Lyapunov stability and poisson structure of the thermal TDHF and RPA equations

Journal Article · Wed Nov 01 04:00:00 EST 1989 · Annals of Physics (New York); (USA)

DOI:https://doi.org/10.1016/0003-4916(89)90247-9· OSTI ID:7243277

Balian, R ^[1]; Veneroni, M ^[2]

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1. Service de Physique, Theorique de Saclay, F-91191 Gif-sur-Yvette Cedex, (FR)
2. Division de Physique Theorique, Institut de Physique Nucleaire, 91406 Orsay Cedex, (FR)

The thermal TDHF equation is analyzed in the Liouville representation of quantum mechanics, where the matrix elements of the single-particle (s.p) density {rho} behave as classical dynamical variables. By introducing the Lie--Poisson bracket associated with the unitary group of the s.p. Hilbert space, we show that TDHF has a Hamiltonian, but non-canonical, classical form. Within this Poisson structure, either the s.p. energy or the s.p. grand potential {Omega}({rho}) act as a Hamilton function. The Lyapunov stability of both the TDHF and RPA equations around a HF state then follows, since the HF approximation for thermal equilibrium is determined by minimizing {Omega}({rho}). The RPA matrix in the Liouville space is expressed as the product of the Poisson tensor with the HF stability matrix, interpreted as a metric tensor generated by the entropy. This factorization displays the roles of the energy and entropy terms arising from {Omega}({rho}) in the RPA dynamics, and it helps to construct the RPA modes. Several extensions are considered. {copyright} 1989 Academic Press, Inc.

• OSTI ID: 7243277
• Journal Information: Annals of Physics (New York); (USA), Journal Name: Annals of Physics (New York); (USA) Vol. 195:2; ISSN 0003-4916; ISSN APNYA
• Country of Publication: United States
• Language: English

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