Calculation of relaxation rates from microscopic equations of motion

For classical systems with anharmonic forces, Newton's equations for partic le trajectories are nonlinear, while Liouville's equation for the evolution of functions of position and momentum is linear and is solved by construct ing 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 dete rmine the solution to Liouville's equation for short times; but for long ti mes, the solutions decay exponentially at rates independent of the initial conditions. These are the relaxation rates of irreversible processes, and t hey arise in these calculations as the imaginary parts of the frequencies w here then are singularities in the analytic continuations of solutions to L iouville's equation. These rates are calculated fur two examples: the inver ted oscillator, which can be solved both analytically and numerically, and a charged particle in a periodic magnetic field, which can only be solved n umerically. In these systems, dissipation arises from traveling-wave soluti ons to Liouville's equation that couple low and high wave-number modes allo wing energy to Bow from disturbances that are coherent over large scales to disturbances on ever smaller scales finally becoming incoherent over micro scopic scales. These results suggest that dissipation in large scale motion of the system is a consequence of chaos in the small scale motion. [S1063- 651X(99)07405-X].