379 research outputs found
A Tale of Two Distributions: From Few To Many Vortices In Quasi-Two-Dimensional Bose-Einstein Condensates
Motivated by the recent successes of particle models in capturing the
precession and interactions of vortex structures in quasi-two-dimensional
Bose-Einstein condensates, we revisit the relevant systems of ordinary
differential equations. We consider the number of vortices as a parameter
and explore the prototypical configurations ("ground states") that arise in the
case of few or many vortices. In the case of few vortices, we modify the
classical result of Havelock [Phil. Mag. , 617 (1931)] illustrating
that vortex polygons in the form of a ring are unstable for .
Additionally, we reconcile this modification with the recent identification of
symmetry breaking bifurcations for the cases of . We also briefly
discuss the case of a ring of vortices surrounding a central vortex (so-called
configuration). We finally examine the opposite limit of large and
illustrate how a coarse-graining, continuum approach enables the accurate
identification of the radial distribution of vortices in that limit.Comment: 15 pages, 2 figure
Analysis of a stochastic chemical system close to a sniper bifurcation of its mean field model
A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs for example in the modelling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) is studied. Our approach is based on the chemical Fokker Planck equation. To get some insights into advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, before the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size
Spatially Distributed Stochastic Systems: equation-free and equation-assisted preconditioned computation
Spatially distributed problems are often approximately modelled in terms of
partial differential equations (PDEs) for appropriate coarse-grained quantities
(e.g. concentrations). The derivation of accurate such PDEs starting from finer
scale, atomistic models, and using suitable averaging, is often a challenging
task; approximate PDEs are typically obtained through mathematical closure
procedures (e.g. mean-field approximations). In this paper, we show how such
approximate macroscopic PDEs can be exploited in constructing preconditioners
to accelerate stochastic simulations for spatially distributed particle-based
process models. We illustrate how such preconditioning can improve the
convergence of equation-free coarse-grained methods based on coarse
timesteppers. Our model problem is a stochastic reaction-diffusion model
capable of exhibiting Turing instabilities.Comment: 8 pages, 6 figures, submitted to Journal of Chemical Physic
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