For the periodic Lorenz gas Sinai showed that rescaling the trajectory of the tracer particle yields Brownian motion in the limit. Does there exist a similar result for the random Lorenz gas? If not, do people believe that there is such a limit?


By the random Lorenz gas I mean: take circular scatterers distributed uniformly at random in the plane conditioned on the scatterers not overlapping. The scatterers are fixed. The tracer particle is a point that moves with constant speed and has perfectly elastic collisions with the scatterers. An initial condition is chosen at random (say, by picking an initial point away from a scatterer and then picking the initial angle uniformly on $[0,2\pi)$.)


The numerical experiments in [Dettmann and Cohen, 2000](https://doi.org/10.1023/A:1026477605331) suggest that there is diffusive behaviour for the random Lorenz gas. This [article by Bunimovich](http://www.bookrags.com/tandf/lorentz-gas-tf/) states that it is believed that velocity autocorrelation decays polynomially, but does not mention whether it decays fast enough for there to be a finite diffusion coefficient.



Does the random Lorenz gas have a non-trivial diffusion coefficient?
I don't think such a theorem has been proved for the random Lorentz gas. First I want to point out that Sinai proved those scaling limit results for the case of (2D) periodic Lorentz gas with *finite horizon* (finite maximum free path for the particle). The case of periodic Lorentz gas with unbounded horizon was studied until recently, and a lot of places reference the work by Szasz and Varju (["Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon"](https://doi.org/10.1007/s10955-007-9367-0)). One gets analogous diffusion results but the scaling factor is different here.


A very recent article which seems to do a good job at surveying these results is ["New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis"](https://arxiv.org/abs/1103.1225) by C.P. Dettmann, which considers higher dimensional Lorentz gases. There is also this [talk](https://web.archive.org/web/20161118032954/http://osp.open.ac.uk/802576AB005B745A/%28httpAssets%29/3DCCD0E5F8AAF4428025785C0042C8FB/$file/Carl_Dettmann.pdf) with the same title summarising the results, it says there that the periodic case is the only one which has been treated rigorously so far, and that the random Lorentz case is expected to have similar diffusion properties, but we have no proof.


