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s(k→, M(k) will also use this compact notation (s=1,2,3). Of course, labels of points in the Brillouin zone k→, still stand for the actual wave vector k→=!k→!k̂. To emphasize that textural nonuniformity results from the large-scale compatibility properties of the ("anisotropic continuum") solid, rather than from trivial small-scale discreteness, we use the long wavelength U(k̂) in simulations.
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46
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33646610579
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note
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ℓ contribution.) The generalized BG equation is then n = Δ→·J→. Integrating over the box yields normal components of J→that add up to zero: the box surfaces have no net frictional losses. (Since periodic boundary conditions are foldings of opposite edges onto each other, there are no real boundaries.) The required condition (Ref. 6) d(T + V)/dt = - 2R, that energy loss is determined only by the bulk Rayleigh dissipation rate, follows at once, using the strain-acceleration expression (4.4a) of BG dynamics.
-
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47
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33646632624
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note
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For finite systems the no-defect compatibility condition holds, both in the bulk and at the surfaces. Periodic boundary conditions on strains reveal the bulk compatibility potential from internal non-OP strains. The surfaces induce extra non-OP strains that can be shown to produce additional surface compatibility potentials (Ref. 10), which decrease with increasing system size, and can also be written in terms of the bulk OP strains. These will be considered elsewhere, and we focus on periodic boundary conditions, and bulk compatibility, for simplicity.
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private communication
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x contour integrals, and taking the relevant real parts of the residues. Yuri B. Gaididei (private communication).
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Time delays are also contained in the TR case dynamics, but with mutually nonlinear couplings in both OP strain components there could be a spectrum of mutually driving frequencies.
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52
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i(k→= 0) = 0.
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note
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2 independent degrees of freedom.
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70
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33646612748
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note
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ℓ of order unity, (varying by a few tenths of a percent over time near the end of a run).
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71
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note
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0 = 0, we get blobs instead of well-defined textures.
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72
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0 of the two quasitwin pictures are δ=0,0391 and 0.0392. For yet another random-number seed a uniform state results, with no domain walls (not shown).
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2.
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80
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note
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0λ ≪ 1. Thus for λ ≪ 1 convective terms are ignored for consistency.
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