5,290 research outputs found
Dynamics of the Desai-Zwanzig model in multiwell and random energy landscapes
We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. {\bf 19}1-24 (1978)]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multi-well potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov equation), the minimization of the free energy functional and a continuation algorithm for the stationary solutions
An exploding glass ?
We propose a connection between self-similar, focusing dynamics in nonlinear
partial differential equations (PDEs) and macroscopic dynamic features of the
glass transition. In particular, we explore the divergence of the appropriate
relaxation times in the case of hard spheres as the limit of random close
packing is approached. We illustrate the analogy in the critical case, and
suggest a ``normal form'' that can capture the onset of dynamic self-similarity
in both phenomena.Comment: 8 pages, 2 figure
Effect of Noise on Excursions To and Back From Infinity
The effect of additive white noise on a model for bursting behavior in large
aspect-ratio binary fluid convection is considered. Such bursts are present in
systems with nearly square symmetry and are the result of heteroclinic cycles
involving infinite amplitude states created when the square symmetry is broken.
A combination of numerical results and analytical arguments show how even a
very small amount of noise can have a very large effect on the amplitudes of
successive bursts. Large enough noise can also affect the physical
manifestations of the bursts. Finally, it is shown that related bursts may
occur when white noise is added to the normal form equations for the Hopf
bifurcation with exact square symmetry.Comment: 17 pages, 9 figure
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
Breathing Spots in a Reaction-Diffusion System
A quasi-2-dimensional stationary spot in a disk-shaped chemical reactor is
observed to bifurcate to an oscillating spot when a control parameter is
increased beyond a critical value. Further increase of the control parameter
leads to the collapse and disappearance of the spot. Analysis of a bistable
activator-inhibitor model indicates that the observed behavior is a consequence
of interaction of the front with the boundary near a parity breaking front
bifurcation.Comment: 4 pages RevTeX, see also http://chaos.ph.utexas.edu/ and
http://t7.lanl.gov/People/Aric
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