GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: Computation of stationary solutions

The crystallization conjecture: a review

“…Since in R d the points x i will have a typical distance of order N −1/d , the measure σ plays the same role as the macroscopic measure M discussed in (13)- (14). For instance, in the case of the Thomson problem for which V ext confines the particles to the unit sphere S 2 and V (|x|) = |x| −1 , the solution is unique and equal to the uniform measure σ = (4π)…”

“…The microscopic scale convergence (16) does not give any information, in principle, about the behavior at the macroscopic scale, such as the convergence of the dilated measure M N defined by (14). Conversely, the convergence of M N does not give any clue about that of µ N .…”

“…The macroscopic object and its microscopic structure. We mentioned above the question of proving the existence of a macroscopic measure M, obtained as the weak limit (14). This is related to the formation of a macroscopic object.…”

“…The knowledge about the positions of particles is very crude, because of the dilation of factor N −1/d , which does not account for the local behavior of the system. Finding conditions on the potential which imply the existence of the weak limit (14) is an important problem. However, it has never been, to our knowledge, studied mathematically.…”

Modeling and Computation of Bose-Einstein Condensates: Stationary States, Nucleation, Dynamics, Stochasticity

“…To show how GEPLab is powerful, we now consider a few numerical examples. Other interesting situations (with downloadable source files) are given in the GPELab user guide and the associated papers [11,13].…”