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Title : Stability of thermodynamic equilibria for the Hartree-Fock equation with exchange term
Abstract
In the Hartree-Fock equation, which models the evolution of a particle system under symmetry assumptions, the energy exchange term between particles is often neglected in favor of the so-called mean field term. Indeed, under certain structural assumptions about the interaction between particles, these two terms corroborate each other, as is the case for point interaction potentials, i.e., when the Hartree-Fock equation reduces to the Schrödinger equation. On the other hand, the distinction between these two terms has no impact on the locally well-posedness of the equation. Nevertheless, for the global problem, and especially for the problem of asymptotic stability of non-localized equilibria of the equation, the two terms play a very different role and modify the analysis of the linearized equation around the equilibrium under consideration. This talk will present the Hartree-Fock equation with exchange term, its heuristic derivation and its associated equilibria. Finally, a result on the asymptotic stability of thermodynamic equilibria will be presented. This is a collaborative result with Charles Collot (CYU), Elena Danesi (Padova) and Cyril Malézé (Ecole Polytechnique).
