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Functional-analytic techniques in the study of time-dependent electron swarms in weakly ionized gases

Part of the Lecture Notes in Mathematics book series (LNM,volume 1460)

Abstract

The time-evolution of charged particle swarms in a weakly ionized gas can be suitably modelled by the linear Boltzmann equation. In this work we discuss the time-dependent problem, the stationary problem as well as the long time behaviour of the particle distribution.

The paper is divided in three main parts. The first part is devoted to a simplified one-dimensional Boltzmann model of the Kač type, to study the velocity distribution of a spatially uniform diluted guest population of electrons moving within a host medium under the influence of a D.C. electric field. Necessary conditions and sufficient conditions are established for the existence, uniqueness and attractivity of a stationary nonnegative distribution corresponding to a specified concentration level. Conditions for the onset of the runaway process are established and the long time behaviour of the velocity distribution is studied within the framework of scattering theory.

The second part is devoted to the study of a non-homogeneous model where the collision frequency and the scattering kernel depend also on the space coordinates. A definition of “runaway” and a necessary condition for the suppression of runaways are given. The time-dependent problem is discussed and the long time behaviour of the solution is investigated. Also in this case, under physically reasonable assumptions on the collision frequency, we prove the existence of wave operators and the corresponding existence of travelling waves.

Finally, in the third part, we report some results about three-dimensional velocity systems.

Keywords

  • Collision Frequency
  • Wave Operator
  • Collision Operator
  • Runaway Electron
  • Positive Linear Operator

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. K. Kumar, H.R. Skullerud and R.E. Robson, Kinetic theory of charged particle swarms in neutral gases, Aust. J. Phys. 33, 343–448 (1980)

    CrossRef  ADS  MathSciNet  Google Scholar 

  2. K. Kumar, The physics of swarms and some basic questions of kinetic theory, Phys. Rep. 112, 319–375 (1984).

    CrossRef  ADS  Google Scholar 

  3. H. Dreicer, Electrons and ion runaway in a fully ionized gas. I, Phys. Rev. 115, 238–249 (1959)

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  4. H. Dreicer, Electrons and ion runaway in a fully ionized gas. II, Phys. Rev. 117, 329–342 (1960).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  5. V.V. Parail and O.P. Pogutse, Runaway electrons in a plasma. In: M.A. Leontovich (Ed.), “Reviews of plasma physics,” Vol. 11, Consultants Bureau, New York, 1986, pp. 1–63.

    Google Scholar 

  6. M.C. Mackey, Kinetic theory model for ion movement through biological membranes, Biophys. J. 11, 75–95 (1971).

    CrossRef  ADS  Google Scholar 

  7. W. Fawcett, A.D. Boardman and S. Swain, Monte Carlo determination of electron transport properties in gallium arsenide, J. Phys. Chem. Solids. 31, 1963–1990 (1970).

    CrossRef  ADS  Google Scholar 

  8. M.O. Vassell, Calculation of high-field distribution functions in semiconductors, J. Math. Phys. 11, 408–412 (1970).

    CrossRef  ADS  Google Scholar 

  9. N. Corngold and D. Rollins, Diffusion with varying drag. The Runaway problem. I, Phys. Fluids 29, 1042–1048 (1986).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  10. N. Corngold and D. Rollins, Diffusion with varying drag. The runaway problem. II, Phys. Fluids 30, 393–398 (1987).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  11. G. Frosali, C.V.M. van der Mee, and S.L. Paveri-Fontana, Conditions for runaway phenomena in the kinetic theory of particle swarms, J. Math. Phys. 30, 1177–1186 (1989).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  12. G. Frosali, and C.V.M. van der Mee, Scattering theory relevant to the linear transport of particle swarms, J. Stat. Phys. 56, 139–148 (1989).

