Equivalence of kinetic-theory and random-matrix approaches to Lyapunov spectra of hard-sphere systems

Abstract

In the study of chaotic behaviour of systems of many hard spheres, Lyapunov exponents of small absolute value exhibit interesting characteristics leading to speculations about connections to non-equilibrium statistical mechanics. Analytical approaches to these exponents so far can be divided into two groups, macroscopically oriented approaches, using kinetic theory or hydrodynamics, and more microscopically oriented random-matrix approaches in quasi-one-dimensional systems. In this paper, I present an approach using random matrices and weak disorder expansion in an arbitrary number of dimensions. Correlations between subsequent collisions of a particle are taken into account. It is shown that the results are identical to those of a previous approach based on an extended Enskog-equation. I conclude that each approach has its merits, and provides different insights into the approximations made, which include the Stoßzahlansatz, the continuum limit, and the long-wavelength approximation. The comparison also gives insight into possible connections between Lyapunov exponents and fluctuations.