We present results from a comprehensive analytical and numerical study of
nonequilibrium dynamics in the 2-dimensional complex Ginzburg-Landau (CGL)
equation. In particular, we use spiral defects to characterize the domain
growth law and the evolution morphology. An asymptotic analysis of the
single-spiral correlation function shows a sequence of singularities --
analogous to those seen for time-dependent Ginzburg-Landau (TDGL) models with
O(n) symmetry, where $n$ is even.Comment: 11 pages, 5 figure

A. Weber

A.J. Bray

A.N. Pargellis

B. Yurke

F. Rojas

G.F. Mazenko

H. Chate

H. Toyoki

I.S. Aranson

K. Binder

M. C. Cross

M.C. Cross

P.S. Hagan

S. K. Das

S. Puri

Sanjay Puri

Subir K. Das

T. Ohta

W. van Saarloos

Y. Kuramoto

English

arXiv

Results from a comprehensive analytical and numerical study of nonequilibrium dynamics in the two-dimensional complex Ginzburg-Landau equation have been presented. In particular, spiral defects have been used to characterize the domain growth law and the evolution morphology. An asymptotic analysis of the single-spiral correlation function shows a sequence of singularities-analogous to those seen for time-dependent Ginzburg-Landau models with

O(n) symmetry, where n is even

Puri, Sanjay

Das, Subir K.

Cross, M. C.

Publications of the IAS Fellows

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Nonequilibrium Dynamics in the Complex Ginzburg-Landau
Equation
Sanjay Puri1, Subir K. Das1 and M.C. Cross2
1School of Physical Sciences, Jawaharlal Nehru University
New Delhi – 110067, India.
2Department of Physics, California Institute of Technology
Pasadena, California 91125, U.S.A.
We present results from a comprehensive analytical and numerical study
of nonequilibrium dynamics in the 2-dimensional complex Ginzburg-Landau
(CGL) equation. In particular, we use spiral defects to characterize the do-
main growth law and the evolution morphology. An asymptotic analysis of
the single-spiral correlation function shows a sequence of singularities – analo-
gous to those seen for time-dependent Ginzburg-Landau (TDGL) models with
O(n) symmetry, where n is even.
1
Much recent interest has focused on pattern formation in the complex Ginzburg-Landau
(CGL) equation:
∂ψ(~r, t)
∂t
= ψ(~r, t) + (1 + iα)∇2ψ(~r, t)− (1 + iβ)|ψ(~r, t)|2ψ(~r, t), (1)
where ψ(~r, t) is a complex order-parameter field which depends on space (~r) and time (t).
In Eq. (1), α and β are real parameters. The CGL equation arises in a range of diverse
contexts, as reviewed by Cross and Hohenberg [1]. This universality arises from the fact
that the CGL equation provides a generic description of oscillations in a spatially-extended
system near a Hopf bifurcation [2].
The CGL equation exhibits rich dynamical behavior with variation of the parameters
α and β, and the “phase diagram” has been investigated (mostly numerically) in various
studies [3]. In a large range of parameter space, the emergence and interaction of spiral
defect structures characterizes the morphology. In this letter, we study the nonequilibrium
dynamics of the CGL equation resulting from a small-amplitude random initial condition.
In general, this nonequilibrium evolution is referred to as “phase ordering dynamics” or
“domain growth”, and constitutes a well-studied example of far-from-equilibrium statistical
physics [4,5]. Our analytical understanding of phase ordering systems has depended critically
upon modeling the dynamics of defects in these systems (e.g., interfaces, vortices, monopoles,
etc.) [5]- [8]. In this letter, we use spiral defect structures to characterize the evolution
morphology in the CGL equation. Many novel features emerge in our study, which should
be of great relevance for both experiments and subsequent numerical simulations.
For simplicity, we will focus on the CGL equation with α = 0 and dimensionality d = 2.
However, the results presented here are also relevant for the cases with α 6= 0 and d > 2, as
the underlying paradigm does not change, i.e., spirals continue to determine the morphology
in large regions of parameter space. Following the work of Hagan [9], Aranson et al. [10],
and Chate and Manneville [3], let us briefly discuss the phase diagram of the d = 2 CGL
equation with α = 0. The limiting case β = 0 corresponds to the dynamical XY model,
which is well understood. The appropriate (point) defects are vortices, and domain growth
2
is driven by the attraction and annihilation of vortex-antivortex pairs. The relevant growth
law for the characteristic length scale is L(t) ∼ (t/ ln t)1/2 [11,5]; and the analytic form
of the time-dependent correlation function (which characterizes the evolving morphology)
has been obtained by Bray and Puri, and (independently) Toyoki [7]. Without loss of
generality, we focus on the case with β ≥ 0. For 0 ≤ β ≤ β1 (β1 ≃ 1.397 [9]), spirals
(which are asymptotically plane-waves) are linearly stable to fluctuations. For β1 < β ≤ β2
(β2 ≃ 1.82 [10,3]), spirals are linearly unstable to fluctuations, but the growing fluctuations
are advected away, i.e., the spiral structure is globally stable. Finally, for β2 < β, the spirals
are globally unstable and cannot exist for extended times [10]. Our results correspond to
the parameter regime with β ≤ β2.
Figure 1 shows the typical evolution from a small-amplitude random initial condition
for the case with β = 0.75. Our numerical simulations were performed by implementing an
isotropic Euler-discretization of Eq. (1) on N2-lattices (N = 256 for Figure 1), with periodic
boundary conditions in both directions. The discretization mesh sizes were ∆t = 0.01 and
∆x = 1.0. In Figure 1, we plot constant-phase regions and the relevant color-coding is
provided in the figure caption. The evolving morphology is characterized by spirals and
antispirals, and there is a typical length scale L, e.g., inter-spiral spacing or the square root
of inverse defect density, which is the definition we will use subsequently.
Figure 2(a) plots ln[L(t)] vs. ln t for 5 representative values of β. The length-scale
data was obtained from 5 independent runs on N2-lattices with N = 1024. After an initial
transient period, the length scale L(t) should saturate to an equilibrium value (Ls) because
of an effective spiral-antispiral repulsive potential [1]. This should be contrasted with the
β = 0 case, where vortices continue to anneal (at zero temperature) as t→∞. As a matter
of fact, the data for β = 0.25, 0.5 in Figure 2(a) does not exhibit this morphological freezing
on the time-scales of our simulation, though signs of the crossover are evident for β = 0.5.
To understand this crossover behavior, we recall the analytical solution for an m-armed
spiral due to Hagan [9]:
3
ψ(~r, t) = ρ(r) exp [−iωt+ imθ − iφ(r)] , (2)
where ~r ≡ (r, θ); and ω = β(1− q2), where q is a constant which is determined by β [9]. The
limiting forms of the functions ρ(r) and φ(r) are
ρ(r)→
√
1− q2, φ′(r)→ q, as r →∞,
ρ(r)→ arm, φ′(r)→ r, as r → 0, (3)
where a is a constant which is determined by finiteness conditions. We will focus on the case
with m = ±1, as only 1-armed spirals are stable in the evolution [9]. Furthermore, we are
only interested in distances r ≫ ξ, where ξ is the defect core. Thus, we consider the spiral
form in Eq. (2) with ρ(r) =
√
1− q2 and φ(r) = qr (appropriate for r →∞).
We expect that spirals behave similarly to vortices for L < