Tensor for the isentropic adiabatic segments. We explore the equivalence of the microscopic and macroscopic views of Carnot's cycle for simple fluids here, beginning with the ideal Knudsen gas and extending the analysis to a prototypical simple fluid.