Heat conduction in systems with Kolmogorov-Arnold-Moser phase space structure

TL;DR

This paper investigates heat conduction in a sinusoidal billiard channel, analyzing how regular and chaotic dynamics influence temperature profiles and heat flux, with a focus on the transition to chaos and anomalous diffusion behaviors.

Contribution

It provides analytical results for heat conduction in systems with KAM structures and proposes a universal temperature profile formula applicable across ripple amplitudes.

Findings

Regular dynamics lead to ballistic heat conduction.

Transition to chaos occurs as ripple amplitude increases.

Anomalous diffusion characterized by t ln(t) growth when no KAM curves are present.

Abstract

We study heat conduction in a billiard channel formed by two sinusoidal walls and the diffusion of particles in the corresponding channel of infinite length; the latter system has an infinite horizon, i.e., a particle can travel an arbitrary distance without colliding with the rippled walls. For small ripple amplitudes, the dynamics of the heat carriers is regular and analytical results for the temperature profile and heat flux are obtained using an effective potential. The study also proposes a formula for the temperature profile that is valid for any ripple amplitude. When the dynamics is regular, ballistic conductance and ballistic diffusion are present. The Poincar\'e plots of the associated dynamical system (the infinitely long channel) exhibit the generic transition to chaos as ripple amplitude is increased.When no Kolmogorov-Arnold-Moser (KAM) curves are present to forbid the connection of all chaotic regions, the mean square displacement grows asymptotically with time t as tln(t).