American Institute of Mathematical Sciences

Advanced
September  2008, 1(3): 331-354. doi: 10.3934/krm.2008.1.331

Diffusion and guiding center approximation for particle transport in strong magnetic fields

Naoufel Ben Abdallah 1, and  Raymond El Hajj 1,

1. 

Institut de Mathématiques de Toulouse, Université de Toulouse and CNRS, Université Paul Sabatier, 31062 Toulouse Cedex 9, France, France

Received  April 2008 Revised  June 2008 Published  August 2008

The diffusion limit of the linear Boltzmann equation with a strong magnetic field is performed. The giration period of particles around the magnetic field is assumed to be much smaller that the collision relaxation time which is supposed to be much smaller than the macroscopic time. The limiting equation is shown to be a diffusion equation in the parallel direction while in the orthogonal direction, the guiding center motion is obtained. The diffusion constant in the parallel direction is obtained through the study of a new collision operator obtained by averaging the original one. Moreover, a correction to the guiding center motion is derived.
Keywords: Diffusion limit, high magnetic field., guiding-center approximation.
Mathematics Subject Classification: Primary: 35B25, 35B40, 82C40, 82C70, Secondary: 82D1.
Citation: Naoufel Ben Abdallah, Raymond El Hajj. Diffusion and guiding center approximation for particle transport in strong magnetic fields. Kinetic & Related Models, 2008, 1 (3) : 331-354. doi: 10.3934/krm.2008.1.331
[1]

Jean-Philippe Braeunig, Nicolas Crouseilles, Michel Mehrenberger, Eric Sonnendrücker. Guiding-center simulations on curvilinear meshes. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 271-282. doi: 10.3934/dcdss.2012.5.271
[2]

Naoufel Ben Abdallah, Hédia Chaker. Mixed high field and diffusion asymptotics for the fermionic Boltzmann equation. Kinetic & Related Models, 2009, 2 (3) : 403-424. doi: 10.3934/krm.2009.2.403
[3]

Jianjun Chen, Wancheng Sheng. The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2018, 17 (1) : 127-142. doi: 10.3934/cpaa.2018008
[4]

Pierluigi Colli, Gianni Gilardi, Pavel Krejčí, Jürgen Sprekels. A vanishing diffusion limit in a nonstandard system of phase field equations. Evolution Equations & Control Theory, 2014, 3 (2) : 257-275. doi: 10.3934/eect.2014.3.257
[5]

Nicolas Crouseilles, Mohammed Lemou. An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits. Kinetic & Related Models, 2011, 4 (2) : 441-477. doi: 10.3934/krm.2011.4.441
[6]

Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086
[7]

Jaume Llibre, Dana Schlomiuk. On the limit cycles bifurcating from an ellipse of a quadratic center. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1091-1102. doi: 10.3934/dcds.2015.35.1091
[8]

Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115
[9]

Hanming Zhou. Lens rigidity with partial data in the presence of a magnetic field. Inverse Problems & Imaging, 2018, 12 (6) : 1365-1387. doi: 10.3934/ipi.2018057
[10]

Gilles Carbou, Stéphane Labbé, Emmanuel Trélat. Smooth control of nanowires by means of a magnetic field. Communications on Pure & Applied Analysis, 2009, 8 (3) : 871-879. doi: 10.3934/cpaa.2009.8.871
[11]

Nurlan Dairbekov, Gunther Uhlmann. Reconstructing the metric and magnetic field from the scattering relation. Inverse Problems & Imaging, 2010, 4 (3) : 397-409. doi: 10.3934/ipi.2010.4.397
[12]

Ariadna Farrés, Àngel Jorba. On the high order approximation of the centre manifold for ODEs. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 977-1000. doi: 10.3934/dcdsb.2010.14.977
[13]

Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627
[14]

Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236
[15]

Iliya D. Iliev, Chengzhi Li, Jiang Yu. Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Communications on Pure & Applied Analysis, 2010, 9 (3) : 583-610. doi: 10.3934/cpaa.2010.9.583
[16]

Mingqi Xiang, Patrizia Pucci, Marco Squassina, Binlin Zhang. Nonlocal Schrödinger-Kirchhoff equations with external magnetic field. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1631-1649. doi: 10.3934/dcds.2017067
[17]

Amer Rasheed, Aziz Belmiloudi, Fabrice Mahé. Dynamics of dendrite growth in a binary alloy with magnetic field effect. Conference Publications, 2011, 2011 (Special) : 1224-1233. doi: 10.3934/proc.2011.2011.1224
[18]

Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic & Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011
[19]

Yernat M. Assylbekov, Hanming Zhou. Boundary and scattering rigidity problems in the presence of a magnetic field and a potential. Inverse Problems & Imaging, 2015, 9 (4) : 935-950. doi: 10.3934/ipi.2015.9.935
[20]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences