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3.Similarly, three-dimensional stripes (lamellae) behave like two-dimensional condensates. Both display power spectra with α = 1.

14.2.2 Stochastic Drifts

Since the temporal power spectra for stripe patterns diverge as 1/ω3/2, the stochastic drift of stripes should be subdi usive, as follows from (14.15) and (14.17). We tested this prediction about stochastic drift of the stripe pattern by numerically solving the Swift–Hohenberg equation (14.21) in 1D. We calculated the displacement of the stripe pattern as a function of time. The displacement x(t) of the stripe position for the SH equation is directly proportional to the phase of the order parameter B(x, t) at the corresponding spatial location in the amplitude equation (14.22). Figure 14.7a shows the power spectrum of the displacement, which follows an ω−3/2 law, in accordance with the analytical predictions. Figure 14.7b shows the power spectrum of the variation (the temporal derivative x(t) − x(t − ∆t)) of the displacement, which follows an ω1/2 law. The average square displacement of the stripe position x(t), averaged over many realizations, is shown in Fig. 14.7c. The predicted slope of 1/2 is clearly seen for times up to t ≈ 1000. For very large times, the usual (Brownian) stochastic drift is obtained. This behavior for large times (corresponding to small frequencies) is, however, an artifact of the numerical space discretization. A subdi usive stochastic drift of kinks (fronts) in 1D systems (for small times, however) was recently found in [12].

The above discussion of stochastic drifts concerns large times: the variance of the position of 1D stripes of the form t1/2 is related to the ω−3/2 power spectrum at small frequencies. The ω−1.75 spectrum at large frequencies (ω ≥ ωc ≈ 4k20) predicts, equally, a t3/4 law for the stochastic drift at small times. The results of numerical calculations in Fig. 14.7 do not, however, take account of small timescales (t ≤ 2π/ωc), and thus the small-time drift law was not observed numerically.

The stochastic drift (although subdi usive) of the order parameter means that for large times the fluctuations become, on average, of the order of magnitude of the order parameter itself. The long-range order eventually breaks up even for a small temperature. In general, for a 1/ωα power spectrum with α > 1, such finite perturbations occur for times t ≥ tc T −1/(α−1). We tested this dependence on 1D stripes, where the critical time is tc T −2. For this purpose, we prepared numerically an o -resonance stripe pattern using the SH equation for 1D without a stochastic term. The o -resonance stripe was stable (it was within the Eckhaus stability range). We then switched on the stochastic term and waited until the fluctuations of the stripe pattern grew and destroyed it locally. We observed that after the stripe pattern was destroyed in some place, a resonant stripe appears there and invades the whole pattern in the form of propagating switching waves. The state of the system changes in this way from a local potential minimum (o -resonance stripe) to

222 14 Patterns and Noise

Fig. 14.7. Statistical properties of the position of a stripe pattern as obtained by numerical integration of the SH equation in 1D with p = 1 and ∆ = 0.7. (a) The power spectrum of the displacement x(t). The dashed line with a slope corresponding to ω−3/2 serves to guide the eye. (b) The power spectrum of the variation of the displacement x(t) − x(t − ∆t). The dashed line with a slope corresponding to ω1/2 serves to guide the eye. (c) The average square displacement of the stripe position, as averaged over 1000 realizations

14.2 Noisy Stripes

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the global potential minimum (resonant stripe) as a result of triggering by a local perturbation.

In Fig. 14.8, the numerically calculated lifetime of the o -resonance stripe pattern is plotted as a function of the temperature of the stochastic force. Again, as predicted by analytic calculations, the dependence tc T −2 is obtained. This shows that in spatially extended systems, the switching from a local potential minimum to a deeper global minimum does not depends exponentially on time as in zero-dimensional (compact) systems, but obeys a power law. In particular, for stripe patterns, the switching time is ts T −2 in 1D.

Fig. 14.8. Lifetime of an o -resonant stripe pattern as a function of the noise temperature T , as obtained by numerical integration of the SH equation in 1D with p = 1. A resonant stripe pattern with k02 = 1 was excited for ∆ = 1. The detuning value was then reduced to 0.75, and the time was measured until the new resonant stripe pattern took over. Every point was obtained by averaging over 10 realizations

14.2.3 Consequences

To conclude, we recall that simple models for stripe patterns (the stochastic Swift–Hohenberg equation for the order parameter, and the stochastic Newell–Whitehead–Segel equation for the envelope of the stripes) allow one to calculate spatio-temporal noise power spectra, and to predict the following properties of stripe patterns in the presence of noise:

1.An anisotropic form of the singularities in the spatial power spectrum.

2.Stability conditions that depend on the number of spatial dimensions.

3.A 1/ωα temporal power spectrum with an exponent that depends explicitly on the number of spatial dimensions.

4.Subdi usive stochastic drift.