Domain formation in transitions with noise
and time-dependent bifurcation parameter


Physical Review E 53, R4271-4274 (1996)

G.D. Lythe

Optique nonlinéaire théorique, Université Libre de Bruxelles CP231,
Bruxelles 1050 BELGIUM

Abstract

The characteristic size for spatial structure, that emerges when the bifurcation parameter in model partial differential equations is slowly increased through its critical value, depends logarithmically on the size of added noise. Numerics and analysis are presented for the real Ginzburg-Landau and Swift-Hohenberg equations.
PACS: 02.50-r, 64.60Ht, 05.70Fh, 47.54.+r

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Many physical systems undergo a transition from a spatially uniform state to one of lower symmetry. Classical examples are the formation of magnetic domains and the Rayleigh-Benard instability [1]. Such systems are commonly modeled by a simple differential equation, having a bifurcation parameter with a critical value at which the spatially uniform state loses stability. Noise is often assumed to provide the initial symmetry-breaking perturbation permitting the system to choose one of the available lower-symmetry states, but is not often explicitly included in mathematical models. However, when the bifurcation parameter is slowly increased through its critical value it is necessary to consider noise explicitly.

The phenomenon of delayed bifurcation and its sensitivity to noise has been reported in the case of non-autonomous stochastic ordinary differential equations [2]; here the corresponding phenomenon is examined in partial differential equations. A characteristic length for the spatial pattern is demonstrated from a stochastic partial differential equation (SPDE), supported by numerical simulations. Noise is added in such a way that it has no correlation length of its own (white in space and time) and a finite difference algorithm is used whose continuum limit is an SPDE.

The mathematical description of transitions is in terms of a space-dependent order parameter Y and a bifurcation parameter g. Because it is the simplest model with the essential features, the real Ginzburg-Landau equation (GL) is considered first. Results are also presented for the Swift-Hohenberg equation (SH), that is more explicitly designed to model Rayleigh-Benard convection.

When the bifurcation parameter g is constant the following is found. For g<0, in both GL and SH, the solution with Y everywhere 0 is stable. In GL for g>0 one sees a pattern of regions where Y is positive and regions where Y is negative (domains) separated by narrow transition layers. In SH for g>0 there is a structure resembling a pattern of parallel rolls, interrupted by defects.

When g is slowly increased through 0 in the presence of noise a characteristic length is produced as follows. The field Y remains everywhere small until well after g passes through 0. At g=gc where
eqnarray19
tex2html_wrap_inline523 is the rate of increase of g and tex2html_wrap_inline527 is the amplitude of the noise, Y at last becomes tex2html_wrap_inline531 and the spatial pattern present is frozen in by the nonlinearity. Thereafter one observes spatial structure with characteristic size proportional to tex2html_wrap_inline533. In GL this length is the typical size of the domains; in SH it is the typical distance betwen defects.

The results reported here were obtained by solving SPDEs [3] of the following dimensionless form for stochastic processes Y depending on x and t:

eqnarray26

The equations were solved as initial value problems, with tex2html_wrap_inline541 slowly increased from -1 to 1. Here tex2html_wrap_inline547, tex2html_wrap_inline549 is a probability space and W is the Brownian sheet [4], the generalisation of the Wiener process (standard Brownian motion) to processes dependent on both space and time. Periodic boundaries in x are used so that any spatial structure is not a boundary effect. The constants tex2html_wrap_inline523, tex2html_wrap_inline527 and tex2html_wrap_inline559 are all tex2html_wrap_inline561. Results are reported for tex2html_wrap_inline563 (GL) and tex2html_wrap_inline565 (SH) where tex2html_wrap_inline567, the Laplacian in tex2html_wrap_inline569.

In the first order finite difference algorithm for numerical realisations of the lattice version of (2), tex2html_wrap_inline571 is generated from tex2html_wrap_inline573 as follows:
eqnarray39
In (3), tex2html_wrap_inline573 is numerical approximation to the value of Y at site i at time t and tex2html_wrap_inline583 is the discrete version of tex2html_wrap_inline585. The tex2html_wrap_inline587 are Gaussian random variables with unit variance, independent of each other, of the values at other sites, and of the values at other times.

