Decay of correlations, Lyapunov exponents and anomalous diffusion in the Sinai billiard

We compute the decay of the velocity autocorrelation function, the Lyapunov exponent and the diffusion constant for the Sinai billiard within the framework of dynamical zeta functions. The asymptotic decay of the velocity autocorrelation function is found to be $C(t)\sim c(R)\mathrm{/}t$. The Lyapunov exponent for the corresponding map agrees with the conjectured limit ${\lambda }_{map}\to -2\log (R)+C$ as $R\to 0$ where $C=1-4\log 2+27/(2\pi^2)\cdot \zeta(3)$. The diffusion constant of the associated Lorentz gas is found to be divergent $D(t)\sim \log t$.