The Equivalence of Dissipation from Gibbs' Entropy Production with Phase-Volume Loss in Ergodic Heat-Conducting Oscillators
arXiv:1511.03201 · doi:10.1142/S0218127416500899
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 phase-volume loss with 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 precisely consistent with irreversible thermodynamics.
21 pages with five figures, accepted by the International Journal of Bifurcation and Chaos on 6 December 2015. This version begins with a historical glossary of statistical mechanical background as an aid to the mathematical readership