# Stochastic PDEs: domain formation in dynamic transitions

###### Abstract

Spatiotemporal evolution in the real Ginzburg-Landau equation is studied with space-time noise and a slowly increasing critical parameter. Analytical estimates for the characteristic size of the domains formed in a slow sweep through the critical point agree with the results of finite difference solution of the stochastic PDEs.

Proceedings of the VIII Reunión de Física Estadística,
FISES ’97

Anales de Física, Monografías RSEF Vol. 4,
55–63 (1998).

Noise is an indispensable part of the description of symmetry-breaking transitions, although it is normally included only implicitly in mathematical models. This work examines a dynamic transition (“quench”) in a spatially extended system. The model contains a bifurcation parameter having a critical value at which the state with zero order parameter becomes unstable to two equivalent states with non-zero order parameter. Above this critical value, the whole system does not choose the same state. Rather, a pattern of domains and defects (“kinks”) is formed [19, 3, 16]. Here it is shown that the characteristic size of the domains formed when the bifurcation parameter is made explicitly time-dependent depends logarithmically on the magnitude of additive noise: additive noise is thus a determining influence even when extremely small. The effect of low-intensity multiplicative noise is much less dramatic. Noise is added in such a way that it has no correlation length of its own (white in space and time). Numerical results are reported, obtained using a finite difference algorithm whose continuum limit is an SPDE. The simplest model with the essential features, the real Ginzburg-Landau equation, is considered here. The dynamics take the system from a potential with a single minimum to one with two minima (Figure 11). Similar results have been found for the Swift-Hohenberg equation, which is more explicitly designed to model Rayleigh-Benard convection [19].

## The Ginzburg-Landau SPDE

The order parameter is a function of space and time and the bifurcation parameter . For fixed, the state is stable, and only noise prevents the system from approaching it arbitrarily closely. For fixed, one sees a pattern of regions in which is positive and regions in which is negative (domains) separated by narrow transition layers (kinks). The subject of this paper is not the slow merging of domains or the nucleation of new kinks - these happen on timescales much longer than those considered here [11, 3]. This paper derives the number of domains formed when the parameter is explicitly time-dependent, starting from and ending with .

The SPDE describing the dynamics is written in the following dimensionless form:

(1) |

Here is a probability space and is the Brownian sheet. The equations were solved as initial value problems, with ,

(2) |

slowly increased from to . Periodic boundaries in are used so that any spatial structure is not a boundary effect. The constants , and are all . The spatial operator where , the Laplacian in .

## The Brownian Sheet

To add noise to an ordinary differential equation, one adds increments of the Wiener process at each time step; the idea is of a particle continuously subject to small impulses [9]. To add noise to a partial differential equation (PDE) one adds increments of the Brownian sheet, that are random in space and time [24, 20]. One can imagine, for example, the motion of a flexible sheet in a sandstorm. Typically the motivation for adding noise to PDEs is to make allowances for small-scale, rapidly-varying effects with mean zero that have been neglected in the derivation of an equation.

The Wiener process can be thought of as assigning to each successive interval of the time axis a Gaussian random variable with variance proportional to the length of the interval. The Brownian sheet assigns to each volume element in a Gaussian random variable whose variance is proportional to the volume of the element [24]. More precisely, it is possible to define a map from to a probability space such that for each , is a Gaussian random variable with mean zero and where is the Lebesgue measure [24]. The Brownian sheet, so called because its realisations in look like a ruffled bed-sheet tucked in on two adjacent sides, is defined as , where is the element (interval, square, cube, ) with opposite corners at the origin and at . Thus

(4) |

The set is the set of labels for realisations; averages over realisations are denoted by angled brackets. Each is a real-valued Gaussian random variable with mean zero and variance . Stochastic processes will be denoted in bold with the subscript . For example, the Wiener process is denoted and is constructed by defining . We denote processes satisfying SPDEs with upper case bold Roman letters with subscript and in brackets.

The action of as an integrator is easily described in the case where the integrand is a deterministic function. If and are continuous functions on , where , then

are Gaussian random variables with , and

## Numerical Solution

An approximate numerical solution of an SPDE is produced by solving a large set of SDEs. The finite difference method for a parabolic SPDE consists of replacing the infinite dimensional system (1) by ordinary SDEs on a grid of equally-spaced points in separated by . The SDE at site is [19]

(5) |

where the discrete Laplacian is defined by

(6) |

and the sum is over the nearest neighbours of . The random variables added at neighbouring grid points are independent no matter how small is. Stochastic PDEs like (1) use space-time white noise along with a deterministic operator that includes some spatial coupling (here the Laplacian) to generate fields with non-delta-function correlations in space and time. Linear SPDEs can be very efficiently solved in Fourier space [7, 8]. However, the finite difference method is more easily adapted to nonlinear SPDEs. One finite-difference numerical realisation of an SPDE generates an approximate solution for one , at a discrete set of times and positions . The time-stepping used to generate the figures in this work is the stochastic analogue of the second-order Runge-Kutta method [10, 14, 13].

