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Theory of One-Dimensional Vlasov-Maxwell Equilibria pp 41–67Cite as

The Use of Hermite Polynomials for the Inverse Problem in One-Dimensional Vlasov-Maxwell Equilibria

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Abstract

In this chapter, the aim is to make a contribution to the theory of exact equilibrium solutions to the Vlasov-Maxwell system, in 1D Cartesian geometry. In particular, we consider a solution method for the inverse problem in collisionless equilibria, namely that of calculating a VM equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans’ theorem (Jeans 1915), the equilibrium DFs are expressed as functions of the constants of motion, in the form of a stationary Maxwellian multiplied by an unknown function of the two conserved canonical momenta. In this case it is possible to reduce the inverse problem to inverting Weierstrass transforms, which we achieve by using expansions over Hermite polynomials. A sufficient condition on the pressure tensor is found which guarantees the convergence of the candidate solution when satisfied, and as a result the existence of velocity moments of all orders. This condition is obtained by elementary means, and it is clear how to put it into practice. We also argue that for a given pressure tensor for which our method applies, there always exists a non-negative DF for a sufficiently magnetised plasma. This argument is in fact proven for certain classes of DFs, and in the form of conjecture for others.

Boltzmann’s is still the most beautiful equation in the world, but Vlasov’s isn’t too shabby!

Cédric Villani

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References

  • I.G. Abel, G.G. Plunk, E. Wang, M. Barnes, S.C. Cowley, W. Dorland, A.A. Schekochihin, Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows. Reports Prog. Phys. 76(11), 116201 (2013)

    Article  ADS  Google Scholar 

  • B. AbrahamspsShrauner, Exact, stationary wave solutions of the nonlinear vlasov equation. Phys. Fluids 11, 1162–1167 (1968). June

    Article  ADS  Google Scholar 

  • O. Allanson, T. Neukirch, S. Troscheit, F. Wilson, From onedimensional fields to Vlasov equilibria: theory and application of Hermite polynomials. J. Plasma Phys. 82.3, p. 905820306 (2016)

    Google Scholar 

  • O. Allanson, T. Neukirch, F.Wilson, S. Troscheit, An exact collisionless equilibrium for the Force-Free Harris Sheet with low plasma beta. Phys. Plasmas 22.10, 102116 (2015)

    Google Scholar 

  • O. Allanson, S. Troscheit, T. Neukirch, On the inverse problem for Channell collisionless plasma equilibria. IMA J. Appl. Math., hxy026 (2018)

    Google Scholar 

  • W. Alpers, Steady state charge neutral models of the magnetopause. Astrophys. Space Sci. 5, 425–437 (1969). Dec

    Article  ADS  Google Scholar 

  • G.B. Arfken, H.J. Weber, Mathematical Methods for Physicists, 5th edn. (Harcourt/Academic Press, Burlington, MA, 2001)

    MATH  Google Scholar 

  • R.G. Bartle, D.R. Sherbert, Introduction to real Analysis (JohnWiley & Sons Limited, Canada, 2000)

    MATH  Google Scholar 

  • W.H. Bennett, Magnetically self-focussing streams. Phys. Rev. 45(12), 890–897 (1934)

    Article  ADS  Google Scholar 

  • G.G. Bilodeau, TheWeierstrass transform and Hermite polynomials. Duke Math. J. 29(2), 293–308 (1962)

    Article  MathSciNet  Google Scholar 

  • E. Camporeale, G.L. Delzanno, G. Lapenta, W. Daughton, New approach for the study of linear Vlasov stability of inhomogeneous systems. Phys. Plasmas 13.9, 092110, p. 092110 (2006)

    Google Scholar 

  • P.J. Channell, Exact Vlasov-Maxwell equilibria with sheared magnetic fields. Phys. Fluids 19, 1541–1545 (1976). Oct

    Article  ADS  Google Scholar 

  • Clement Mouhot and Cedric Villani, On Landau damping. Acta Math. 207(1), 29–201 (2011)

    Article  MathSciNet  Google Scholar 

  • B. Coppi, G. Laval, R. Pellat, Dynamics of the geomagnetic Tail. Phys. Rev. Lett. 16, 1207–1210 (1966). June

    Article  ADS  Google Scholar 

  • W. Daughton, The unstable eigenmodes of a neutral sheet. Phys. Plasmas 6, 1329–1343 (1999). Apr

    Article  ADS  Google Scholar 

  • J.F. Drake, Y.C. Lee, Kinetic theory of tearing instabilities. Phys. Fluids 20, 1341–1353 (1977). Aug

