1,905 research outputs found
High density limit of the stationary one dimensional Schrödinger-Poisson system
International audienceThe stationary one dimensional Schrödinger-Poisson system on a bounded interval is considered in the limit of a small Debye length (or small temperature). Electrons are supposed to be in a mixed state with the Boltzmann statistics. Using various reformulations of the system as convex minimization problems, we show that only the first energy level is asymptotically occupied. The electrostatic potential is shown to converge towards a boundary layer potential with a profile computed by means of a half space Schrödinger-Poisson system
Invariant measures of the 2D Euler and Vlasov equations
We discuss invariant measures of partial differential equations such as the
2D Euler or Vlasov equations. For the 2D Euler equations, starting from the
Liouville theorem, valid for N-dimensional approximations of the dynamics, we
define the microcanonical measure as a limit measure where N goes to infinity.
When only the energy and enstrophy invariants are taken into account, we give
an explicit computation to prove the following result: the microcanonical
measure is actually a Young measure corresponding to the maximization of a
mean-field entropy. We explain why this result remains true for more general
microcanonical measures, when all the dynamical invariants are taken into
account. We give an explicit proof that these microcanonical measures are
invariant measures for the dynamics of the 2D Euler equations. We describe a
more general set of invariant measures, and discuss briefly their stability and
their consequence for the ergodicity of the 2D Euler equations. The extension
of these results to the Vlasov equations is also discussed, together with a
proof of the uniqueness of statistical equilibria, for Vlasov equations with
repulsive convex potentials. Even if we consider, in this paper, invariant
measures only for Hamiltonian equations, with no fluxes of conserved
quantities, we think this work is an important step towards the description of
non-equilibrium invariant measures with fluxes.Comment: 40 page
Systems of Points with Coulomb Interactions
Large ensembles of points with Coulomb interactions arise in various settings
of condensed matter physics, classical and quantum mechanics, statistical
mechanics, random matrices and even approximation theory, and give rise to a
variety of questions pertaining to calculus of variations, Partial Differential
Equations and probability. We will review these as well as "the mean-field
limit" results that allow to derive effective models and equations describing
the system at the macroscopic scale. We then explain how to analyze the next
order beyond the mean-field limit, giving information on the system at the
microscopic level. In the setting of statistical mechanics, this allows for
instance to observe the effect of the temperature and to connect with
crystallization questions.Comment: 30 pages, to appear as Proceedings of the ICM201
Efficient description of Bose-Einstein condensates in time-dependent rotating traps
Quantum sensors based on matter-wave interferometry are promising candidates
for high-precision gravimetry and inertial sensing in space. The favorable
source for the coherent matter waves in these devices are Bose-Einstein
condensates. A reliable prediction of their dynamics, which is governed by the
Gross-Pitaevskii equation, requires suitable analytical and numerical methods
which take into account the center-of-mass motion of the condensate, its
rotation and its spatial expansion by many orders of magnitude. In this
chapter, we present an efficient way to study their dynamics in time-dependent
rotating traps that meet this objective. Both, an approximate analytical
solution for condensates in the Thomas-Fermi regime and dedicated numerical
simulations on a variable adapted grid are discussed. We contrast and relate
our approach to previous alternative methods and provide further results, such
as analytical expressions for the one- and two-dimensional spatial density
distributions and the momentum distribution in the long-time limit that are of
immediate interest to experimentalists working in this field of research.Comment: 49 pages, 7 figures, preprint submitted to Advances in Atomic,
Molecular, and Optical Physics Volume 6
Double scale analysis of a Schrödinger-Poisson system with quantum wells and macroscopic nonlinearities in dimensions 2 and 3
We consider the stationary Schrödinger-Poisson model with a background potential describing a quantum well. The Hamiltonian of this system composes of contributions the background potential well plus a nonlinear repulsive term which extends on different length scales with ratio parametrized by the small parameter h. With a partition function which forces the particles to remain in the quantum well, the limit h?0 in the nonlinear system leads to different asymptotic behaviours, including spectral renormalization, depending on the dimensions 1, 2 or 3
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