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On the distribution of free path lengths for the periodic Lorentz gas II

François Golse;  Bernt Wennberg — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

Consider the domain Z ϵ = { x ∈ ℝ n ; d i s t ( x , ϵ ℤ n ) > ϵ γ } and let the free path length be defined as τ ϵ ( x , v ) = inf { t > 0 ; x - t v ∈ Z ϵ } . In the Boltzmann-Grad scaling corresponding to γ = n n - 1 , it is shown that the limiting distribution φ ϵ of τ ϵ is bounded from below by an expression of the form , for some . A numerical study seems to indicate that asymptotically for large , φ ϵ ∼ C / t . This is an extension of a previous work [J. Bourgain , (1998) 491-508]. As a consequence, it is proved that the linear Boltzmann type transport equation is inappropriate to describe the...

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