We discuss deterministic diffusion in periodic billiard models in terms of the central limit theorem, i.e. the convergence of rescaled distributions to a limiting normal distribution; this is stronger than the usual requirement that the mean square displacement grow asymptotically linearly in time.
In a chaotic Lorentz gas, where the scatterers are discs, we find that one-dimensional position densities exhibit an oscillatory fine structure, which we show has its origin in the geometry of the billiard domain. Using this we conjecture how to strengthen the standard central limit theorem, and give a physical estimate of the rate of convergence. We also show that using a gaussian distribution of velocities changes the limiting shape of the distributions to a non-gaussian one.
The same methods show numerically that a non-chaotic polygonal channel also obeys the central limit theorem, which provides further information on the extent to which weakly-chaotic models can exhibit strong statistical properties.