Solving the Liouville equation to probe relaxation in strongly nonlinear systems

Abstract

An established formal way to solve the Liouville equation for dynamical systems with hermitian Hamiltonians is to frame the problem via parametrization in a canonical ensemble. The goal is to solve for the dynamics in the canonical ensemble. This approach allows one to examine temperature-dependent relaxation functions as a part of the construction of the full solution to the Liouville equation. Of course, a full solution to the Liouville equation may not always be necessary as it is the relaxation function which often tends to be of interest. The formalism involves evaluation of infinite continued fractions and this can at times be an especially challenging problem. Here we explore the relaxation function in near-equilibrium systems in the presence of strong nonlinearity. Due to computational challenges, we consider small systems. However, even our small system considerations may be adequate to infer about the relaxation in large systems. We show that the nonlinear systems tend to lead to nonconvergent infinite continued fractions. These continued fractions cannot be approximated perturbatively. We close by presenting a new strategy to evaluate them, thereby demonstrating a successful way to examine relaxation processes in strongly nonlinear systems.