38 research outputs found
Stochastic Turing patterns in the Brusselator model
A stochastic version of the Brusselator model is proposed and studied via the
system size expansion. The mean-field equations are derived and shown to yield
to organized Turing patterns within a specific parameters region. When
determining the Turing condition for instability, we pay particular attention
to the role of cross diffusive terms, often neglected in the heuristic
derivation of reaction diffusion schemes. Stochastic fluctuations are shown to
give rise to spatially ordered solutions, sharing the same quantitative
characteristic of the mean-field based Turing scenario, in term of excited
wavelengths. Interestingly, the region of parameter yielding to the stochastic
self-organization is wider than that determined via the conventional Turing
approach, suggesting that the condition for spatial order to appear can be less
stringent than customarily believed.Comment: modified version submitted to Phys Rev. E. 5. 3 Figures (5 panels)
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Dynamical Reduction of Discrete Systems Based on the Renormalization Group Method
The renormalization group (RG) method is extended for global asymptotic
analysis of discrete systems. We show that the RG equation in the discretized
form leads to difference equations corresponding to the Stuart-Landau or
Ginzburg-Landau equations. We propose a discretization scheme which leads to a
faithful discretization of the reduced dynamics of the original differential
equations.Comment: LaTEX. 12pages. 1 figure include
Influence of through-flow on linear pattern formation properties in binary mixture convection
We investigate how a horizontal plane Poiseuille shear flow changes linear
convection properties in binary fluid layers heated from below. The full linear
field equations are solved with a shooting method for realistic top and bottom
boundary conditions. Through-flow induced changes of the bifurcation thresholds
(stability boundaries) for different types of convective solutions are deter-
mined in the control parameter space spanned by Rayleigh number, Soret coupling
(positive as well as negative), and through-flow Reynolds number. We elucidate
the through-flow induced lifting of the Hopf symmetry degeneracy of left and
right traveling waves in mixtures with negative Soret coupling. Finally we
determine with a saddle point analysis of the complex dispersion relation of
the field equations over the complex wave number plane the borders between
absolute and convective instabilities for different types of perturbations in
comparison with the appropriate Ginzburg-Landau amplitude equation
approximation. PACS:47.20.-k,47.20.Bp, 47.15.-x,47.54.+rComment: 19 pages, 15 Postscript figure
Chemical morphogenenis: recent experimental advances in reaction-diffusion system design and control
In his seminal 1952 paper, Alan Turing predicted that diffusion could spontaneously drive an initially uniformsolution of reacting chemicals to develop stable spatially periodic concentration patterns. It took nearly 40 years before the first two unquestionable experimental demonstrations of such reaction-diffusion patterns could be made in isothermal single phase reaction systems. The number of these examples stagnated for nearly 20 years.We recently proposed a design method that made their number increase to six in less than 3 years. In this report, we formally justify our original semi-empiricalmethod and support the approach with numerical simulations based on a simple but realistic kinetic model. To retain a number of basic properties of real spatial reactors but keep calculations to a minimal complexity, we introduce a new way to collapse the confined spatial direction of these reactors.Contrary to similar reduced descriptions, we take into account the effect of the geometric size in the confinement direction and the influence of the differences in the diffusion coefficient on exchange rates of species with their feed environment. We experimentally support the method by the observation of stationary patterns in red-ox reactions not based on oxihalogen chemistry. Emphasis is also brought on how one of these new systems can process different initial conditions and memorize them in the form of localized patterns of different geometries