From near-integrable to far-from-integrable: A unified picture of thermalization and heat transport

Whether and how a system approaches equilibrium is a central issue in nonequilibrium statistical physics, underpinning our understanding of both thermalization and heat transport. Bogoliubov's three-stage (initial, kinetic, and hydrodynamic) evolution hypothesis provides a qualitative framework, but quantitative advances have mostly been restricted to near-integrable systems such as dilute gases. In this work, we investigate the relaxation dynamics of a one-dimensional diatomic hard-point gas and present a phase diagram that comprehensively characterizes relaxation behavior across the full parameter space, from near-integrable to far-from-integrable regimes. We first analyze thermalization (local energy relaxation in nonequilibrium states) and identify three universal dynamical regimes: (1) In the near-integrable regime, kinetic processes dominate (initial <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mo>+</a:mo> </a:math> kinetic phase), local energy relaxation decays exponentially, and the thermalization time <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>τ</b:mi> </b:math> scales with perturbation strength <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>δ</c:mi> </c:math> as <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>τ</d:mi> <d:mo>∝</d:mo> <d:msup> <d:mi>δ</d:mi> <d:mrow> <d:mo>−</d:mo> <d:mn>2</d:mn> </d:mrow> </d:msup> </d:mrow> </d:math> . (2) In the far-from-integrable regime, hydrodynamic effects dominate (initial <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mo>+</e:mo> </e:math> hydrodynamic phase), local energy relaxation follows power-law decay, and thermalization time scales linearly with system size <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>N</f:mi> </f:math> . (3) In the intermediate regime, Bogoliubov phase emerges, marked by the transition from kinetic to hydrodynamic relaxation. The phase diagram further reveals that hydrodynamic behavior can emerge even in small systems when the system is sufficiently far from the integrable regime, challenging the conventional view that such effects are only significant in large systems. In the thermodynamic limit, the system's relaxation behavior depends on the order in which the limits ( <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>N</g:mi> <g:mo>→</g:mo> <g:mi>∞</g:mi> </g:mrow> </g:math> or <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mi>δ</h:mi> <h:mo>→</h:mo> <h:mn>0</h:mn> </h:mrow> </h:math> ) are taken. We then turn to heat transport (decay of heat-current fluctuations in equilibrium) and demonstrate that its behavior is fully consistent with the thermalization process, leading to the first unified theoretical description of thermalization and heat transport. Our results establish a comprehensive view of relaxation in classical many-body systems and suggest a route for studying similar dynamics in quantum systems, highlighting universal scaling laws and the interplay of kinetic and hydrodynamic processes across integrability regimes.