The Equivalence of Dissipation from Gibbs’ Entropy Production with Phase-Volume Loss in Ergodic Heat-Conducting Oscillators

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.