Dynamics of energy transport in a Toda ring

We present results on the relationships between persistent currents and the known conservation laws in the classical Toda ring. We also show that perturbing the integrability leads to a decay of the currents at long times with a time scale that is determined by the perturbing parameter. We summarize several known results concerning the Toda ring in one dimension, and present new results relating to the frequency, average kinetic and potential energy, and mean-square displacement in the cnoidal waves, as functions of the wave vector and a parameter that determines the non linearity.