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

Abstract

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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 C/t, for some C> 0. A numerical study seems to indicate that asymptotically for large t, φ ϵ ∼ 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)].

How to cite

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Golse, 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

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  1. C. Boldrighini, L.A. Bunimovich and Ya.G. Sinai, On the Boltzmann equation for the Lorentz gas. J. Statist. Phys.32 (1983) 477-501.  
  2. 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.  
  3. L.A. Bunimovich and Ya.G. Sinai, Markov Partitions of Dispersed Billiards. Comm. Math. Phys.73 (1980) 247-280.  
  4. 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.  
  5. L.A. Bunimovich, Ya.G. Sinai and N.I. Chernov, Markov partitions for two-dimensional hyperbolic billiards. Russian Math. Surveys45 (1990) 105-152.  
  6. L.A. Bunimovich, Ya.G. Sinai and N.I. Chernov, Statistical properties of two-dimensional hyperbolic billiards. Russian Math. Surveys46 (1991) 47-106.  
  7. 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.  
  8. G. Gallavotti, Rigorous theory of the Boltzmann equation in the Lorentz gas. Nota Interna No. 358, Istituto di Fisica, Università di Roma (1972).  
  9. F. Golse, Transport dans les milieux composites fortement contrastés. I. Le modèle du billard. Ann. Inst. H. Poincaré Phys. Théor.61 (1994) 381-410.  
  10. H. Spohn, The Lorentz flight process converges to a random flight process. Comm. Math. Phys.60 (1978) 277-290.  

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