It is also possible to introduce multiplicative noise, for example to make g a random function of space and time [5,6]. The effect in that case is proportional to the magnitude of the noise and is thus less dramatic at small intensities than that of additive noise.

The timing of the emergence of spatial structure can be understood by deriving the stochastic ordinary differential equation for the most unstable Fourier mode, which is of the form [7]


eqnarray47

where w is the Wiener process. Trajectories of (4) remain close to y=0 until well after g=0, then jump abruptly towards one of the new attractors (Figure 1). The value of g at the jump can be determined by solving the linearised version ; for tex2html_wrap_inline599 it is a random variable with mean approximately tex2html_wrap_inline601 and standard deviation proportional to tex2html_wrap_inline523 [8].

  figure49
Figure 1: Dynamic pitchfork bifurcation with noise.
The dotted lines are the loci of stable fixed points of tex2html_wrap_inline605 as a function of g.
The solid lines are solutions of the non-autonomous SDE (4) with tex2html_wrap_inline609, for noise levels tex2html_wrap_inline611, tex2html_wrap_inline613, tex2html_wrap_inline615, tex2html_wrap_inline617. (In each case tex2html_wrap_inline619 and the initial condition is y=1.0 at g=-1.0.)

The Ginzburg-Landau equation is a simple model of a spatially extended system where a uniform state loses stability to a collection of non-symmetric states. When g is fixed and positive in this equation, a pattern of domains is found. In each domain, Y is close either to tex2html_wrap_inline629 or to tex2html_wrap_inline631. The gradual merging of domains on extremely long timescales [9] is not the subject of this paper; here the focus is on how the domains are formed by a slow increase of the bifurcation parameter through 0. An example is depicted in Figure 2: a pattern of domains emerges when Y is everywhere small and is frozen in at tex2html_wrap_inline521. When Y is small an excellent approximation to the correlation function, tex2html_wrap_inline641, can be calculated from the solution of the linearised version of (2) (that is, without the cubic term). The correlation length at tex2html_wrap_inline643 becomes the characteristic length for spatial structure after tex2html_wrap_inline643.

For GL, the solution of the linearised version of (2) is:
eqnarray60

  figure67
Figure 2: Dynamic transition, GL, one space dimension. Four configurations, tex2html_wrap_inline647, are shown from one numerically-generated realisation of the SPDE (note the different vertical scales). Nonlinear terms become important when tex2html_wrap_inline649; their effect is to freeze in the spatial structure. (L=300, tex2html_wrap_inline619, tex2html_wrap_inline655.)

where tex2html_wrap_inline657
displaymath659

with x-v understood modulo tex2html_wrap_inline663. The first term, dependent on the initial data f(x), relaxes quickly to very small values and remains negligible if tex2html_wrap_inline667.

The correlation function is therefore obtained from the second integral in (5). The mean of the product of two such stochastic integrals is an ordinary integral [4]. Performing the integration over space [7], assuming that tex2html_wrap_inline669, gives
eqnarray79

Before g approaches 0, the correlation function differs by only tex2html_wrap_inline675 from its static (g=constant) form [7]; it remains well-behaved as g passes through 0 and, for tex2html_wrap_inline683, is well approximated by:
eqnarray89

For tex2html_wrap_inline685, typical values of Y(x) increase exponentially fast and the correlation length is proportional to tex2html_wrap_inline689. Effectively noise acts for tex2html_wrap_inline691 to provide an initial condition for the subsequent evolution. At a value of g that is a random variable with mean tex2html_wrap_inline521 and standard deviation proportional to tex2html_wrap_inline523, the cubic nonlinearity becomes important. Its effect is to freeze in the spatial structure; no perceptible changes occur between tex2html_wrap_inline643 and g=1.

