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Theoretical methods in pattern formation in Physics, Chemistry and Biology

Cross, M. C. (1988) Theoretical methods in pattern formation in Physics, Chemistry and Biology. In: Far from Equilibrium Phase Transitions. Lecture Notes in Physics. No.319. Springer Berlin Heidelberg , Berlin, Heidelberg, pp. 45-74. ISBN 9783540506430. https://resolver.caltech.edu/CaltechAUTHORS:20200930-113055493

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Abstract

In these lectures some simple ideas common to theories of pattern formation in diverse areas of Physics, Chemistry and Biology are reviewed. The unifying theme of these theoretical models is the existence of an instability (bifurcation) in a system driven far from equilibrium to a state with spatial variation at a non-zero wave vector. The basic pattern forming tendency is understood by a linear stability analysis of the spatially uniform system. The subsequent saturation of the exponential growth by non-linearities together with boundary effects (the influence of the finite size of the domain) leads to the complexity and subtlety in the understanding of non-equilibrium structures. Some elements of the theory of linear instability, non-linearity and boundary effects will be briefly reviewed in these lectures. The emphasis is on common features of the theoretical modeling of the phenomena. No attempt will be made to assess the detailed applicability of these models to the different fields, or the main aims of the study of structure formation in the different fields. There are clearly great differences between the carefully controlled laboratory models of the physicist studying hydrodynamic instabilities the chemist studying transient evolution of complicated chemical reactions, and the in vivo observations of the biologist studying developing organisms, differences in the ability to control the system, the understanding of the underlying microscopic processes, and particularly the goals of the endeavor. At the current elementary understanding of the general phenomena, these disparities are not very evident in the theoretical modeling. In the second part of the lectures I will report on two examples of the application of some of the above methods. These lectures are derived from a longer forthcoming review with P.C. Hohenberg. A general overview has been presented elsewhere (Hohenberg and Cross, 1986): The present lectures provide a more detailed presentation of a narrower range of topics.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/3-540-50643-8_33DOIArticle
https://rdcu.be/b7U9ZPublisherFree ReadCube access
Additional Information:© Springer-Verlag 1988.
Subject Keywords:Rayleigh Number; Couette Flow; Phase Equation; Linear Instability; Amplitude Equation
Series Name:Lecture Notes in Physics
Issue or Number:319
Record Number:CaltechAUTHORS:20200930-113055493
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200930-113055493
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:105686
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:30 Sep 2020 19:32
Last Modified:30 Sep 2020 19:32

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