216 14 Patterns and Noise

itself. Defects must then appear in the ordered state, even for a small temperature.

For high-dimensional systems with D > 2, α < 1, in contrast, the integral over the temporal power spectrum diverges at large frequencies. It may be expected that large fluctuations of the order parameter occur at small times, given by

ωmax

|a(t)|2 ≈ S(ω)dω , (14.18)

0

where ωmax = 2π/t is the upper cuto boundary of the temporal spectra. This results in the di usion law

|a(t)|2 1 (14.19) t1−α

for α < 1, which diverges for small times. The fluctuations of the order parameter in a 3D system (α = 0.5) should diverge as

a(t)	2	= t−1/2	(14.20)
|	|

for small times. This means that a continuous creation and annihilation of pairs of defects in the ordered state can be predicted for D > 2. These defects are termed “virtual defects”, since they appear on a short timescale only and do not have any dynamical significance.

The case D = 2 is marginal. The integral over the spectrum diverges weakly (logarithmically) in the limits of both small and large frequencies. A specific such low-frequency divergence in this kind of 2D patterns was investigated in [10], and is termed the Kosterlitz–Thouless transition.

14.2 Noisy Stripes

An analysis of the stochastic dynamics of stripes may be performed by solving a stochastic Swift–Hohenberg equation [4],

∂A		= pA − A3 − ∆ + 2	2
			A + Γ(r, t) ,	(14.21)
∂t

for the temporal evolution of the real-valued order parameter A(r, t), defined in the D-dimensional space r. Again, p is the control parameter (the stripe

formation instability occurs at p = 0); ∆ is the detuning parameter, deter-

√

mining the resonant wavenumber of the stripe pattern given by k20 = ∆; and Γ(r, t) is an additive noise, δ-correlated in space and time, and of temperature T, defined as in (14.2).

Analytical results are obtained by solving the stochastic amplitude equation for the stripes,

		14.2 Noisy Stripes	217
∂B	= pB − |B|2 B − 2ik0 + 2	2
		B + Γ(r, t) ,	(14.22)
∂t

for the slowly varying complex-valued envelope B(r, t) of the stripe pattern corresponding to the resonant wavevector k0. The amplitude equa-

tion (14.22) can be obtained directly from (14.21), by inserting A(r, t) =

√

[B(r, t) exp(ik0 · r)+c.c.]/ 3, or directly from the microscopic equations of various stripe-forming systems (e.g. [11]). Equation (14.22) can also be obtained phenomenologically on the basis of symmetry considerations for arbitrary stripe patterns [5].

14.2.1 Spatio-Temporal Noise Spectra

We again assume that the system is su ciently far above the stripe-forming transition, and that p T . The homogeneous component dominates in (14.22) (and, correspondingly, one stripe component dominates in (14.21)), and one can look for a solution of (14.22) in the form of a perturbed homogeneous state, B(r, t) = B0 + b(r, t). After linearization of (14.22) around B0

and diagonalization, we obtain the linear stochastic equations for the pertur-
√			√
bations of the amplitude, b+ = (b + b )/	2	, and phase, b− = (b − b )/		2:

∂b+		ˆ
		= −2pb+ + L+( )b+ + Γ+(r, t) ,
∂t
∂b−		ˆ
∂t		= L−( )b− + Γ−(r, t) .

Here the nonlocality operators are given by

ˆ±(k0, ) = −p + (2k0 )2 − 4 p2 − 4k0 3 2 , L

and their spectra by

(14.23a)

(14.23b)

(14.24)

ˆ	, k) = −p − (2k0 dk)	2	− dk	4		p	2	3	2	,	(14.25)
L±(k0								− (4k0 dk)

as obtained by the substitution →i dk, where dk = k−k0 is the wavevector of the perturbation mode in (14.22).

Asymptotic values of the the nonlocality operator ˆ ( ) for phase per-

L−

turbations can be found in two opposite limits, namely the strongand weakpump limits, where

Lˆ

−(k0

,

) =

(2k0 )2 − 4

2for

4k0 3 p ,

(14.26a)

Lˆ

−

(k

,

) =

2ik

0

+

2

for 4k

3

p .

(14.26b)

0

−

0

Equation (14.23a) is an equation

for the amplitude fluctuations b

+ cor-

responding to the modulation amplitude of the stripe pattern, while (14.23b) is an equation for the phase fluctuations b− corresponding to parallel translation of the stripes. Equation (14.25) indicates that the phase fluctuations

218

14 Patterns and Noise

ˆ

,

=

(2k

2

dk

4

in the strong-pump limit, or

decay at a rate

L

−(k0

k)

−

0

dk)

−

2

ˆ

, k) = −(2k0 dk+dk ) in the weak-pump limit. This means that the

L−(k0

long-wavelength phase perturbation modes decay asymptotically slowly, with a decay rate approaching zero as dk → 0, which is a consequence of the phase invariance of the system.

