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Erzeugende Funktionen für die exakte Lösung kinetischer Gleichungen vom Vlasov‐Typ
Author(s) -
Vojta Günter
Publication year - 1970
Publication title -
beiträge aus der plasmaphysik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.531
H-Index - 47
eISSN - 1521-3986
pISSN - 0005-8025
DOI - 10.1002/ctpp.19700100402
Subject(s) - vlasov equation , series (stratigraphy) , mathematics , partial differential equation , differential equation , mathematical physics , series expansion , hermite polynomials , mathematical analysis , physics , quantum mechanics , plasma , paleontology , biology
Kinetic equations of the Vlasov type play a dominant role in the theories of gaseous and semiconductor plasmas (self‐consistent field approximation), in the dynamics of quantum systems with collective interactions (Hartree‐Fock approximation), also of spin systems Weiss molecular field approximation, in the theory of liquids and in astrophysics. Here a new general exact method of solving the linearized Vlasov equation is given in the language of classical plasma theory. This method is based on a series expansion of the distribution function in terms of Hermite polynomials in velocity space. A partial differential equation for the generating function of the time‐dependent expansion coefficients is derived and solved yielding a closed formula for the general term of the series expansion. Thus the problem of solving the kinetic integro‐differential equation is reduced to the more tractable problem of solving a single differential equation, and the mathematical structure of the series expansion becomes very clear. The full equivalence of the solution given with the solution of van Kampen und Case in terms of singular normal modes can be demonstrated.