When heating isn’t cooling in reverse: Nosé–Hoover thermostat fluctuations from equilibrium symmetry to nonequilibrium asymmetry

Recent laboratory experimental work has shown that heating and cooling processes exhibit intrinsic asymmetry, with heating occurring more efficiently than cooling. Two decades earlier, nonequilibrium molecular dynamics simulations addressed the topic of heating one side of a computational cell while cooling the other side by applying two different thermostats, producing a sinusoidal temperature profile. We revisit the theory underlying those computer calculations and show how it accurately predicts the laboratory results. Recent realizations of a simple two-dimensional one-particle cell model give surprisingly relevant results, where two Nosé–Hoover thermostats are applied to the two directions (x and y) at two temperatures, Tx ≥ Ty. At equilibrium, the thermostatting rate variables ξx and ξy are identical, while under nonequilibrium temperature differences, the heating variable (ξx) is weaker than the cooling one (ξy), demonstrating the ratio of thermostat effort to thermal bias, just as predicted by theory: ⟨ξx⟩/⟨ξy⟩ = −Ty/Tx. We relate this to the negative rate of change in entropy of the nonequilibrium system that accompanies the contraction of phase space onto a fractal strange attractor of lower dimension. Histograms of the thermostat variables reveal the stark differences between equilibrium and nonequilibrium heat flow. At equilibrium, the cell-model ξ-distributions are both Gaussians centered at zero. We redid much earlier many-body simulations of a Nosé–Hoover thermostatted fluid at equilibrium, which reported that the distribution was biased toward heating; we discovered that the prior work suffered from systematic integration error. In fact, we find that the distribution at equilibrium is a totally symmetric Gaussian for many-body systems, in agreement with the cell model. Under nonequilibrium conditions, when Tx > Ty, the simple 2D cell model clearly demonstrates the microscopic origin of heating–cooling asymmetry, thereby strongly confirming the results of real-world laboratory experiments.