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CH09_PATTERNS - Chapter 9 Pattern Formation Patterning is a...

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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 Reaction-Diffusion 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 } ) , (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 } ) . (9.4) Let us expand about a homogeneous solution to the local dynamics, R ( { φ i bracerightbig ) = 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
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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 ± 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 ) 0 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. If Re ω 1 ( q = Q ; λ c ) = 0 but Im ω 1 ( q = Q ; λ c ) negationslash = 0 we expect a Hopf bifurcation to a spatiotemporal pattern structure. In the vicinity of a bifurcation, space and time scales associated with the unstable mode(s) tend to infinity. This indicates a critical slowing down . If the unstable modes evolve very slowly, the faster, non-critical modes may be averaged over ( i.e. ‘adiabatically eliminated’).
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9.1. TURING AND HOPF INSTABILITIES 3 For the most unstable mode ω ω 1 , we envisage the following possibilities: ω = ǫ A q 2
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