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.