François Castella

Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems

We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker–Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit; however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.

SOME CONSIDERATIONS ON THE DERIVATION OF THE NONLINEAR QUANTUM BOLTZMANN EQUATION.

In this paper we analyze a system of Nidentical quantum particles in a weak-coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermis Golden Rule. However, we are not able to prove that the quantity neglected while retaining a subseries of the complete original perturbative expansion, indeed vanishes in the limit: we only give plausibility arguments in this direction. The present study holds in any space dimension d≥2.

ON THE WEAK-COUPLING LIMIT FOR BOSONS AND FERMIONS

In this paper we consider a large system of bosons or fermions. We start with an initial datum which is compatible with the Bose–Einstein, respectively Fermi–Dirac, statistics. We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at time t, agrees with the analogous expansion for the solution to the Uehling–Uhlenbeck equation. This paper follows the same spirit as the companion work,2 where the authors investigated the weak-coupling limit for particles obeying the Maxwell–Boltzmann statistics: here, they proved a (much stronger) convergence result towards the solution of the Boltzmann equation.

From the Von-Neumann Equation to the Quantum Boltzmann Equation in a Deterministic Framework

In this paper, we investigate the rigorous convergence of the Density Matrix Equation (or Quantum Liouville Equation) towards the Quantum Boltzmann Equation (or Pauli Master Equation). We start from the Density Matrix Equation posed on a cubic box of size L with periodic boundary conditions, describing the quantum motion of a particle in the box subject to an external potential V. The physics motivates the introduction of a damping term acting on the off-diagonal part of the density matrix, with a characteristic damping time α−1. Then, the convergence can be proved by letting successivelyL tend to infinity and α to zero. The proof relies heavily on a lemma which allows to control some oscillatory integrals posed in large dimensional spaces. The present paper improves a previous announcement [CD].

Modeling of multiwavelength Raman fiber lasers using a new and fast algorithm

We describe a numerical model for a multiwavelength Raman fiber laser. It uses an original algorithm which makes the model robust and fast. We show a comparison between simulated and measured Raman laser output powers, characterized by their slope efficiency and their threshold.

From the von Neumann Equation to the Quantum Boltzmann Equation II: Identifying the Born Series

In a previous paper [Ca1], the author studied a low density limit in the periodic von Neumann equation with potential, modified by a damping term. The model studied in [Ca1], considered in dimensions d≥3, is deterministic. It describes the quantum dynamics of an electron in a periodic box (actually on a torus) containing one obstacle, when the electron additionally interacts with, say, an external bath of photons. The periodicity condition may be replaced by a Dirichlet boundary condition as well. In the appropriate low density asymptotics, followed by the limit where the damping vanishes, the author proved in [Ca1] that the above system is described in the limit by a linear, space homogeneous, Boltzmann equation, with a cross-section given as an explicit power series expansion in the potential. The present paper continues the above study in that it identifies the cross-section previously obtained in [Ca1] as the usual Born series of quantum scattering theory, which is the physically expected result. Hence we establish that a von Neumann equation converges, in the appropriate low density scaling, towards a linear Boltzmann equation with cross-section given by the full Born series expansion: we do not restrict ourselves to a weak coupling limit, where only the first term of the Born series would be obtained (Fermis Golden Rule).

Stroboscopic Averaging for the Nonlinear Schrödinger Equation

In this paper, we are concerned with an averaging procedure, namely Stroboscopic averaging, for highly oscillatory evolution equations posed in a (possibly infinite dimensional) Banach space, typically partial differential equations in a high-frequency regime where only one frequency is present. We construct a high-order averaged system whose solution remains exponentially close to the exact one over long time intervals, possesses the same geometric properties (structure, invariants, ...) as compared to the original system and is non-oscillatory. We then apply our results to the nonlinear Schrödinger equation on the

High-order averaging schemes with error bounds for thermodynamical properties calculations by molecular dynamics simulations.

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