# Does the random Lorenz gas have a non-trivial diffusion coefficient?

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 suggest that there is diffusive behaviour for the random Lorenz gas. This article by Bunimovich 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.

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## 2 Answers

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"). 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" by C.P. Dettmann, which considers higher dimensional Lorentz gases. There is also this talk 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.

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I have been looking at random (and periodic) Lorentz gas models recently in preparing a review which has just appeared at arxiv:1402.7010. As far as I know, there are rigorous results only for the low density (Boltzmann-Grad) limit, and for models where the scatterers are known not to overlap (for example placing a scatterer or empty space randomly at each site of a lattice). In the fixed density case there does not even appear to be a proof that "conditional on the scatterers not overlapping" converges. Choosing a point outside a scatterer at random is problematic since the measure is infinite; better to fix the point at the origin and then choose the scatterers conditional on not overlapping each other or the origin. The low density limit suggests that the velocity autocorrelation decays as $t^{-d/2-1}$, which is fast enough for the diffusion coefficient (its integral) to exist.

Correction: After the above paper appeared online (Commun. Theor. Phys. 62 521-540 (2014).), and following discussions with D. Szasz, I discovered references that do indeed show that the non-overlapping condition converges. See A variational principle for the equilibrium of hard sphere systems, Gallavotti, G and Miracle-Sole, S, Annales IHP A 8 287-299 (1968); appendix B of Observables at infinity and states with short range correlations in statistical mechanics, Lanford III, OE and Ruelle, David, Commun. Math. Phys. 13 194-215 (1969). But the diffusion question remains open.

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