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are very slow, the characteristic buildup time being of the order of τb ≈ 1/k2, as can be seen from (14.6) and (14.8), and this time diverges as k → 0. Thus one has to average for a very long time to obtain the correct statistics for the long waves. On the other hand, the characteristic buildup times for short wavelengths are very small, since the same relation τb ≈ 1/k2 holds. Here, correspondingly, in order to obtain the correct statistics of the mode occupation, one has to decrease the size of the temporal step as k → ∞. We thus come to the conclusion that one can never obtain the analytically predicted (correct) 1/k2 statistical distribution in a single numerical run with finite temporal steps (i.e. with a limited time resolution). A spectrum calculated with a fixed temporal step is shown in Fig. 14.4. In a log − log representation (Fig. 14.4a), a sharp decrease of the occupation of the large wavenumbers occurs. In a representation of the logarithm of the spectral density versus k2 (Fig. 14.4b), a straight line indicating an exponential decrease is obtained for large wavenumbers. The spectrum shown in Fig. 14.4, curiously enough, is thus precisely a Bose–Einstein distribution, which decays with a power law for long wavelengths, i.e. S(k → 0) k−2, and exponentially for short wavelengths, i.e. S(k → ∞) exp(k−2).

Fig. 14.4. The total spatial spectrum as obtained by numerical integration of the CGL equation in 1D for a fixed temporal step of ∆t = 0.05, but combined from four calculations with di erent sizes of integration region. Averaging was performed over a time t = 106 . Plot (a) shows the spectrum in a log −log representation, and the dashed line corresponds to a 1/k2 dependence, and (b) shows the spectrum in a single-log representation and the dashed line corresponds to an exp −k2 dependence

We note that the linear stability analysis does not lead to the Bose– Einstein distribution found numerically with finite temporal steps. The finite

214 14 Patterns and Noise

temporal step ∆t is equivalent to a particular cuto frequency ωmax of the temporal spectrum, with ωmax = 2π/∆t. In order to account for this finite temporal resolution, the integration of (14.9) should be performed not over all frequencies, but over [0, ωmax]. This integration, however, leads to a power-law decay for short wavenumbers, and not to the expected exponential decay. We have no explanation for this discrepancy between the analytical and numerical results.

We performed a series of numerical calculations in which the size of the temporal step was varied, in order to interpolate the spectra over the total range of spatial frequencies. The result can be represented as

Here C is a constant of order one. Equation (14.13) reproduces correctly the numerically obtained spectra in both asymptotic limits of k → 0 and k → ∞. For intermediate values of wavelength, a transition between a power law and an exponential decay is predicted by (14.11), exactly as found in the numerical calculations. In this way, the numerical results show that the spatial spectrum of the CGL equation in the case of limited temporal resolution coincides precisely with a Bose–Einstein distribution, whereas the spectrum in the case of unlimited temporal resolution follows a power law.

14.1.3 Consequences

To conclude this section, we show analytically and numerically that the power spectra of spatially extended systems with order–disorder transitions obey power laws: the spatial noise spectra are of 1/k2 form, thus being Bose– Einstein-like. The temporal noise spectra of the CGL equation are shown to be of 1/ωα form, with the exponent α = 2 − D/2 depending explicitly only on the dimension of the space D. Spatially extended systems with order– disorder transitions are described by a CGL equation with stochastic forces (14.1); this equation accounts for the symmetries of the phase space (Hopf bifurcation) and the symmetries of the physical space (rotational and translational invariance).

All ordered states in nature are, presumably, oneto three-dimensional. This corresponds to exponents of the 1/ωα noise satisfying 1/2 < α < 3/2, according to our model, which corresponds well with the experimentally observed exponents of 1/ωα noise (for reviews of 1/f noise, see [7]). The exponent is found experimentally to lie in the range 0.6 < α < 1.4 [7], depending on the particular system. Another prominent feature of 1/ω noise is that the spectrum usually extends over many decades of frequency with constant α, which also follows simply and naturally from our model.

The model presented here for 1/ω noise comprises the two most accepted models for 1/ω noise. In [8], 1/ω noise is interpreted as a result of a superposition of Lorentzian spectra, requiring a somewhat unphysical assumption of

a specific distribution of damping rates. In our model, the 1/ω spectrum also results formally from a superposition of stochastic spatial modes (see (14.5) and (14.6)). However, the distribution of the damping rates f(γ) (γ = k2 in our case) results naturally from the dimensionality of the space and is universally valid.

There is also a relation to the model of self-organized criticality [9], in that the phase variable in our model is always in a critical state, as (14.3b) indicates. This analogy with self-organized criticality for the phase variable is a consequence of the phase invariance in the Hopf bifurcation. Consequently, one would expect that the noise power spectra of models of self-organized criticality would show the same dependence on the spatial dimension, α = 2 − D/2, as found here. To our knowledge, no detailed investigations of the dependence of α on the dimension of the space have been performed for self-organized criticality.

The above dependence of α on the dimension of the space leads to general conclusions concerning the stability of the ordered state of the system. The integral of the 1/ωα power spectrum always diverges in the limit of either large or small frequency, indicating a breakup of the ordered state in the limit of small or of large times, respectively. For example, in the case of a low-dimensional system with D < 2, α > 1, the integral of the temporal power spectrum diverges at low frequencies, which means that the average size of the fluctuations of the order parameter grows to infinity for large times. The average size of a fluctuation is

∞

|a(t)|2 ≈ S(ω)dω , (14.14)

ωmin

where ωmin = 2π/t is the lower cuto boundary of the temporal spectrum; thus this average size grows as

with increasing time. This generalizes the Wiener stochastic di usion process,

well known for zero-dimensional systems, and predicts that di usion in spatially extended systems is weaker than in zero-dimensional systems. For example, the fluctuations of the order parameter in a 1D system (α = 1.5) should di use as

This also means that for large times, the fluctuations of the order parameter become, on average, of the order of magnitude of the order parameter