Energy transfer in a fast-slow Hamiltonian system

TL;DR

This paper derives a nonlinear diffusion equation describing energy transfer in a lattice of weakly interacting geodesic flows, advancing the understanding of macroscopic behavior from microscopic Hamiltonian dynamics.

Contribution

It introduces a method to connect microscopic Hamiltonian systems with macroscopic diffusion equations in weakly coupled geodesic flows.

Findings

Energy evolution follows a nonlinear diffusion equation

Rescaling interactions and time reveals macroscopic energy behavior

First step toward deriving macroscopic equations from microscopic Hamiltonian dynamics

Abstract

We consider a finite region of a lattice of weakly interacting geodesic flows on manifolds of negative curvature and we show that, when rescaling the interactions and the time appropriately, the energies of the flows evolve according to a non linear diffusion equation. This is a first step toward the derivation of macroscopic equations from a Hamiltonian microscopic dynamics in the case of weakly coupled systems.