Next, we consider only the phase perturbations. These perturbations determine the stochastic dynamics of the stripe pattern above the stripe formation threshold, i.e. for p > 0. More precisely, the amplitude fluctuations are

(14.26).		0		p, as follows from
small compared with the phase fluctuations if		4k3 dk

We calculate the spatio-temporal power spectra of the phase fluctuations by rewriting (14.23b) in terms of the spatial and temporal Fourier compo-

	nents,
	b(r, t) =	b−(k, ω)eiωt−ik·r dω dk ,						(14.27)
	and
	S(k, ω) =	b	(k, ω)	2	=	|Γ−(k, ω)|2	.	(14.28)
				|
						ω2 + |L−(k0, k)|2
		| −

Assuming δ-correlated noise in space and time, |Γ−(k, ω)|2 is simply proportional to the temperature T of the random force.

The spatial power spectrum is obtained by integration of (14.28) over all temporal frequencies:

S(k) =	∞	ω2 + |L−(k0, k)|2 dω =	2 |L−(k0, k)| .	(14.29)
		T	T π

−∞

This results in a divergence of the spatial spectrum as dk → 0 (and, equivalently, in a divergence of the spatial spectrum of a roll pattern obtained from (14.21) as k → k0). As follows from (14.29), perturbations of the stripe pattern dk diverge di erently, depending on whether the perturbations are parallel or perpendicular to the wavevector of the stripe pattern k0. This follows from the isotropic form of the nonlocality operator (14.25). The parallel perturbations (corresponding to compression and undulation of the stripes) diverge as dk−2, while the perpendicular perturbations (corresponding to a zigzagging of the stripes) diverge as dk−4. This results in an anisotropic form of the singularity at dk = 0, which can actually be expected from the anisotropic form of the amplitude equation for rolls (14.22). Figure 14.5 shows the spatial power spectrum of the noise of the stripe pattern as obtained from a numerical integration of the SH equation (14.21) and illustrates the anisotropy. The anisotropy results in the fact that the stability conditions of the stripes depend on the number of spatial dimensions. The

14.2 Noisy Stripes

219

ky

kx

Fig. 14.5. Spatial noise power spectrum of stripes in 2D obtained numerically by solving the stochastic SH equation (14.1) with p = 1 and ∆ = 0.7. The averaging time was tav = 5000. The intensity of the spatial spectral components is represented logarithmically

integral of (14.29) over the spatial wavenumbers dk diverges for spatial dimensions D < 4, and converges for D ≥ 4 only. Only for four (or more) dimensions of space are the stripes absolutely stable against additive noise. This is in contrast to a well-known theorem concerning the stability of a “condensate”: a condensate (a homogeneous distribution) is known to be stable for all spatial dimensions larger than two.

The temporal power spectra are obtained by integration of (14.28) over all possible wavevectors dk:

	∞			T
S(ω) =						dk .		(14.30)
		ω	2	+ |L−(k0, k)|	2
	−∞
However, this has no analytic form, even for one spatial dimension.
2							→	0, when the term
Asymptotically, in the limit of small frequencies ω

(2k0 dk) dominates in the denominator of the integral (14.30), an analyt-

ical integration is possible, and leads to the following results. For 1D, the
spectrum is S1D(ω) = c1DT ω−	3/2	√		2
		, with a coe cient c1D = π/(2 2k0). For

2D, S2D(ω) = c2DT ω−1.25; for 3D, S3D(ω) = c3DT ω−1; and in the general case of D dimensions, SD (ω) = cDT ω−α, where α = 1 + (3 − D)/4 and the coe cients cD is of order unity.

The integral (14.30) has been evaluated numerically, and the results for one, two and three dimensions are given in Fig. 14.6; 1/ωα dependences are obtained. In the small-frequency limit ω → 0, the exponents obey α = 1 + (3 − D)/4; in the large-frequency limit ω → ∞, the spectra also show a power-law form, but with exponents α = 1 + (4 − D)/4. The exponents change abruptly from the small-frequency value to the large-frequency value at a critical frequency ωc ≈ 4k20, as follows from an analysis of (14.29), and as seen from Fig. 14.6.

220 14 Patterns and Noise

Fig. 14.6. Temporal spectra obtained by numerical calculation of the integral (14.9) with p = 1 and ∆ = 1. (a) 1D case. The phase power spectrum (obtained from integration of (14.23b)), the amplitude power spectrum (obtained from integration of (14.23a)) and the total spectrum are shown. (b) The phase power spectra as calculated for one, two and three spatial dimensions

Comparing these results with the noise spectra of condensates (Sect. 14.1) one can conclude that:

1.One-dimensional stripes have the same exponent of noise power spectra as one-dimensional condensates. This is plausible, since the amplitude equation for stripes is similar to a complex Ginzburg–Landau equation,

and the two equations coincide in the limit of dk → 0.

2. Two-dimensional stripes behave like noisy condensates of dimension D = 1.5, if one judges from the exponents of the noise spectra in the low-frequency limit. As discussed above (see also Fig. 14.5), the singularity of the spatial noise spectrum is strongly squeezed in the direction along the stripes. It is then plausible that the noise characteristics of this anisotropic system are between those of isotropic oneand twodimensional systems.