    CrossRef  ADS  MathSciNet  Google Scholar 

  13. L. Arlotti, On the asymptotic behaviour of electrons in an ionized gas. In: P. Nelson et al. (Eds.), “Transport Theory, Invariant Imbedding, and Integral Equations,” Lecture Notes in Pure and Appl. Math., vol. 115, M. Dekker, New York, 1989, pp.81–96.

    Google Scholar 

  14. R. Beals and V. Protopopescu, Abstract time dependent transport equations, J. Math. Anal. Appl. 121, 370–405 (1987).

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. W. Greenberg, C.V.M. van der Mee and V. Protopopescu, “Boundary value problems in abstract kinetic theory,” Basel, Birkhäuser OT 23, 1986.

    Google Scholar 

  16. B. Sherman, The difference-differential equation of electron energy dfistribution in a gas, J. Math. Anal. Appl. 1, 342–354 (1960).

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. J.L. Delcroix, “Physique des plasmas,” Tome 2, Paris, 1966.

    Google Scholar 

  18. G. Cavalleri and S.L. Paveri-Fontana, Drift velocity and runaway phenomena for electrons in neutral gases, Phys. Rev. A 6, 327–333 (1972).

    CrossRef  ADS  Google Scholar 

  19. G.L. Braglia, Motion of electrons and ions in a weakly ionized gas in a field 1. Foundations of the integral theory, Plasmaphysik 3, 147–194 (1980)

    CrossRef  ADS  Google Scholar 

  20. M.A. Krasnoselskii, “Positive solutions of operator equations,” Groningen, Noordhoff, 1964=Moscow, Fizmatgiz, 1962 [Russian].

    Google Scholar 

  21. V.C. Boffi and V.G. Molinari, Nonlinear transport problems by factorization of the scattering probability, Nuovo Cimento B 65, 29–44 (1981)

    CrossRef  ADS  Google Scholar 

  22. V.C. Boffi, V.G. Molinari, and R. Scardovelli, Kinetic approach to the propagation of electromagnetic waves in a weakly ionized plasma, Nuovo Cimento D, 1, 673–687 (1982).

    CrossRef  ADS  Google Scholar 

  23. R. Nagel (Ed.), “One-parameter semigroups of positive operators,” Lecture Notes in Mathematics 1184, Berlin, Springer, 1986.

    MATH  Google Scholar 

  24. J. Hejtmanek, Scattering theory of the linear Boltzmann operator, Commun. Math. Phys. 43, 109–120 (1975).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  25. B. Simon, Existence of the scattering matrix for the linearized Boltzmann equation, Commun. Math. Phys. 41, 99–108 (1975).

    CrossRef  ADS  MathSciNet  Google Scholar 

  26. W. Schappacher, Scattering theory for the linear Boltzmann equation, Ber. Math.-Statist. Sekt. Forschungszentrum Graz n. 69, 14 pp. (1976).

    Google Scholar 

  27. J. Voigt, On the existence of the scattering operator for the linear Boltzmann equation, J. Math. Anal. Appl. 58, 541–558 (1977).

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. C.V.M. van der Mee, Trace theorems and kinetic equations for non divergence free external forces, Applicable Anal., to appear.

    Google Scholar 

  29. F.A. Molinet, Existence, uniqueness and properties of the solutions of the Boltzmann kinetic equation for a weakly ionized gas. I, J. Math. Phys. 18, 984–996 (1977).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  30. T. Kato, “Perturbation theory for linear operators,” Springer Verlag, Berlin, 1966.

    CrossRef  MATH  Google Scholar 

  31. L. Arlotti, and G. Frosali, Long time behaviour of particle swarms in runaway regime, preprint

    Google Scholar 

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© 1991 Springer-Verlag

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Frosali, G. (1991). Functional-analytic techniques in the study of time-dependent electron swarms in weakly ionized gases. In: Toscani, G., Boffi, V., Rionero, S. (eds) Mathematical Aspects of Fluid and Plasma Dynamics. Lecture Notes in Mathematics, vol 1460. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091364

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  • DOI: https://doi.org/10.1007/BFb0091364

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