Lax dynamics

Abstract

A novel approach is proposed to characterize the dynamics of perturbed many-body integrable systems. Focusing on the paradigmatic case of the Toda chain under non-integrable Hamiltonian perturbations, this study introduces a method based the time evolution of the Lax eigenvalues ${\lambda }_{\alpha }$ as a proxy of the quasi-particles velocities and of the perturbed Toda actions. A set of exact equations of motion for the ${\lambda }_{\alpha }$ is derived that closely resemble those for eigenenergies of a quantum problem (also known as the Pechukas-Yukawa gas). Numerical simulations suggest that the invariant measure of such dynamics is basically the thermal density of states of the Toda lattice, regardless of the form of the perturbation.

arXiv commentTo appear in EPL

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