35 research outputs found
A constrained optimization problem in quantum statistical physics
In this paper, we consider the problem of minimizing quantum free energies
under the constraint that the density of particles is fixed at each point of
Rd, for any d 1. We are more particularly interested in the
characterization of the minimizer, which is a self-adjoint nonnegative trace
class operator, and will show that it is solution to a nonlinear
self-consistent problem. This question of deriving quantum statistical
equilibria is at the heart of the quantum hydrody-namical models introduced by
Degond and Ringhofer. An original feature of the problem is the local nature of
constraint, i.e. it depends on position, while more classical models consider
the total number of particles in the system to be fixed. This raises
difficulties in the derivation of the Euler-Lagrange equations and in the
characterization of the minimizer, which are tackled in part by a careful
parametrization of the feasible set
Vortex patterns in the almost-bosonic anyon gas
We study theoretically and numerically the ground state of a gas of 2D
abelian anyons in an external trapping potential. We treat anyon statistics in
the magnetic gauge picture, perturbatively around the bosonic end. This leads
to a mean-field energy functional, whose ground state displays vortex lattices
similar to those found in rotating Bose-Einstein condensates. A crucial
difference is however that the vortex density is proportional to the underlying
matter density of the gas
Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity
International audienceThe aim of this chapter is first to give an introduction to the derivation of the Gross-Pitaevskii Equations (GPEs) that arise in the modeling of Bose-Einstein Condensates (BECs). In particular, we describe some physical problems related to stationary states, dynamics, multi-components BECs and the possibility of handling stochastic effects into the equation. Next, we explain how to compute the stationary (and ground) states of the GPEs through the imaginary time method (also called Conjugate Normalized Gradient Flow) and finite difference or pseudo-spectral dis-cretization techniques. Examples are provided by using GPELab which is a Mat-lab toolbox dedicated to the numerical solution of GPEs. Finally, we explain how to discretize correctly the time-dependent GPE so that the schemes are physically admissible. We again provide some examples by using GPELab. Furthermore, extensions of the discretization schemes to some classes of stochastic (in time) GPEs are described and analyzed
Stochastic regularization effects of semi-martingales on random functions
International audienceIn this paper we address an open question formulated in [17]. That is, we extend the ItĂ´-Tanaka trick, which links the time-average of a deterministic function f depending on a stochastic process X and F the solution of the Fokker-Planck equation associated to X, to random mappings f. To this end we provide new results on a class of adpated and non-adapted Fokker-Planck SPDEs and BSPDEs
Acceleration of the imaginary time method for spectrally computing the stationary states of Gross-Pitaevskii equations
International audienceThe aim of this paper is to propose a simple accelerated spectral gradient flow formulation for solving the Gross-Pitaevskii Equation (GPE) when computing the stationary states of Bose-Einstein Condensates. The new algorithm, based on the recent iPiano minimization algorithm [35], converges three to four times faster than the standard implicit gradient scheme. To support the method, we provide a complete numerical study for 1d-2d-3d GPEs, including rotation and dipolar terms
Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose-Einstein condensates
International audienceWe consider the Backward Euler SPectral (BESP) scheme proposed in [10] for computing the stationary states of Bose-Einstein Condensates (BECs) through the Gross-Pitaevskii equation. We show that the fixed point approach introduced in [10] fails to converge for fast rotating BECs. A simple alternative approach based on Krylov subspace solvers with a Laplace or Thomas-Fermi preconditioner is given. Numerical simulations (obtained with the associated freely available Matlab toolbox GPELab) for complex configurations show that the method is accurate, fast and robust for 2D/3D problems and multi-components BECs