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

François Golse; Bernt Wennberg

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 34, Issue: 6, page 1151-1163
- ISSN: 0764-583X

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topGolse, François, and Wennberg, Bernt. "On the distribution of free path lengths for the periodic Lorentz gas II." ESAIM: Mathematical Modelling and Numerical Analysis 34.6 (2010): 1151-1163. <http://eudml.org/doc/197407>.

@article{Golse2010,

abstract = {
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 shown
that the limiting distribution $\phi_\epsilon$ of $\tau_\epsilon$ is bounded from below
by an expression of the form C/t, for some C> 0. A numerical study seems to
indicate 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 a
consequence, it is proved that the linear Boltzmann type transport equation is inappropriate to describe
the Boltzmann-Grad limit of the periodic Lorentz gas, at variance with the usual
case of a Poisson distribution of scatterers treated in [G. Gallavotti (1972)].
},

author = {Golse, François, Wennberg, Bernt},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Lorentz gas; Boltzmann-Grad limit; kinetic theory; mean free path.; mean free path; Boltzmann-Grad scaling; limiting distribution; linear Boltzmann type transport equation; periodic Lorentz gas},

language = {eng},

month = {3},

number = {6},

pages = {1151-1163},

publisher = {EDP Sciences},

title = {On the distribution of free path lengths for the periodic Lorentz gas II},

url = {http://eudml.org/doc/197407},

volume = {34},

year = {2010},

}

TY - JOUR

AU - Golse, François

AU - Wennberg, Bernt

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 34

IS - 6

SP - 1151

EP - 1163

AB -
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 shown
that the limiting distribution $\phi_\epsilon$ of $\tau_\epsilon$ is bounded from below
by an expression of the form C/t, for some C> 0. A numerical study seems to
indicate 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 a
consequence, it is proved that the linear Boltzmann type transport equation is inappropriate to describe
the Boltzmann-Grad limit of the periodic Lorentz gas, at variance with the usual
case of a Poisson distribution of scatterers treated in [G. Gallavotti (1972)].

LA - eng

KW - Lorentz gas; Boltzmann-Grad limit; kinetic theory; mean free path.; mean free path; Boltzmann-Grad scaling; limiting distribution; linear Boltzmann type transport equation; periodic Lorentz gas

UR - http://eudml.org/doc/197407

ER -

## References

top- C. Boldrighini, L.A. Bunimovich and Ya.G. Sinai, On the Boltzmann equation for the Lorentz gas. J. Statist. Phys.32 (1983) 477-501.
- J. Bourgain, F. Golse and B. Wennberg, On the distribution of free path lengths for the periodic Lorentz gas. Comm. Math. Phys.190 (1998) 491-508.
- L.A. Bunimovich and Ya.G. Sinai, Markov Partitions of Dispersed Billiards. Comm. Math. Phys.73 (1980) 247-280.
- L.A. Bunimovich and Ya.G. Sinai, Statistical properties of the Lorentz gas with periodic configurations of scatterers. Comm. Math. Phys.78 (1981) 479-497.
- L.A. Bunimovich, Ya.G. Sinai and N.I. Chernov, Markov partitions for two-dimensional hyperbolic billiards. Russian Math. Surveys45 (1990) 105-152.
- L.A. Bunimovich, Ya.G. Sinai and N.I. Chernov, Statistical properties of two-dimensional hyperbolic billiards. Russian Math. Surveys46 (1991) 47-106.
- H.S. Dumas, L. Dumas and F. Golse, Remarks on the notion of mean free path for a periodic array of spherical obstacles. J. Statist. Phys.87 (1997) 943-950.
- G. Gallavotti, Rigorous theory of the Boltzmann equation in the Lorentz gas. Nota Interna No. 358, Istituto di Fisica, Università di Roma (1972).
- F. Golse, Transport dans les milieux composites fortement contrastés. I. Le modèle du billard. Ann. Inst. H. Poincaré Phys. Théor.61 (1994) 381-410.
- H. Spohn, The Lorentz flight process converges to a random flight process. Comm. Math. Phys.60 (1978) 277-290.

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