Pattern Size in Gaussian Fields from Spinodal Decomposition

  • We study the two-dimensional snake-like pattern that arises in phase separation of alloys described by spinodal decomposition in the Cahn-Hilliard model. These are somewhat universal pattern due to an overlay of eigenfunctions of the Laplacian with a similar wave-number. Similar structures appear in other models like reaction-diffusion systems describing animal coats' patterns or vegetation patterns in desertification. Our main result studies random functions given by cosine Fourier series with independent Gaussian coefficients, which are taken over domains in Fourier space that grow and scale with aparameter of order 1/ε. Using a theorem by Edelman and Kostlan and ergodic theory, we show that on any straight line through the spatial domain the average distance of zeros of the series is asymptotically of order ε with a precisely given constant.

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Author:Luigi Amedeo Bianchi, Dirk BlömkerGND, Philipp Düren
Frontdoor URL
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2015-13)
Year of first Publication:2015
Publishing Institution:Universität Augsburg
Release Date:2015/10/08
Tag:Gaussian fields; pattern formation; spinodal decomposition; ergodic theorem
GND-Keyword:Gauß-Zufallsfeld; Musterbildung; Spinodale Entmischung; Ergodentheorie; Cahn-Hilliard-Gleichung
Erschienen in SIAM Journal on Applied Mathematics, 77, 4, S. 1292-1319,
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht mit Print on Demand