Physics 313: Lecture 6
Wednesday, 9/10/08
Comments to the Class
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You should finish reading Chapter 3 of Cross-Greenside
and start Chapter 4.
Topics:
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Type I, II, and III instabilities, each 's' or 'o': six
categories in all.
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Chapter 3: Turing-instability of two interacting
chemicals.
Vocabulary Check
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Extended, confined coordinates?
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Primary, secondary, tertiary instabilities?
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Reduced bifurcation parameter?
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Neutral stability curve?
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Coherence length?
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Type “s” or type “o” instability?
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Translationally or rotationally invariant system, equation,
state?
Type I, II, III instabilities
Stationary “s” or oscillatory “o”
Note: growth rate Re[q] typically has
multiple extrema but typically only one
extremum crosses zero first. Need to
vary two parameters to get two extrema
to cross at same time.
Seashells As Illustration of Instability Types
Neutral Stability Region Re[q] =0
For Swift-Hohenberg
Linear Instability of Single-Reagent
Reaction-Diffusion System
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How does analysis change if we allow two
extended coordinates, c=c(x,y,t)?
Growth Rate for Isotropic Systems
Pattern of Random Critical States
nstates = 8
angles = Table[ Random[ Real, {0, 2 Pi} ], {nstates} ]
phases = Table[ Random[ Real, {0, 2 Pi} ], {nstates} ]
amps = Table[ Random[ Real, {0, 1.0} ], {nstates} ]
f2d[x_, y_] :=
Sum[ amps[[i]] Cos[ Cos[angles[[i]]] x + Sin[angles[[i]]] y + phases[[i]]] ,
{i, 1, nstates} ]
DensityPlot[ f2d[x,y], {x,0,10Pi}, {y,0,10Pi}, PlotPoints -> 50 ]
Growth Rate Surfaces For Uniaxial
Anisotropic Medium
Nematic liquid crystal
Growth Rate for Anisotropic Uniaxial System