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SYSS: Structure Formation and Self-Organization in non-equilibrium Systems
SYSS 4: Structure Formation and Self-Organization in non-equilibrium Systems III
SYSS 4.2: Hauptvortrag
Freitag, 31. März 2006, 10:45–11:15, HSZ 04
Coarsening versus lengthscale persistence in nonequilibrium pattern-forming systems — •Chaouqi Misbah1 and Paolo Politi2 — 1Laboratoire de Spectrométrie Physique, CNRS, Univ. J. Fourier, Grenoble 1, BP87, F-38402 Saint Martin d’Hères, France — 2Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy
Global evolution of nonequlibrium pattern-formming systems can be broadly classified into two important classes: (i) those which present a persistent length scale, (ii) those which undergo a perpetual coarsening. A general criterion about coarsening for a class of nonlinear evolution equations describing one dimensional pattern-forming systems will be presented. This criterion allows one to discriminate between the situation where a coarsening process takes place and the one where the wavelength is fixed in the course of time. An intermediate scenario may occur, namely ‘interrupted coarsening’. The power of the criterion lies in the fact that the statement about the occurrence of coarsening, or selection of a length scale, can be made by only inspecting the behavior of the branch of steady-state periodic solutions. The criterion states that coarsening occurs if λ′(A)>0 while a lengthscale selection prevails if λ′(A)<0, where λ is the wavelength of the pattern, and A the amplitude of the profile (prime refers to differentiation). This is established thanks to the analysis of the phase diffusion equation of the pattern. The phase diffusion coefficient (which carries a kinetic information)is connected to λ′(A), which refers to a pure steady-state property. The relationship between kinetics and the behavior of the branch of steady-state solutions, is established fully analytically for a class of equations. Another result which emerges from this study is that the exploitation of the phase diffusion equation enables us to determine in a rather straightforward manner the dynamical coarsening exponent. Our calculation is exemplified on several nonlinear equations, showing that the exact exponent is captured. Contrary to many situations where the one dimensional character has proven essential for the derivation of the coarsening exponent, the present idea can be used, in principle, at any dimension. Some speculations about the extension of the present results will be outlined.