# American Institute of Mathematical Sciences

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:

show all references

##### References:
 [1] Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012 [2] Zengjing Chen, Qingyang Liu, Gaofeng Zong. Weak laws of large numbers for sublinear expectation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 637-651. doi: 10.3934/mcrf.2018027 [3] Zengjing Chen, Weihuan Huang, Panyu Wu. Extension of the strong law of large numbers for capacities. Mathematical Control & Related Fields, 2019, 9 (1) : 175-190. doi: 10.3934/mcrf.2019010 [4] Freddy Dumortier. Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1465-1479. doi: 10.3934/dcds.2012.32.1465 [5] María del Mar González, Regis Monneau. Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1255-1286. doi: 10.3934/dcds.2012.32.1255 [6] Vadim Kaushansky, Christoph Reisinger. Simulation of a simple particle system interacting through hitting times. Discrete & Continuous Dynamical Systems - B, 2017, 12 (11) : 1-22. doi: 10.3934/dcdsb.2019067 [7] S.-I. Ei, M. Mimura, M. Nagayama. Interacting spots in reaction diffusion systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 31-62. doi: 10.3934/dcds.2006.14.31 [8] Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581 [9] David Cowan. Rigid particle systems and their billiard models. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 111-130. doi: 10.3934/dcds.2008.22.111 [10] Matthias Büger. Torsion numbers, a tool for the examination of symmetric reaction-diffusion systems related to oscillation numbers. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 691-708. doi: 10.3934/dcds.1998.4.691 [11] Manfred G. Madritsch, Izabela Petrykiewicz. Non-normal numbers in dynamical systems fulfilling the specification property. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4751-4764. doi: 10.3934/dcds.2014.34.4751 [12] Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control & Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435 [13] Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258 [14] Eliot Fried. New insights into the classical mechanics of particle systems. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1469-1504. doi: 10.3934/dcds.2010.28.1469 [15] Salah-Eldin A. Mohammed, Tusheng Zhang. Large deviations for stochastic systems with memory. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 881-893. doi: 10.3934/dcdsb.2006.6.881 [16] Péter Koltai, Alexander Volf. Optimizing the stable behavior of parameter-dependent dynamical systems --- maximal domains of attraction, minimal absorption times. Journal of Computational Dynamics, 2014, 1 (2) : 339-356. doi: 10.3934/jcd.2014.1.339 [17] Barton E. Lee. Consensus and voting on large graphs: An application of graph limit theory. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1719-1744. doi: 10.3934/dcds.2018071 [18] Nicolo' Catapano. The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit. Kinetic & Related Models, 2018, 11 (3) : 647-695. doi: 10.3934/krm.2018027 [19] Feimin Huang, Yeping Li. Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 455-470. doi: 10.3934/dcds.2009.24.455 [20] Nicolas Forcadel, Cyril Imbert, Régis Monneau. Homogenization of some particle systems with two-body interactions and of the dislocation dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 785-826. doi: 10.3934/dcds.2009.23.785

2018 Impact Factor: 1.38