Mixing, Lyapunov instability, and the approach to equilibrium in a hard-sphere gas

We present maximum Lyapunov exponents lambda(1) and related Kolmogorov-Sinai entropies h(KS) for a gas of hard spheres at various densities. The time scales defined by lambda(1) and h(KS) are compared with the collision time, the decay time of typical autocorrelation functions, and the relaxation time of a one-particle distribution. At low densities the Lyapunov time tau(lambda)=1/lambda(1) is much smaller than the collision time tau(c), whereas at high densities we find tau(lambda)much greater than tau(c). We discuss consequences for kinetic theory and numerical simulations. The mixing properties in phase space are investigated for the two-dimensional Lorentz gas. It is numerically verified that the Kolmogorov-Sinai time tau(KS)=1/h(KS) is the characteristic time for this relaxation process.