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Unformatted text preview: Chapter 9 Pattern Formation Patterning is a common occurrence found in a wide variety of physical systems, including chemically active media, fluids far from equilibrium, liquid crystals, etc. In this chapter we will touch very briefly on the basic physics of patterning instabilities. 9.0.1 ReactionDiffusion Dynamics Revisited Let braceleftbig φ i ( r ,t ) bracerightbig denote a set of scalar fields satisfying ∂ t φ i + ∇ · J i = R i , (9.1) where J α i = − D αβ ij ∂ β φ j (9.2) is the α component of the current density of species i . We assume that the local reaction kinetics is given by R i = R i ( { φ j } ,λ ) , (9.3) where λ is a control parameter, or possibly a set of control parameters. Thus, ∂ t φ i = ∂ α D αβ ij ∂ β φ j + R i ( { φ j } ,λ ) . (9.4) Let us expand about a homogeneous solution to the local dynamics, R ( { φ ∗ i bracerightbig ,λ ) = 0, writing φ i ( r ,t ) = φ ∗ i + η i ( r ,t ) . (9.5) We then have ∂ t η i = ∂R i ∂φ j vextendsingle vextendsingle vextendsingle vextendsingle φ ∗ η j + ∂ α D αβ ij ∂ β η j . (9.6) Assuming D αβ ij is constant in space, we obtain the linear equation ∂ t ˆ η ( q ,t ) = L ij ( q ; λ ) ˆ η j ( q ,t ) , (9.7) 1 2 CHAPTER 9. PATTERN FORMATION Figure 9.1: Instabilities in linear systems ˙ η = L η occur when the real part of the largest eigenvalue of L crosses zero. If L is a real operator, its eigenvalues are either real or come in complex conjugate pairs. where L ij ( q ; λ ) = ∂R i ∂φ j vextendsingle vextendsingle vextendsingle vextendsingle φ ∗ − D αβ ij q α q β . (9.8) Let P ( ω ) = det ( ω I − L ) (9.9) be the characteristic polynomial for L ( q ; λ ). The eigenvalues ω a ( q ; λ ) satisfy P ( ω a ) = 0. If we assume that L ij ( q ; λ ) ∈ R is real, then P ( ω ∗ ) = bracketleftbig P ( ω ) bracketrightbig ∗ , which means that the eigenvalues ω a are either purely real or else come in complex conjugate pairs ω a, 1 ± iω a, 2 . The eigenvectors ψ a i ( q ; λ ) need not be real, since L is not necessarily Hermitian. The general solution is then η i ( q t ) = summationdisplay a C a ψ a i ( q ; λ ) e ω a ( q ; λ ) t . (9.10) Modes with Re ω a > 0 are stabilized by nonlinear terms, e.g. ˙ A = rA − A 3 . Let’s assume the eigenvalues are ordered so that Re ( ω a ) ≥ Re ( ω a +1 ), and that Re ( ω 1 ) ≤ for λ ≤ λ c . • If ω 1 ( q = 0; λ c ) = 0, we expect a transition between homogeneous ( q = 0) states at λ = λ c . • If ω 1 ( q = Q ; λ c ) = 0, we expect a transition to a spatially modulated structure with wavevector Q . • If Re ω 1 ( q = 0; λ c ) = 0 but Im ω 1 ( q = 0; λ c ) negationslash = 0 we expect a Hopf bifurcation and limit cycle behavior....
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This note was uploaded on 09/30/2011 for the course PHYS 221a taught by Professor Staff during the Fall '11 term at UCSD.
 Fall '11
 staff
 Physics

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