When Heating Isn't Cooling in Reverse: Nosé-Hoover Thermostat Fluctuations from Equilibrium Symmetry to Nonequilibrium Asymmetry

Abstract: Recent laboratory experiments suggest an intrinsic asymmetry between heating and cooling, with heating occurring more efficiently. Two decades earlier, molecular dynamics (MD) simulations had examined a related setup - heating one side of a computational cell while cooling the other via distinct thermostats. We revisit those calculations, recapitulating the underlying theory and showing that earlier MD results already hinted at the observed laboratory asymmetry. Recent realizations of a simple two-dimensional single-particle model, thermostatted in $x$ and $y$ at different temperatures, reproduces key features: at equilibrium, thermostat variables were identical, but under nonequilibrium conditions, the heating variable is weaker than the cooling one. At the same time, MD simulations from four decades ago by Evans and Holian reported a surprising skew in the Nose--Hoover thermostat variable $\xi$ under equilibrium - indicating a statistical bias in energy injection versus extraction. We revisit those results with exact reproduction of their setup. We show that when (1) the center-of-mass velocity is set to zero, (2) integration is done carefully with finite differencing, and (3) sampling is sufficiently long, the distribution of $\xi$ is symmetric and Gaussian with zero mean, as predicted by theory and validated by two independent error estimates. However, in the two-temperature cell, the distribution of thermostat variables become asymmetric, the cold bath requires significantly stronger damping than the hot bath requires anti-damping, with $\langle \xi_x \rangle / \langle \xi_y \rangle = -T_y/T_x$. This exact analytic relation links thermostat effort to thermal bias and the negative rate of change in the entropy of the system. These results identify the microscopic origin of heating-cooling asymmetry as a genuine nonequilibrium effect, consistent with experimental findings.