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example, the far-field distributions of hexagonal patterns show distributions of the form (k − ki)−2, where ki (i = 1, ..., 6) are the locations of singularities of the far field, arranged on the vertices of a hexagon.

•The spatial power spectra of the noise show a 1/fα distribution, where the exponent α is close to unity, and depends on the dimensionality and symmetry of the pattern.

•The stochastic drift of the patterns is sub-Brownian: it is well known that the stochastic drift of the position of a Brownian particle obeys a square root law,

x (t)2 t1/2 .

We show that the stochastic drift of nonlinear patterns is in general di erent (weaker) than the Brownian; for example the stripe pattern has a root mean wandering t1/4 in the case of one spatial dimension.

The analysis of noisy stripes is performed by solving the stochastic Swift– Hohenberg equation as the order parameter equation for a stripe pattern in a spatially isotropic system [4], or the Newell–Whitehead–Segel equation as an amplitude equation for perturbations of stripe a pattern [5]. However, we start from an analysis of a noisy homogeneous state or, in other words, of a nonzero-temperature condensate. The main results (spatio-temporal spectra) in the case of the condensate (the first part of the chapter), are then applied to calculate the noise properties of stripe patterns.

14.1 Noise in Condensates

An order–disorder transition in a condensate or, in general, in a spatially extended nonlinear system can be described in the lowest order by a complex Ginzburg–Landau equation with a stochastic term:

∂A	= pA − (1 + ic) |A|2 A + (1 + ib) 2A + Γ(r, t) .	(14.1)
∂t

Here A(r, t) is a complex-valued order parameter defined in an n-dimensional space r, evolving with time t. The control parameter is p (the order–disorder transition occurs at p = 0). The Laplace operator 2A represents the nonlocality in the system, and Γ(r, t) is an additive noise, δ-correlated in space and time and of temperature T , such that

Γ(r1, t1)Γ (r2, t2) = 2T δ(r1 − r2)δ(t1 − t2) .

(14.2)

Below the transition threshold (p < 0), the CGL equation (14.1) yields a disordered state: the order parameter A(r, t) is essentially noise filtered in space and time, with an exponential (thermal) intensity distribution. Above

14.1 Noise in Condensates

207

the transition threshold (p > 0), (14.1) yields an ordered, or coherent, state (or a condensate) in modulationally stable cases, with the intensity dis-

tributed around its mean value |A|2 = p.

The CGL equation (14.1), with complex-valued coe cients, has been investigated in the previous chapters to describe the dynamics of optical vortices in the case of zero or negative detuning. Here, however, we consider the CGL equation as a universal model describing an order–disorder phase transition. The first two terms, pA − |A|2 A, approximate to the lowest order a supercritical Hopf bifurcation, a bifurcation that brings the system from a trivial state to a state with phase invariance of the order parameter A(r, t). The complex-valued character of the order parameter is important, since every ordered, or coherent, state, both in classical and in quantum mechanics, is characterized not only by the modulus of the order parameter, but also by its phase. The di usion term 2A describes the simplest possible nonlocality term in a spatially isotropic and translationally invariant system.

The real Ginzburg–Landau equation was introduced [6] as the normal form for a second-order phase transition between two arbitrary spatially extended states and can be derived systematically for many systems, as well as phenomenologically from symmetry considerations. The real Ginzburg– Landau equation does not contain information about the coherence properties of the system. Thus, analogously, we try to find a simple model for the order–disorder phase transition, a normal form that can be derived systematically for particular systems, as well as phenomenologically from symmetry considerations. The complex Ginzburg–Landau equation is just that. It describes, as a normal form, systems characterized by (1) a supercritical phase transition between a disordered and an ordered state, (2) phase invariance of the order parameter, and (3) isotropy and homogeneity in space.

14.1.1 Spatio-Temporal Noise Spectra

For the analytical treatment, we assume that the system is su ciently far above the order–disorder transition, i.e. p >> T . Then the homogeneous component |A0| = √p dominates, and we can look for a solution of (14.1) in the form of a perturbed homogeneous state, A(r, t) = A0 + a(r, t). After linearization of (14.1) around A0 and diagonalisation, we obtain the linear

stochastic equations for the amplitude and phase perturbations, b+ = (a +
	√					√
a )/		2			and b− = (a − a )/		2	, respectively:
	∂b+			2
					= −2pb+ +	b+ + Γ+(r, t) ,			(14.3a)
	∂t
	∂b−			2
					= b− + Γ−(r, t) .				(14.3b)
	∂t

Equation (14.3a) gives the evolution of the amplitude fluctuations b+, which decay at a rate λ+ = −2p − k2 (where k is the spatial wavenumber

208 14 Patterns and Noise

of the perturbation). Asymptotically, long-lived amplitude perturbations are possible only at the Hopf bifurcation point (in the critical state), but never above or below it. Equation (14.3b) is the equation for phase fluctuations b− which decay at a rate λ− = −k2 above the Hopf bifurcation point. This means that the long-wavelength modes decay asymptotically slowly, with a decay rate approaching zero as k → 0, which is a consequence of the phase invariance of the system. The phase, as a result, is in a critical state for all p > 0.

