American Institute of Mathematical Sciences

Advanced
September  2008, 1(3): 387-404. doi: 10.3934/krm.2008.1.387

On the slowing down of charged particles in a binary stochastic mixture

Jean-François Clouet 1, , François Golse 2, , Marjolaine Puel 3, and  Rémi Sentis 1,

1. 

CEA/Bruyères, Bruyères-le-Chatel, 91297 Arpajon Cedex, France, France
2. 

Centre de Mathématiques, Ecole Polytechnique, 91128 Palaiseau Cedex, France
3. 

Laboratoire MIP, Université paul Sabatier, 31062 Toulouse, France

Received  January 2008 Revised  June 2008 Published  August 2008

A kinetic equation is addressed for the straight line slowing-down of charged particles, the geometrical domain consists of randomly distributed spherical grains of dense material imbedded in a light material. The dense material is assumed to be a Boolean medium (the sphere centers are sampled according to a Poisson random field). We focus on the fraction of particles $P$ which stop in the light medium. After setting some properties of the Boolean medium, we perform an asymptotic analysis in two extreme cases corresponding to grain radius very small and very large with respect to the stopping distance of the dense material. A fitted analytic formula is proposed for the quantity $P$ and results of numerical simulations are presented in order to validate the proposed formula.
Keywords: Particle slowing-down, Stochastic geometry., Poisson germ-grain random set, Boolean medium, Underwood theorem, Partial differential equation in random media, Mean chord length.
Mathematics Subject Classification: Primary: 35Q72, 60D05 . Secondary: 65C0.
Citation: Jean-François Clouet, François Golse, Marjolaine Puel, Rémi Sentis. On the slowing down of charged particles in a binary stochastic mixture. Kinetic & Related Models, 2008, 1 (3) : 387-404. doi: 10.3934/krm.2008.1.387
[1]

James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167
[2]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
[3]

Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715
[4]

Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473
[5]

Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1
[6]

Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.

[7]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473
[8]

Julia Calatayud, Juan Carlos Cortés, Marc Jornet. On the random wave equation within the mean square context. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021082
[9]

Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951
[10]

Guillaume Bal, Tomasz Komorowski, Lenya Ryzhik. Kinetic limits for waves in a random medium. Kinetic & Related Models, 2010, 3 (4) : 529-644. doi: 10.3934/krm.2010.3.529
[11]

Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503
[12]

Dan Stanescu, Benito Chen-Charpentier. Random coefficient differential equation models for Monod kinetics. Conference Publications, 2009, 2009 (Special) : 719-728. doi: 10.3934/proc.2009.2009.719
[13]

Galina Kurina, Vladimir Zadorozhniy. Mean periodic solutions of a inhomogeneous heat equation with random coefficients. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1543-1551. doi: 10.3934/dcdss.2020087
[14]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093
[15]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9
[16]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[17]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329
[18]

Yanfeng Guo, Jinqiao Duan, Donglong Li. Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1701-1715. doi: 10.3934/dcdss.2016071
[19]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[20]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy