From Near-Integrable to Far-from-Integrable: A Unified Picture of Thermalization and Heat Transport

Whether and how a system approaches equilibrium is central in nonequilibrium statistical physics, crucial to understanding thermalization and transport. Bogoliubov's three-stage (initial, kinetic, and hydrodynamic) evolution hypothesis offers a qualitative framework, but quantitative progress has focused on near-integrable systems like dilute gases. In this work, we investigate the relaxation dynamics of a one-dimensional diatomic hard-point (DHP) gas, presenting a phase diagram that characterizes relaxation behavior across the full parameter space, from near-integrable to far-from-integrable regimes. We analyze thermalization (local energy relaxation in nonequilibrium states) and identify three universal dynamical regimes: (i) In the near-integrable regime, kinetic processes dominate, local energy relaxation decays exponentially, and the thermalization time $τ$ scales as $τ\propto δ^{-2}$. (ii) In the far-from-integrable regime, hydrodynamic effects dominate, energy relaxation decays power-law, and thermalization time scales linearly with system size $N$. (iii) In the intermediate regime, the Bogoliubov phase emerges, characterized by the transition from kinetic to hydrodynamic relaxation. The phase diagram also shows that hydrodynamic behavior can emerge in small systems when sufficiently far from the integrable regime, challenging the view that such effects occur only in large systems. In the thermodynamic limit, the system's relaxation depends on the order in which the limits ($N \to \infty$ or $δ\to 0$) are taken. We then analyze heat transport (decay of heat-current fluctuations in equilibrium), demonstrating its consistency with thermalization, leading to a unified theoretical description of thermalization and transport. Our approach provides a pathway for studying relaxation dynamics in many-body systems, including quantum systems.