## Fourier decomposition

For , let the Fourier coefficient be defined by

(7) |

Each is a complex-valued stochastic process satisfying the following SDE when :

(8) |

Since is real-valued, for we obtain from the relation = . The imaginary part of is always ; the real part satisfies

(9) |

Each is an independent complex-valued Wiener process. When for all , the nonlinear terms that involve products of s are unimportant and the SDEs (8) can be solved separately. It then becomes necessary to evaluate the value of at which these nonlinear terms become important. This is called an exit value problem in a dynamic bifurcation [18, 21, 22, 23].

## Dynamic bifurcation

We consider a stochastic ordinary differential equation slightly more general than (9), including also multiplicative noise:

(10) |

where and are independent real-valued Wiener processes, and the initial condition is at . , . The solution of (10) is

(11) |

The second term in (11) has zero mean and variance given by

(12) |

For , this variance is well approximated by

(13) |

Comparing the mean and standard deviation of for , we find that, for , and , the standard deviation is much larger than the mean if . Then the dynamic bifurcation is noise-dominated [18], remains small until well after passes through , and finally attains an magnitude at a value of with mean value and standard deviation proportional to where

(14) |

Stochastic equations like (1) and (10) with constant and have attracted much attention because multiplicative noise acts in a way that can be interpreted as a shift of the bifurcation point from [2]. However, in all cases the shift is proportional to compared and is dwarfed by the large dynamic delay of the bifurcation of interest here. In (13) we see that, for the variance of , is replaced by . Note, however, that for the distribution of is not Gaussian. A slight non-Gaussianity is the main effect of multiplicative noise in the SDE and SPDE systems of interest here and will not be further discussed.

## Space-time evolution

Now return to the spatially extended system. We first make the hypothesis that, as in the dynamic bifurcation, remains everywhere small for . Then the emerging pattern of domains can be studied from the linearised version of (1) (i.e. without the cubic term). The solution of the linearised version of (1) is:

(15) |

where

(16) |

The first term in (15), dependent on the initial data , relaxes quickly to very small values. The correlation function is therefore obtained from the second, stochastic, integral in (15). Performing the integration over space, assuming that , gives

(17) |

Since satisfies a non-autonomous SPDE, the correlation function is explicitly a function of time. In the early part of the evolution, however, the deviation from that obtained from the corresponding static (constant) equation is small. We consider this quasi-static period by making the change of variables in (17). Then

(18) |

and

(19) |

Thus

(20) |

where is the modified Bessel function of order . For example, when ,

(21) |

The first term in (20) is the (long-time) correlation function for the SPDE obtained by fixing [15]. Clearly the expansion in is no longer useful for . The correlation function itself, however, remains well-behaved as passes through ; the divergences associated with critical slowing down are not present.

In one space dimension, the solution of the SPDE (1) is a stochastic process with values in a space of continuous functions [24, 6]. That is, for fixed and , that is a continuous function of . This can be pictured as the shape of a string at time that is constantly subject to small random impulses all along its length. In more than one space dimension, however, the are not continuous functions but only distributions [24, 5] and the correlation function diverges at . In the non-autonomous equations studied here, however, the divergent part does not grow exponentially for , and by it is only apparent on extremely small scales, beyond the resolution of any feasible finite difference algorithm. , one obtains a configuration,

We now examine the evolution for , where we can approximate (17) using Laplace’s formula:

(22) |

Thus typical values of increase exponentially fast and the correlation length at time is . Once , the noise no longer greatly influences the evolution; its effect can be thought of as wiping out the memory of the initial condition at and replacing it with an effective random initial condition. From (22), we see that, at , and the cubic nonlinearity can no longer be ignored. (Figure 2.)

## Density of kinks

The second hypothesis that enables the characteristic domain size to be estimated is that the effect of the cubic nonlinearity, when it finally makes itself felt, is to freeze in the spatial structure. This is indeed the case in numerical simulations: no perceptible changes occur between and [17, 19]. Thus the correlation length at ,

(23) |

becomes the characteristic length for spatial structure after . In one space dimension it is possible to put the scenario just described to quantitative test by producing numerous realisations of the non-autonomous SPDE (1) and recording , the number of times that crosses upwards through in the domain at . The average can then be compared with the mean number of upcrossings for Gaussian random field with correlation function (17) at .

Consider a homogeneous Gaussian random field with correlation function . Then is a Gaussian random variable with mean zero and variance , where

(24) |

The probability that has an upcrossing of between and is given by

(25) |

Now consider a grid of total length made up of sites separated by . Let with fixed, i.e. let . When , the number of zero crossings of defined on this grid is proportional to for . When, as in (17), , the mean number of zero crossings per unit length approaches a finite number as given by [1, 12]:

(26) |

In Fig.3 this average is displayed as a function of the sweep rate for the case (22), evaluated at :

(27) |

## Summary

When the critical parameter in the Ginzburg-Landau equation is slowly increased through a characteristic length is produced as follows. The field remains everywhere small until well after the critical value for the loss of stability of the uniform symmetric state. Meanwhile, the correlation length grows proportional to . At , where is the rate of increase of the parameter and is the amplitude of the noise, the field at last becomes and the spatial pattern present is frozen in by the nonlinearity. Thereafter one observes spatial structure with characteristic size proportional to

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