Non-linear Liouville and Shrödinger equations in phase space

M.C.B. Fernandes, F.C. Khanna, M.G.R. Martins, A.E. Santana and J.D.M. Vianna

Physica A: Statistical Mechanics and its Applications, 2010, vol. 389, issue 17, 3409-3419

Abstract: Unitary representations of the Galilei group are studied in phase space, in order to describe classical and quantum systems. Conditions to write in general form the generator of time translation and Lagrangians in phase space are then established. In the classical case, Galilean invariance provides conditions for writing the Liouville operator and Lagrangian for non-linear systems. We analyze, as an example, a generalized kinetic equation where the collision term is local and non-linear. The quantum counter-part of such unitary representations are developed by using the Moyal (or star) product. Then a non-linear Schrödinger equation in phase space is derived and analyzed. In this case, an association with the Wigner formalism is established, which provides a physical interpretation for the formalism.

Keywords: Galilei group; Kinetic theory; Non-linear equations in phase space (search for similar items in EconPapers)
Date: 2010
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:389:y:2010:i:17:p:3409-3419

DOI: 10.1016/j.physa.2010.04.030

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