We consider the Wigner equation corresponding to a nonlinear Schrödinger evolution of the Hartree type in the semiclassical limit → 0.Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in L 2 to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology.The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the L 2 norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which -as it is well known -is not pointwise positive in general.
We consider a quantum system constituted by N identical particles interacting by means of a mean-field Hamiltonian. It is well known that, in the limit N -> 8, the one-particle state obeys to the Hartree equation. Moreover, propagation of chaos holds. In this paper, we take care of the h dependence by considering the semiclassical expansion of the N-particle system. We prove that each term of the expansion agrees, in the limit N -> 8, with the corresponding one associated with the Hartree equation. We work in the classical phase space by using the Wigner formalism, which seems to be the most appropriate for the present problem
Abstract. We consider the Lorentz model in a slab with two mass reservoirs at the boundaries. We show that, in a low density regime, there exists a unique stationary solution for the microscopic dynamics which converges to the stationary solution of the heat equation, namely to the linear profile of the density. In the same regime the macroscopic current in the stationary state is given by the Fick's law, with the diffusion coefficient determined by the Green-Kubo formula.
The temporal evolution of a perturbation of the equilibrium distribution of a
condensed Bose gas is investigated using the kinetic equation which describes
collision between condensate and noncondensate atoms. The dynamics is studied
in the low momentum limit where an analytical treatment is feasible. Explicit
results are given for the behavior at large times in different temperature
regimes.Comment: 25 pages, 3 figures. Typos corrected. Final version to appear in
Annals of Physic
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