In one space dimension it is possible to put the scenario just described to quantitative test by producing numerous realisations like that of Fig.2 and recording the number of times Y crosses upwards through 0 in the domain [0,L] at g=1. In Fig.3 the average number of upcrossings is displayed as a function of the sweep rate tex2html_wrap_inline523. The solid line is the expected number of upcrossings of zero,

eqnarray100

for a field with correlation function (7) at tex2html_wrap_inline643 [10]. The hypothesis that the spatial pattern does not change after tex2html_wrap_inline643 is succesful.

  figure109
Figure 3: Number of zero crossings after a dynamic transition. The dots are the mean number of upcrossings of 0 at g=1 in numerical realisations of GL in one space dimension. The solid line is the prediction based on the assumption that the correlation function (7) is valid until tex2html_wrap_inline643, after which time the spatial pattern does not change. (tex2html_wrap_inline723 and L=800.)

In one space dimension, the solution of the SPDE (2) is a stochastic process with values in a space of continuous functions [3,12,13]. That is, for fixed tex2html_wrap_inline727 and tex2html_wrap_inline729, one obtains a configuration, tex2html_wrap_inline647, that is a continuous function of x. This can be pictured as the shape of a string at time t that is constantly subject to small random impulses all along its length. In more than one space dimension, however, the tex2html_wrap_inline647 are not necessarily continuous functions but only distributions [3,12]. Typically the correlation function c(x) diverges at x=0. In the dynamic equations studied here, however, the divergent part does not grow exponentially for g>0, and by tex2html_wrap_inline643 it is only apparent on extremely small scales, beyond the resolution of any feasible finite difference algorithm. Figure 4 depicts configurations at g=1 from realisations of (2) in two space dimensions. In Figures 4(a) and 4(b) (GL) one sees that a faster rate of increase of g results in a smaller average domain size. The SPDEs were simulated on a grid of tex2html_wrap_inline751 points with second order timestepping [13].

 
(a)(b)

(c)(d)

Figure 4: Two-dimensional pattern at g=1: smaller tex2html_wrap_inline523 means larger characteristic length. In black regions Y<0; in white or grey regions Y>0. In GL, (a) and (b), the typical domain size decreases with tex2html_wrap_inline523 , the rate of increase of g. In SH, (c) and (d), where there is a short-range structure resembling parallel rolls, the effect of reducing tex2html_wrap_inline523 is to reduce the number of defects. 4(a): GL, L=300, tex2html_wrap_inline769, tex2html_wrap_inline771. 4(b): GL, L=300, tex2html_wrap_inline769, tex2html_wrap_inline777. 4(c): SH, L=200, tex2html_wrap_inline769, tex2html_wrap_inline619. 4(d): SH, L=200, tex2html_wrap_inline769, tex2html_wrap_inline789.

The essential difference between the Swift-Hohenberg and Ginzburg-Landau models is that the first spatial Fourier mode to become unstable has k=1 rather than k=0. Hence there is a preferred small-scale pattern that resembles the parallel rolls seen in experiments. However, there is no preferred orientation of the roll pattern and when the correlation length is smaller than the system size, many defects are found, separating regions where the rolls have different orientations. When g is increased through 0, the number of defects resulting decreases when tex2html_wrap_inline523 decreases - Fig.4(c) and (d). Here a grid of tex2html_wrap_inline801 points was used with first order timestepping.

A notable feature of dynamic bifurcations and dynamic transitions is that the evolution for g>0 is independent of the initial conditions (provided they are such that that the system descends into the noise). Noise acts, near g=0, to wipe out the memory of the system and to provide an initial condition for the subsequent evolution. The correlation function (7) is, for example, a natural initial condition for studying the dynamics of defects and phase separation because it emerges from a slow increase to supercritical of the bifurcation parameter in the presence of space-time noise, mimicking an idealised experimental situation.

In summary, dynamic transitions are analysed in models of spatially extended systems with white noise. The correlation length that emerges from the noise during a slow sweep past g=0 is frozen in by the nonlinearity as a characteristic length proportional to tex2html_wrap_inline533 where tex2html_wrap_inline523 is the rate of increase of the bifurcation parameter and tex2html_wrap_inline527 is the amplitude of the noise.


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