Chaos in nonequilibrium two-temperature ( <i>T</i> <i>x</i> , <i>T</i> <i>y</i> ) Nosé–Hoover cell models

Abstract

We revisit a two-temperature Nosé–Hoover wanderer particle embedded in a two-dimensional periodic 2 × 2 cell with four smooth repulsive corners at (x, y) = (±1, ±1) to explore chaos with anisotropic thermostatting. The model employs separate thermostats in the x and y directions, enabling controlled deviations from equilibrium. By integrating the full six-dimensional equations of motion and computing the complete Lyapunov spectrum, we confirm chaos and quantify phase-space contraction from the fully resolved six-dimensional Lyapunov spectrum. The total contraction rate, interpreted as entropy production, increases nonlinearly with the thermostat anisotropy, deviating from the quadratic dependence expected from linear-response theory, Λ ∝ δ2. We analyze two functional forms for the entropy-production rate, Λ(δ) (with δ = 0.5 − Ty): (i) a quadratic-plus-quartic expansion, consistent with linear-response expectations, and (ii) a power law, Λ ∝ δ2.44. While the latter captures the low-driving regime slightly better, the former more accurately describes the strongly driven regime and remains consistent with linear-response theory near equilibrium. An empirical linear relation between dissipation and phase-space dimensionality loss is also identified, Λ ≈ (DKY − 6)/3, where DKY is the approximate Kaplan–Yorke dimension. Momentum statistics show a significant non-Gaussian behavior under strong driving. Despite its dissipative nature, the model remains strictly time-reversible, offering a pedagogically rich example of microscopic reversibility coexisting with macroscopic entropy production.