    Article  ADS  Google Scholar 

  • A.S. Eddington, On a formula for correcting statistics for the effects of a known error of observation. Mon. Not. Royal Astron. Soc. 73, 359–360 (1913). Mar

    Article  ADS  Google Scholar 

  • L.C. Evans, Partial differential equations. Second. Vol. 19. Graduate Studies in Mathematics. Am. Math. Soc. Providence, RI, pp. xxii+749 (2010)

    Google Scholar 

  • R. Fitzpatrick, Plasma Physics: An Introduction (CRC Press, Taylor & Francis Group, 2014)

    Google Scholar 

  • H. Grad, On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2, 331–407 (1949)

    Article  MathSciNet  Google Scholar 

  • I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, series, and products. Seventh. (Elsevier/Academic Press, Amsterdam, pp. xlviii+1171, 2007)

    Google Scholar 

  • E.G. Harris, On a plasma sheath separating regions of oppositely directed magnetic field. Nuovo Cimento 23, 115 (1962)

    Article  Google Scholar 

  • M.G. Harrison, T. Neukirch, One-dimensional Vlasov-maxwell equilibrium for the force-free harris sheet. Phys. Rev. Lett. 102.13, pp. 135003-+ (2009)

    Google Scholar 

  • D.W. Hewett, C.W. Nielson, D. Winske, Vlasov confinement equilibria in one dimension. Phys. Fluids 19, 443–449 (1976). Mar

    Article  ADS  Google Scholar 

  • G.G. Howes, S.C. Cowley, W. Dorland, G.W. Hammett, E. Quataert, A.A. Schekochihin, Astrophysical gyrokinetics: basic equations and linear theory. Astrophys. J. 651, 590–614 (2006). Nov

    Article  ADS  Google Scholar 

  • J.H. Jeans, On the theory of star-streaming and the structure of the universe. Mon. Not. Royal Astron. Soc. 76, 70–84 (1915). Dec

    Article  ADS  Google Scholar 

  • L.D. Landau, On the vibrations of the electronic plasma. J. Phys. (USSR) 10, 25–34 (1946)

    MathSciNet  MATH  Google Scholar 

  • T.G. Northrop, The guiding center approximation to charged particle motion. Ann. Phys. 15, 79–101 (1961). July

    Article  ADS  Google Scholar 

  • K.B. Quest, F.V. Coroniti, Linear theory of tearing in a high-beta plasma. J. Geophys. Res. 86, 3299–3305 (1981). May

    Google Scholar 

  • G. Sansone, Orthogonal functions. Revised English ed. Translated from the Italian by A. H. Diamond; with a foreword by E. Hille. Pure and Applied Mathematics, Vol. IX. Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, pp. xii+411 (1959)

    Google Scholar 

  • A.A. Schekochihin, J.T. Parker, E.G. Highcock, P.J. Dellar, W. Dorland, G.W. Hammett, Phase mixing versus nonlinear advection in drift-kinetic plasma turbulence. J. Plasma Phys. 82, 905820212, p. 47 (2016)

    Google Scholar 

  • A. Suzuki, T. Shigeyama, A novel method to construct stationary solutions of the Vlasov-Maxwell system. Phys. Plasmas 15.4, p. 042107-+ (2008)

    Google Scholar 

  • G.N. Watson, Notes on generating functions of polynomials: (2) Hermite Polynomials. J. London Math. Soc. s1-8.3, 194–199 (1933)

    Google Scholar 

  • E.W. Weisstein, Hermite Polynomial. From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/HermitePolynomial.html (2017)

  • D.V. Widder, Necessary and sufficient conditions for the representation of a function by a Weierstrass transform. Trans. Am. Math. Soc. 71, 430–439 (1951). Nov

    Article  MathSciNet  Google Scholar 

  • D.V. Widder, The convolution transform. Bull. Am. Math. Soc. 60(5), 444–456 (1954). Sept

    Article  MathSciNet  Google Scholar 

  • F. Wilson, T. Neukirch, A family of one-dimensional Vlasov-Maxwell equilibria for the force-free Harris sheet. Phys. Plasmas 18(8), 082108 (2011). Aug

    Article  ADS  Google Scholar 

  • A. Zocco, Linear collisionless Landau damping in Hilbert space. J. Plasma Phys. 81.4, 905810402, p. 049002 (2015)

    Google Scholar 

Download references

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  1. Department of Meteorology, University of Reading, Reading, UK

    Oliver Allanson

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  1. Oliver Allanson
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Allanson, O. (2018). The Use of Hermite Polynomials for the Inverse Problem in One-Dimensional Vlasov-Maxwell Equilibria. In: Theory of One-Dimensional Vlasov-Maxwell Equilibria. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-97541-2_2

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