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

TL;DR

This paper investigates the asymmetry between heating and cooling in thermostatted systems, revealing that nonequilibrium conditions cause a bias in thermostat variables, linking microscopic dynamics to observed laboratory asymmetries.

Contribution

The study demonstrates that heating-cooling asymmetry arises from nonequilibrium effects, providing an exact analytic relation between thermostat effort and thermal bias, supported by revisited simulations and theory.

Findings

Equilibrium thermostat variables are symmetric and Gaussian.

Nonequilibrium conditions cause thermostat asymmetry with stronger cold bath damping.

Analytic relation links thermostat effort to thermal bias and entropy change.

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.