From (14.3) one can calculate the spatio-temporal noise spectra, by rewriting (14.3) in terms of the spatial and temporal Fourier components,

b±(r, t) = b±(k, ω)eiωt−ikr dω dk , (14.4)

where the coe cients of the fourier components follow directly from (14.5):

b+(k, ω) =			Γ+(k, ω)		,	(14.5a)
			iω + k2 + 2p
b	−	(k, ω) =	Γ−(k, ω)	.		(14.5b)
			iω + k2

The coe cients of the spatio-temporal power spectra (sometimes called

the “structure function”) are
S		(k, ω) =	b	(k, ω)	2	=	|Γ+(k, ω)|2		,	(14.6a)
							ω2 + (2p + k2)2
	+		| +	|
S		(k, ω) =	b	(k, ω)	2	=	|Γ−(k, ω)|2	,		(14.6b)
	−				|
			| −				ω2 + k4

for the amplitude and phase fluctuations, respectively. If we assume δ- correlated noise in space and time, |Γ±(k, ω)|2 are simply proportional to the temperature T of the random force.

Spatial Power Spectra. The spatial spectra are obtained by integration of (14.6) over all the temporal frequencies ω:

S(k) = S+(k) + S−(k) =	S+(k, ω) dω + S−(k, ω) dω .	(14.7)

The total power spectrum here is the sum of the power spectrum of the amplitude, S+(k), and that of the phase, S−(k), since the spectral components b±(r, t) are mutually uncorrelated, as follows from (14.5). For clariry, the integration is performed here separately for the amplitude and phase fluctuations, and yields

S+(k) =	+∞		T			dω =		T π	,	(14.8a)
	−∞	ω2	+ (k2 + 2p)2					k2 + 2p
S−(k) =	+∞		T		T π
				dω =			.			(14.8b)
	−∞	ω2	+ k4		k2

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209

This means that the spectrum of phase fluctuations is of the form 1/k2 (14.8b). The spatial spectrum of amplitude fluctuations is Lorentzian: in the short-wavelength limit, |k|2 2p, the amplitude spectrum is equal to the phase spectrum, i.e. S+(k) = S−(k). The total spectrum is then S(k) = 2S+(k) in the short-wavelength limit. In the long-wavelength limit |k|2 2p, the amplitude fluctuation power spectrum saturates at S+(k ≈ 0) = T π/2p, and is negligibly small compared with phase fluctuation spectrum. Therefore the total spectrum is essentially determined by the phase fluctuations in this long-wavelength limit.

Temporal Power Spectra. The temporal power spectra are obtained by integration of (14.6) over all possible spatial wavevectors k. In the case of one spatial dimension,

S+1D(ω) =	∞	ω2	+ (2p + k2)2 dk = ω				Im (2p − iω)−1/2		, (14.9a)
			T			T π
	−∞
S−1D(ω) =	∞	ω2	+ k4 dk =	21/2ω3/2 .					(14.9b)
			T		T π

−∞

This results in a power spectrum of phase fluctuations (14.9b) of precisely the form ω−3/2 over the entire frequency range. The spectrum of amplitude fluctuations (14.9a) is more complicated: in the limit of large frequencies |ω| 2p it is equal to the phase spectrum, i.e. S+1D(ω) = S−1D(ω). In the limit of small frequencies |ω| 2p, the amplitude power spectrum saturates at

S+1D(ω ≈ 0) =	T π	|ω| 2p .	(14.10)
	2 (2p)3/2 ,

In this way, (14.9a) represents a Lorentz-like spectrum (with an ω−3/2 frequency dependence) for the amplitude fluctuations of a system extended in one-dimensional space.

Integration of (14.6) in two spatial dimensions yields

S+2D(ω) =	T π	π − 2 arctan	2p	, S−2D(ω) =	T π221/2		(14.11)
						.
	2ω		ω		2ω

This results in a power spectrum of the phase fluctuations of the form 1/ω over the entire frequency range, and in a Lorentz-like spectrum of the amplitude fluctuations with an ω−1 frequency dependence.

Finally, integration of (14.6) in three spatial dimensions yields

	2	T π2		(2p + iω)1/2 , S−3D(ω) =	T π221/2
S+3D(ω) =			Im			.	(14.12)
		ω			ω1/2