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

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 $Ω$ 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.