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Abstract
Vlasov–Maxwell equilibria are described by the selfconsistent solutions of the timeindependent Maxwell equations for the realspace dynamics of electromagnetic fields and the Vlasov equation for the phasespace dynamics of particle distribution functions (DFs) in a collisionless plasma. These two systems (macroscopic and microscopic) are coupled via the source terms in Maxwell’s equations, which are sums of velocityspace ‘moment’ integrals of the particle DF. This paper considers a particular subset of solutions of the broad plasma physics problem: ‘the inverse problem for collisionless equilibria’ (IPCE), viz. ‘given information regarding the macroscopic configuration of a collisionless plasma equilibrium, what selfconsistent equilibrium DFs exist?’ We introduce the constants of motion approach to IPCE using the assumptions of a ‘modified Maxwellian’ DF, and a strictly neutral and spatially onedimensional plasma, and this is consistent with ‘Channell’s method’ (Channell, 1976, Exact VlasovMaxwell equilibria with sheared magnetic fields. Phys. Fluids, 19, 1541–1545). In such circumstances, IPCE formally reduces to the inversion of Weierstrass transformations (Bilodeau, 1962, The Weierstrass transform and Hermite polynomials. Duke Math. J., 29, 293–308). These are the same transformations that feature in the initial value problem for the heat/diffusion equation. We discuss the various mathematical conditions that a candidate solution of IPCE must satisfy. One method that can be used to invert the Weierstrass transform is expansions in Hermite polynomials. Building on the results of Allanson et al. (2016, From onedimensional fields to Vlasov equilibria: Theory and application of Hermite polynomials. Journal of Plasma Physics, 82, 905820306), we establish under what circumstances a solution obtained by these means converges and allows velocity moments of all orders. Ever since the seminal work by Bernstein et al. (1957, Exact nonlinear plasma oscillations. Phys. Rev., 108, 546–550) on ‘stationary’ electrostatic plasma waves, the necessary quality of nonnegativity has been noted as a feature that any candidate solution of IPCE will not a priori satisfy. We discuss this problem in the context of Channell equilibria, for magnetized plasmas.
Original language  English 

Pages (fromto)  849873 
Number of pages  25 
Journal  IMA Journal of Applied Mathematics 
Volume  83 
Issue number  5 
Early online date  7 Jun 2018 
DOIs  
Publication status  Published  24 Sept 2018 
Keywords
 Kinetic physics
 Collisionless plasma
 Hermite polynomial
 Inverse problem
 Maxwell's equations
 Vlasov equation
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 2 Finished

Plasma Theory: Solar and Magnetospheric Plasma Theory
Hood, A. W., Mackay, D. H., Neukirch, T., Parnell, C. E., Priest, E. R., Archontis, V., Cargill, P., De Moortel, I. & Wright, A. N.
Science & Technology Facilities Council
1/04/13 → 31/03/16
Project: Standard