September  2011, 4(3): 633-668. doi: 10.3934/krm.2011.4.633

A simple particle model for a system of coupled equations with absorbing collision term

1. 

Université de Lyon and CNRS, UMPA, UMR-CNRS 5669, ENS-Lyon, 46, allée d’Italie, 69364 Lyon Cedex 07, France

2. 

Dipartimento di Metodi e Modelli Matematici, Università di Palermo, Viale delle Scienze Edificio 8, I90128 Palermo, Italy

Received  November 2009 Revised  April 2011 Published  August 2011

We study a particle model for a simple system of partial differential equations describing, in dimension $d\geq 2$, a two component mixture where light particles move in a medium of absorbing, fixed obstacles; the system consists in a transport and a reaction equation coupled through pure absorption collision terms. We consider a particle system where the obstacles, of radius $\varepsilon$, become inactive at a rate related to the number of light particles travelling in their range of influence at a given time and the light particles are instantaneously absorbed at the first time they meet the physical boundary of an obstacle; elements belonging to the same species do not interact among themselves. We prove the convergence (a.s. w.r.t. the product measure associated to the initial datum for the light particle component) of the densities describing the particle system to the solution of the system of partial differential equations in the asymptotics $ a_n^d n^{-\kappa}\to 0$ and $a_n^d \varepsilon^{\zeta}\to 0$, for $\kappa\in(0,\frac 12)$ and $\zeta\in (0,\frac12 - \frac 1{2d})$, where $a_n^{-1}$ is the effective range of the obstacles and $n$ is the total number of light particles.
Citation: Cédric Bernardin, Valeria Ricci. A simple particle model for a system of coupled equations with absorbing collision term. Kinetic & Related Models, 2011, 4 (3) : 633-668. doi: 10.3934/krm.2011.4.633
References:
[1]

C. Boldrighini, L. A. Bunimovich and Ya. G. Sinaĭ, On the Boltzmann equation for the Lorentz gas,, J. Stat. Phys., 32 (1983), 477. doi: 10.1007/BF01008951. Google Scholar

[2]

J. Bourgain, F. Golse and B. Wennberg, On the distribution of free path lengths for the periodic Lorentz gas,, Commun. Math. Phys., 190 (1998), 491. doi: 10.1007/s002200050249. Google Scholar

[3]

L. Desvillettes and V. Ricci, Non-Markovianity of the Boltzmann-Grad limit of a system of random obstacles in a given force field,, Bull. Sci. Math., 128 (2004), 39. doi: 10.1016/j.bulsci.2003.09.003. Google Scholar

[4]

L. Desvillettes and V. Ricci, A rigorous derivation of a linear kinetic of the Fokker-Planck type in the limit of grazing collisions,, J. Stat. Phys., 104 (2001), 1173. doi: 10.1023/A:1010461929872. Google Scholar

[5]

G. Gallavotti, "Rigorous Theory of the Boltzmann Equation in the Lorentz Gas,", Nota interna n. 358, (1972). Google Scholar

[6]

F. Golse, The mean-field limit for the dynamics of large particle systems,, in Journées, (2003). Google Scholar

[7]

F. Golse, On the periodic Lorentz gas and the Lorentz kinetic equation,, Ann. Fac. Sci. Toulouse, 17 (2008), 735. Google Scholar

[8]

F. Golse, Recent results on the periodic Lorentz gas, preprint,, HAL: hal-00390895, (2009), 1. Google Scholar

[9]

R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Vol. 2, (1988). Google Scholar

[10]

P. Malliavin and H. Airault, "Intégration et Analyse de Fourier, Probabilités et Analyse Gaussienne,", 2nd edition, (1994). Google Scholar

[11]

D. Mihalas and B. Weibel Mihalas, "Foundations of Radiation Hydrodynamics,", Oxford University Press, (1984). Google Scholar

[12]

G. Nappo, E. Orlandi and H. Rost, A reaction-diffusion model for moderately interacting particles,, J. Stat. Phys., 55 (1989), 579. doi: 10.1007/BF01041598. Google Scholar

[13]

