The Equivalence of Dissipation from Gibbs’ Entropy Production with Phase-Volume Loss in Ergodic Heat-Conducting Oscillators
Abstract
Gibbs’ thermodynamic entropy is given by the logarithm of the phase volume, which itself responds to heat transfer to and from thermal reservoirs. We compare the thermodynamic dissipation described by (i) phase-volume loss with (ii) heat-transfer entropy production. Their equivalence is documented for computer simulations of the response of an ergodic harmonic oscillator to thermostated temperature gradients. In the simulations one or two thermostat variables control the kinetic energy or the kinetic energy and its fluctuation. All of the motion equations are time-reversible. We consider both strong and weak control variables. In every case, the time-averaged dissipative loss of phase-space volume coincides with the entropy produced by heat transfer. Linear-response theory nicely reproduces the small-gradient results obtained by computer simulation. The thermostats considered here are ergodic and provide simple dynamical models, some of them with as few as three ordinary differential equations, while remaining capable of reproducing Gibbs’ canonical phase-space distribution and are precisely consistent with irreversible thermodynamics.
References
- [1990] “ Canonical ensemble averages from pseudomicrocanonical dynamics,” Phys. Rev. A 42, 5045–5048. Crossref, ISI, Google Scholar
- [1998] “ Note on phase-space contraction and entropy production in thermostated Hamiltonian systems,” Chaos 8, 357–365. Crossref, ISI, Google Scholar
- [1999] “ Entropy production and phase-space volume contraction,” Phys. Rev. E 59, 4000–4006. Crossref, ISI, Google Scholar
- [1985] “ Canonical dynamics: Equilibrium phase-space distributions,” Phys. Rev. A 31, 1695–1697. Crossref, ISI, Google Scholar
- [1996] “ Kinetic moments method for the canonical ensemble distribution,” Phys. Lett. A 211, 253–257. Crossref, ISI, Google Scholar
- [1997] “ Mécanique de nonéquilibre à la californienne,” Physica A 240, 1–11. Crossref, ISI, Google Scholar
- Hoover, W. G., Hoover, C. G. & Uribe, F. J. [2010] “Flexible macroscopic models for dense-fluid shockwaves: Partitioning heat and work; delaying stress and heat flux; two-temperature thermal relaxation,” Proc. Advanced Problems in Mechanics (1–5 July 2010), sponsored by the Institute for Problems in Mechanical Engineering under the patronage of the Russian Academy of Sciences, arXiv:1005.1525. Google Scholar
- [2015] “ Comparison of very smooth cell-model trajectories using five symplectic and two Runge–Kutta integrators,” Commun. Sci. Technol. 23, 109–116. Google Scholar
- [2015a] “ Deterministic time-reversible thermostats: Chaos, ergodicity, and the zeroth law of thermodynamics,” Molec. Phys. 113, 2863–2872. Crossref, ISI, Google Scholar
- [2015b] “ Hard disks and harmonic oscillators near and far from equilibrium,” Molec. Simul., in press. Google Scholar
- [2015c] “ Ergodic time-reversible chaos for Gibbs’ canonical oscillator,” Phys. Lett. A 379, 2935–2940. Crossref, ISI, Google Scholar
- [2016] “ Ergodicity of a singly-thermostated harmonic oscillator,” Commun. Nonlin. Sci. Numer. Simul. 32, 234–240. Crossref, ISI, Google Scholar
- [2010] “ Deterministic thermostats, theories of nonequilibrium systems and parallels with the ergodic condition,” J. Phys. 43, 1–42. Google Scholar
- [1993] “ Finite-temperature properties of sodium clusters,” Phys. Rev. B 48, 2721–2732. Crossref, ISI, Google Scholar
- [1990] “ Canonical ensembles from chaos,” Ann. Phys. 204, 155–185. Crossref, ISI, Google Scholar
- [1992] “ Canonical ensembles from chaos II: Constrained dynamical systems,” Ann. Phys. 214, 180–218. Crossref, ISI, Google Scholar
- [1992] “ Nosé–Hoover chains: The canonical ensemble via continuous dynamics,” J. Chem. Phys. 97, 2635–2643. Crossref, ISI, Google Scholar
- [1984a] “ A molecular dynamics method for simulations in the canonical ensemble,” Molec. Phys. 52, 255–268. Crossref, ISI, Google Scholar
- [1984b] “ A unified formulation of the constant temperature molecular dynamics methods,” J. Chem. Phys. 81, 511–519. Crossref, ISI, Google Scholar
- [2015] “ A general formalism for singly-thermostated Hamiltonian dynamics,” Phys. Rev. E 92, 052138. Crossref, ISI, Google Scholar
- [2000] “ On thermostats and entropy production,” Physica A 279, 21–29. Crossref, ISI, Google Scholar
| Remember to check out the Most Cited Articles! |
|---|
|
Check out our Bifurcation & Chaos |
