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Mean-field evolution of fermionic systems. (English) Zbl 1304.82061
Authors’ abstract: The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems. In this paper, we study the dynamics of initial data close to a Slater determinant, whose reduced one-particle density is an orthogonal projection $$\omega_N$$ with the appropriate semiclassical structure. Assuming some regularity of the interaction potential, we show that the evolution of such an initial data remains close to a Slater determinant, with reduced one-particle density given by the solution of the Hartree-Fock equation with initial data $$\omega_N$$. Our result holds for all (semiclassical) times, and gives effective bounds on the rate of the convergence towards the Hartree-Fock dynamics.

##### MSC:
 82C40 Kinetic theory of gases in time-dependent statistical mechanics 81V70 Many-body theory; quantum Hall effect 35Q40 PDEs in connection with quantum mechanics 35Q41 Time-dependent Schrödinger equations and Dirac equations 35Q55 NLS equations (nonlinear Schrödinger equations) 82B30 Statistical thermodynamics 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 82D20 Statistical mechanics of solids 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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