Energy Dissipation in Hamiltonian Chains of Rotators

TL;DR

This paper investigates how energy dissipates in a chain of Hamiltonian rotators with dissipation at one end, deriving bounds on dissipation rates and providing numerical evidence of their sharpness, linked to KAM theory.

Contribution

It introduces bounds on energy dissipation rates in Hamiltonian rotator chains with dissipation at one end, connecting these to KAM theory and non-resonant term decoupling.

Findings

Dissipation rates can be arbitrarily small in certain regimes.

Numerical evidence supports the sharpness of the derived bounds.

The behavior relates to decoupling of non-resonant terms in KAM theory.

Abstract

We discuss, in the context of energy flow in high-dimensional systems and Kolmogorov-Arnol'd-Moser (KAM) theory, the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numerical evidence that these bounds are sharp. We relate this to the decoupling of non-resonant terms as is known in KAM problems.