Calculation of relaxation rates from microscopic equations of motion

For classical systems with anharmonic forces, Newton's equations for particle trajectories are nonlinear, while Liouville's equation for the evolution of functions of position and momentum is linear and is solved by constructing a basis of functions in which the Liouvillian is a tridiagonal matrix, which is then diagonalized. For systems that are chaotic in the sense that neighboring trajectories diverge exponentially, the initial conditions determine the solution to Liouville's equation for short times; but for long times, the solutions decay exponentially at rates independent of the initial conditions. These are the relaxation rates of irreversible processes, and they arise in these calculations as the imaginary parts of the frequencies where there are singularities in the analytic continuations of solutions to Liouville's equation. These rates are calculated for two examples: the inverted oscillator, which can be solved both analytically and numerically, and a charged particle in a periodic magnetic field, which can only be solved numerically. In these systems, dissipation arises from traveling-wave solutions to Liouville's equation that couple low and high wave-number modes allowing energy to flow from disturbances that are coherent over large scales to disturbances on ever smaller scales finally becoming incoherent over microscopic scales. These results suggest that dissipation in large scale motion of the system is a consequence of chaos in the small scale motion.