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Semigroup representation of the Vlasov evolution

Published online by Cambridge University Press:  01 June 1998

I. PRIGOGINE  and
T. PETROSKY
I. PRIGOGINE
Affiliation:
International Solvay Institutes for Physics and Chemistry, Université Libre de Bruxelles, Campus Plaine CP231, Boulevard de Triomphe, B-1050 Bruxelles, Belgium, and Center for Studies in Statistical Mechanics and Complex Systems, The University of Texas at Austin, Austin, Texas 78712 USA
T. PETROSKY
Affiliation:
International Solvay Institutes for Physics and Chemistry, Université Libre de Bruxelles, Campus Plaine CP231, Boulevard de Triomphe, B-1050 Bruxelles, Belgium, and Center for Studies in Statistical Mechanics and Complex Systems, The University of Texas at Austin, Austin, Texas 78712 USA
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Abstract

The well-known van Kampen–Case treatment of the Vlasov equation leads to a spectrum on the real axis. In this paper we show that, by going to a ‘rigged’ Hilbert space, we can derive a spectral representation that is complex and breaks time symmetry. This leads to a semigroup description in which the decay rates due to the Landau damping appear explicitly in the spectrum. Moreover, we can then define an entropy. In this way, the relation between Landau damping and irreversibility is made explicit. The analogy with the well-known Friedrichs model is stressed.

Type
Research Article
Information
Journal of Plasma Physics , Volume 59 , Issue 4 , June 1998 , pp. 611 - 618
Copyright
© 1998 Cambridge University Press

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