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

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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

Cédric Bernardin ^1, and Valeria Ricci ^2,

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

Abstract
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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.

Keywords: large numbers limit, absorption., Interacting particle systems.

Mathematics Subject Classification: 82C22, 82C21, 60F05, 60K3.

Citation: Cédric Bernardin, Valeria Ricci. A simple particle model for a system of coupled equations with absorbing collision term. Kinetic and 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), 477-501. doi: 10.1007/BF01008951.
[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), 491-508. doi: 10.1007/s002200050249.
[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), 39-46. doi: 10.1016/j.bulsci.2003.09.003.
[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), 1173-1189. doi: 10.1023/A:1010461929872.
[5]	G. Gallavotti, "Rigorous Theory of the Boltzmann Equation in the Lorentz Gas," Nota interna n. 358, Istituto di Fisica, Università di Roma, 1972.
[6]	F. Golse, The mean-field limit for the dynamics of large particle systems, in Journées "Equations aux Dérivées Partielles," Exp. No. IX, 47 pp., Univ. Nantes, Nantes, 2003.
[7]	F. Golse, On the periodic Lorentz gas and the Lorentz kinetic equation, Ann. Fac. Sci. Toulouse, Ser.6, 17 (2008), 735-749.
[8]	F. Golse, Recent results on the periodic Lorentz gas, preprint, HAL: hal-00390895, (2009), 1-62.
[9]	R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," Vol. 2, Functional and Variational Methods, Springer-Verlag, Berlin, 1988.
[10]	P. Malliavin and H. Airault, "Intégration et Analyse de Fourier, Probabilités et Analyse Gaussienne," 2^nd edition, Collection Maîtrise de Mathématiques Pures, Masson, Paris, 1994.
[11]	D. Mihalas and B. Weibel Mihalas, "Foundations of Radiation Hydrodynamics," Oxford University Press, New York, 1984, Reprint, Dover 1999.
[12]	G. Nappo, E. Orlandi and H. Rost, A reaction-diffusion model for moderately interacting particles, J. Stat. Phys., 55 (1989), 579-600. doi: 10.1007/BF01041598.
[13]	V. Ricci and B. Wennberg, On the derivation of a linear Boltzmann equation from a periodic lattice gas, Stochastic Process. Appl., 111 (2004), 281-315. doi: 10.1016/j.spa.2004.01.002.
[14]	L. Schwartz, "Théorie des Distributions," Publications de l'institut de Mathématique de l'Université de Strasbourg, No. IX-X, Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966.
[15]	H. Spohn, The Lorentz process converges to a random flight process, Commun. Math. Phys., 60 (1978), 277-290. doi: 10.1007/BF01612893.
[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), 539-558.
[17]	A. S. Sznitman, Propagation of chaos for a system of annihilating Brownian spheres, CPAM, 40 (1987), 663-690.

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), 477-501. doi: 10.1007/BF01008951.
[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), 491-508. doi: 10.1007/s002200050249.
[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), 39-46. doi: 10.1016/j.bulsci.2003.09.003.
[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), 1173-1189. doi: 10.1023/A:1010461929872.
[5]	G. Gallavotti, "Rigorous Theory of the Boltzmann Equation in the Lorentz Gas," Nota interna n. 358, Istituto di Fisica, Università di Roma, 1972.
[6]	F. Golse, The mean-field limit for the dynamics of large particle systems, in Journées "Equations aux Dérivées Partielles," Exp. No. IX, 47 pp., Univ. Nantes, Nantes, 2003.
[7]	F. Golse, On the periodic Lorentz gas and the Lorentz kinetic equation, Ann. Fac. Sci. Toulouse, Ser.6, 17 (2008), 735-749.
[8]	F. Golse, Recent results on the periodic Lorentz gas, preprint, HAL: hal-00390895, (2009), 1-62.
[9]	R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," Vol. 2, Functional and Variational Methods, Springer-Verlag, Berlin, 1988.
[10]	P. Malliavin and H. Airault, "Intégration et Analyse de Fourier, Probabilités et Analyse Gaussienne," 2^nd edition, Collection Maîtrise de Mathématiques Pures, Masson, Paris, 1994.
[11]	D. Mihalas and B. Weibel Mihalas, "Foundations of Radiation Hydrodynamics," Oxford University Press, New York, 1984, Reprint, Dover 1999.
[12]	G. Nappo, E. Orlandi and H. Rost, A reaction-diffusion model for moderately interacting particles, J. Stat. Phys., 55 (1989), 579-600. doi: 10.1007/BF01041598.
[13]	V. Ricci and B. Wennberg, On the derivation of a linear Boltzmann equation from a periodic lattice gas, Stochastic Process. Appl., 111 (2004), 281-315. doi: 10.1016/j.spa.2004.01.002.
[14]	L. Schwartz, "Théorie des Distributions," Publications de l'institut de Mathématique de l'Université de Strasbourg, No. IX-X, Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966.
[15]	H. Spohn, The Lorentz process converges to a random flight process, Commun. Math. Phys., 60 (1978), 277-290. doi: 10.1007/BF01612893.
[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), 539-558.
[17]	A. S. Sznitman, Propagation of chaos for a system of annihilating Brownian spheres, CPAM, 40 (1987), 663-690.

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