The European Physical Journal D

, Volume 46, Issue 2, pp 295–302 | Cite as

Poisson-Vlasov: stochastic representation and numerical codes

Plasma Physics

Abstract.

A stochastic representation for the solutions of the Poisson-Vlasov equation, with several charged species, is obtained. The representation involves both an exponential and a branching process and it provides an intuitive characterization of the nature of the solutions and its fluctuations. Here, the stochastic representation is also proposed as a tool for the numerical evaluation of the solutions.

PACS.

52.20.-j Elementary processes in plasmas 52.65.Ff Fokker-Planck and Vlasov equation 05.10.Gg Stochastic analysis methods 

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References

  1. R. Courant, K. Friedrichs, H. Lewy, Mat. Ann. 100, 32 (1928) CrossRefMathSciNetGoogle Scholar
  2. H.P. McKean, Comm. Pure Appl. Math. 28, 323 (1975); H.P. McKean, Comm. Pure Appl. Math. 29, 553 (1976) MATHCrossRefMathSciNetGoogle Scholar
  3. E.B. Dynkin, Prob. Theory Rel. Fields 89, 89 (1991) MATHCrossRefMathSciNetADSGoogle Scholar
  4. E.B. Dynkin, Diffusions, Superdiffusions and Partial Differential Equations (AMS Colloquium Pubs., Providence, 2002) Google Scholar
  5. Y. LeJan, A.S. Sznitman, Prob. Theory Relat. Fields 109, 343 (1997) CrossRefMathSciNetGoogle Scholar
  6. E.C. Waymire, Prob. Surveys 2, 1 (2005) CrossRefMathSciNetGoogle Scholar
  7. R.N. Bhattacharya et al., Trans. Amer. Math. Soc. 355, 5003 (2003) CrossRefMathSciNetGoogle Scholar
  8. M. Ossiander, Prob. Theory Relat. Fields 133, 267 (2005) MATHCrossRefMathSciNetGoogle Scholar
  9. R. Vilela Mendes, Zeitsch. Phys. C 54, 273 (1992) CrossRefADSGoogle Scholar
  10. R. Vilela Mendes, F. Cipriano, Commun. Nonlin. Sci. Num. Simul. 13, 221 (2008) MATHCrossRefMathSciNetGoogle Scholar
  11. R. Illner, G. Rein, Math. Meth. Appl. Sci. 19, 1409 (1996) MATHCrossRefMathSciNetGoogle Scholar
  12. P. Braasch, G. Rein, J. Vukadinovic, SIAM J. Appl. Math. 59, 831 (1998) CrossRefMathSciNetGoogle Scholar
  13. J.D. Biggins, N.H. Bingham, Adv. Appl. Prob. 25, 757 (1993) MATHCrossRefMathSciNetGoogle Scholar
  14. K.B. Athreya, Ann. Appl. Prob. 4, 779 (1994) MATHCrossRefMathSciNetGoogle Scholar
  15. P.E. Ney, A.N. Vidyashankar, Ann. Appl. Prob. 14, 1135 (2004) MATHCrossRefMathSciNetGoogle Scholar
  16. K. Fleischmann, V. Wachtel, e-print arXiv:math.PR/0605617 Google Scholar
  17. J. Seixas, R. Vilela Mendes, Nucl. Phys. B 383, 622 (1992) CrossRefADSGoogle Scholar
  18. C.R. Oberman, E.A. Williams, in Handbook of Plasma Physics, edited by M.N. Rosenbluth, R.Z. Sagdeev (North-Holland, Amsterdam, 1985), pp. 279–333 Google Scholar
  19. J.A. Krommes, Phys. Rep. 360, 1 (2002) CrossRefADSGoogle Scholar
  20. P.J. Morrison, Phys. Plasmas 12, 058102 (2005) CrossRefMathSciNetGoogle Scholar
  21. C. Graham, S. Méléard, in ESAIM Proceedings, edited by F. Coquel, S. Cordier (EDP Sciences, Les Ulis, 2001), Vol. 10, pp. 77–126 Google Scholar
  22. J.A. Acebrón, M.P. Busico, P. Lanucara, R. Spigler, SIAM J. Sci. Comput. 27, 440 (2005) MATHCrossRefMathSciNetGoogle Scholar
  23. J.A. Acebrón, R. Spigler, Lect. Notes Comput. Sci. Eng. 55, 475 (2007) Google Scholar
  24. D. Talay, L. Tubaro, Probabilistic models for nonlinear partial differential equations, Lecture Notes in Mathematics (Springer, 1996), p. 1627 Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  1. 1.Centre de Physique Théorique, CNRS LuminyMarseille Cedex 9France
  2. 2.Centro de Fusão Nuclear, EURATOM/IST Association, Instituto Superior TécnicoLisboaPortugal
  3. 3.CMAF, Complexo Interdisciplinar, Universidade de LisboaLisboaPortugal

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