Toward a theory of relaxation in correlated systems: diffusion in the phase space of a chaotic Hamiltonian

Abstract

Relaxation in correlated systems such as interacting ions, entangled polymer chains or viscous liquids requires a time-dependent relaxation rate for a full accounting of the observed phenomena. This has been demonstrated by the coupling scheme for relaxations in correlated systems. A fundamental theory of these necessarily involves non-integrable interactions or constraints which are known to produce chaos in Hamiltonian models. Van Kampen has shown that the transition rates in the master equation for a relaxing Hamiltonian system can be interpreted in terms of diffusion in phase space. In this paper, this approach is generalized to include chaotic Hamiltonian. It is demonstrated that general results from anomalous diffusion in fractals produce all the properties of the time-dependent relaxation rate required to explain the experiments. This leads to a conjecture on the non-Euclidean and possibly fractal phase sphase structure of relaxing chaotic Hamiltonians which would allow a basic understanding of the central results of the coupling scheme.