Thermodynamic Entropy from Sadi Carnot’s Cycle using Gauss’ and Doll’s-Tensor Molecular Dynamics
Hoover William G. *, Hoover Carol G.
Ruby Valley Research Institute
601 Highway Contract 60
Ruby Valley, Nevada 89833, USA∗E-mail: hooverwilliam@yahoo.com
Received:
Received: 29 January 2022; in final form: 25 February 2022; accepted: 26 February 2022; published online: 8 March 2022
DOI: 10.12921/cmst.2022.0000003
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 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 mechanics 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.
Key words:
Carnot cycle, entropy, nonequilibrium molecular dynamics, reversible processes, states
References:
[1] W.G. Hoover, Sec. 1.9: Gauss’ Principle and Nonholonomic Constraints and Sec. 2.5: Second Law of Thermodynamics, [In:] Computational Statistical Mechanics, Elsevier, New York (1991). The latter Section details the Carnot Cycle from the macroscopic viewpoint.
[2] W.G. Hoover, A.J.C. Ladd, R.B. Hickman, B.L. Holian, Bulk Viscosity via Nonequilibrium and Equilibrium Molecular Dynamics, Phys. Rev. A 21, 1756–1760 (1980).
[3] H.A. Posch, W.G. Hoover, F.J. Vesely, Canonical Dynamics of the Nosé Oscillator: Stability, Order, and Chaos, Phys. Rev. A 33, 4253–4265 (1986).
[4] D.J. Evans, W.G. Hoover, B.H. Failor, B. Moran, A.J.C. Ladd, Nonequilibrium Molecular Dynamics via Gauss’ Principle of Least Constraint, Phys. Rev. A 28, 1016–1021 (1983).
[5] W.G. Hoover, H.A. Posch, Shear Viscosity via Global Control of Spatiotemporal Chaos in Two-Dimensional isoenergetic Dense Fluids, Phys. Rev. E 51, 273–279 (1995).
[6] D.M. Gass, Enskog Theory for a Rigid Disk Fluid, J. Chem. Phys. 54, 1898–1902 (1971).
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 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 mechanics 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.
Key words:
Carnot cycle, entropy, nonequilibrium molecular dynamics, reversible processes, states
References:
[1] W.G. Hoover, Sec. 1.9: Gauss’ Principle and Nonholonomic Constraints and Sec. 2.5: Second Law of Thermodynamics, [In:] Computational Statistical Mechanics, Elsevier, New York (1991). The latter Section details the Carnot Cycle from the macroscopic viewpoint.
[2] W.G. Hoover, A.J.C. Ladd, R.B. Hickman, B.L. Holian, Bulk Viscosity via Nonequilibrium and Equilibrium Molecular Dynamics, Phys. Rev. A 21, 1756–1760 (1980).
[3] H.A. Posch, W.G. Hoover, F.J. Vesely, Canonical Dynamics of the Nosé Oscillator: Stability, Order, and Chaos, Phys. Rev. A 33, 4253–4265 (1986).
[4] D.J. Evans, W.G. Hoover, B.H. Failor, B. Moran, A.J.C. Ladd, Nonequilibrium Molecular Dynamics via Gauss’ Principle of Least Constraint, Phys. Rev. A 28, 1016–1021 (1983).
[5] W.G. Hoover, H.A. Posch, Shear Viscosity via Global Control of Spatiotemporal Chaos in Two-Dimensional isoenergetic Dense Fluids, Phys. Rev. E 51, 273–279 (1995).
[6] D.M. Gass, Enskog Theory for a Rigid Disk Fluid, J. Chem. Phys. 54, 1898–1902 (1971).
