Kolmogorov-Sinai entropy and Lyapunov spectra of a hard-sphere gas

Abstract

The mixing behavior of a hard-sphere gas has its origin in the exponential growth of small perturbations in phase space. This instability is characterized by the so-called Lyapunov exponents. In this work, we compute full spectra of Lyapunov exponents for the hard-sphere gas for a wide range of densities ϱ and particle numbers by using a recently developed algorithm. In the dilute-gas regime, the maximum Lyapunov exponent is found to obey the Krylov relation λ ∝ ϱ ln ϱ, a formula exactly derived for the low-density Lorentz gas by Dorfman and van Beijeren. We study the system-size dependence and the effect of the fluid-solid-phase transition on the spectra. In the second part of this work we describe and test a direct simulation Monte Carlo method (DSMC) for the computation of Lyapunov spectra and present results for dilute hard-sphere gases. Excellent agreement is obtained with the results of the deterministic simulations. This suggests that the Lyapunov instability of a hard sphere gas may be analyzed within the framework of kinetic theory.