V. Ricci and B. Wennberg, On the derivation of a linear Boltzmann equation from a periodic lattice gas,, Stochastic Process. Appl., 111 (2004), 281. doi: 10.1016/j.spa.2004.01.002. Google Scholar

[14]

L. Schwartz, "Théorie des Distributions,", Publications de l'institut de Mathématique de l'Université de Strasbourg, (1966). Google Scholar

[15]

H. Spohn, The Lorentz process converges to a random flight process,, Commun. Math. Phys., 60 (1978), 277. doi: 10.1007/BF01612893. Google Scholar

[16]

C. I. Steefel, D. J. De Paolo and P. C. Lichtner, Reactive transport modeling: An essential tool and a new research approach for the Earth sciences,, Earth and Planetary Science Letters, 240 (2005), 539. Google Scholar

[17]

A. S. Sznitman, Propagation of chaos for a system of annihilating Brownian spheres,, CPAM, 40 (1987), 663. Google Scholar

show all references

References:
[1]

C. Boldrighini, L. A. Bunimovich and Ya. G. Sinaĭ, On the Boltzmann equation for the Lorentz gas,, J. Stat. Phys., 32 (1983), 477. doi: 10.1007/BF01008951. Google Scholar

[2]

J. Bourgain, F. Golse and B. Wennberg, On the distribution of free path lengths for the periodic Lorentz gas,, Commun. Math. Phys., 190 (1998), 491. doi: 10.1007/s002200050249. Google Scholar

[3]

L. Desvillettes and V. Ricci, Non-Markovianity of the Boltzmann-Grad limit of a system of random obstacles in a given force field,, Bull. Sci. Math., 128 (2004), 39. doi: 10.1016/j.bulsci.2003.09.003. Google Scholar

[4]

L. Desvillettes and V. Ricci, A rigorous derivation of a linear kinetic of the Fokker-Planck type in the limit of grazing collisions,, J. Stat. Phys., 104 (2001), 1173. doi: 10.1023/A:1010461929872. Google Scholar

[5]

G. Gallavotti, "Rigorous Theory of the Boltzmann Equation in the Lorentz Gas,", Nota interna n. 358, (1972). Google Scholar

[6]

F. Golse, The mean-field limit for the dynamics of large particle systems,, in Journées, (2003). Google Scholar

[7]

F. Golse, On the periodic Lorentz gas and the Lorentz kinetic equation,, Ann. Fac. Sci. Toulouse, 17 (2008), 735. Google Scholar

[8]

F. Golse, Recent results on the periodic Lorentz gas, preprint,, HAL: hal-00390895, (2009), 1. Google Scholar

[9]

R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Vol. 2, (1988). Google Scholar

[10]

P. Malliavin and H. Airault, "Intégration et Analyse de Fourier, Probabilités et Analyse Gaussienne,", 2nd edition, (1994). Google Scholar

[11]

D. Mihalas and B. Weibel Mihalas, "Foundations of Radiation Hydrodynamics,", Oxford University Press, (1984). Google Scholar

[12]

G. Nappo, E. Orlandi and H. Rost, A reaction-diffusion model for moderately interacting particles,, J. Stat. Phys., 55 (1989), 579. doi: 10.1007/BF01041598. Google Scholar

[13]

V. Ricci and B. Wennberg, On the derivation of a linear Boltzmann equation from a periodic lattice gas,, Stochastic Process. Appl., 111 (2004), 281. doi: 10.1016/j.spa.2004.01.002. Google Scholar

[14]

L. Schwartz, "Théorie des Distributions,", Publications de l'institut de Mathématique de l'Université de Strasbourg, (1966). Google Scholar

[15]

H. Spohn, The Lorentz process converges to a random flight process,, Commun. Math. Phys., 60 (1978), 277. doi: 10.1007/BF01612893. Google Scholar

[16]

C. I. Steefel, D. J. De Paolo and P. C. Lichtner, Reactive transport modeling: An essential tool and a new research approach for the Earth sciences,, Earth and Planetary Science Letters, 240 (2005), 539. Google Scholar

[17]

A. S. Sznitman, Propagation of chaos for a system of annihilating Brownian spheres,, CPAM, 40 (1987), 663. Google Scholar

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