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On the distribution of free path lengths for the periodic Lorentz gas II
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- Journal:
- ESAIM: Mathematical Modelling and Numerical Analysis / Volume 34 / Issue 6 / November 2000
- Published online by Cambridge University Press:
- 15 April 2002, pp. 1151-1163
- Print publication:
- November 2000
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- Article
- Export citation
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Consider the domain $Z_\epsilon=\{x\in\mathbb{R}^n ; {dist}(x,\epsilon\mathbb{Z}^n)> \epsilon^\gamma\}$
and let the free path length be defined as $\tau_\epsilon(x,v)=\inf\{t> 0 ; x-tv\in Z_\epsilon\}.$ 
In the Boltzmann-Grad scaling corresponding to $\gamma=\frac{n}{n-1}$ 
, it is shownthat the limiting distribution $\phi_\epsilon$ 
of $\tau_\epsilon$ 
is bounded from belowby an expression of the form C/t, for some C> 0. A numerical study seems toindicate that asymptotically for large t, $\phi_\epsilon\sim C/t$ 
.This is an extension of a previous work [J. Bourgain et al., Comm. Math. Phys.190 (1998) 491-508]. As aconsequence, it is proved that the linear Boltzmann type transport equation is inappropriate to describethe Boltzmann-Grad limit of the periodic Lorentz gas, at variance with the usualcase of a Poisson distribution of scatterers treated in [G. Gallavotti (1972)].
