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 ${\mathrm{h}}_{\mathrm{KS}}$ for a gas of hard spheres at various densities. The time scales defined by ${\lambda }_1$ and ${\mathrm{h}}_{\mathrm{KS}}$ 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 ${\mathrm{\tau }}_{\lambda }$≡1/${\lambda }_1$ is much smaller than the collision time ${\mathrm{\tau }}_{\mathrm{c}}$, whereas at high densities we find ${\mathrm{\tau }}_{\lambda }$≫:${\mathrm{\tau }}_{\mathrm{c}}$. 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 ${\mathrm{\tau }}_{\mathrm{KS}}$≡1/${\mathrm{h}}_{\mathrm{KS}}$ is the characteristic time for this relaxation process.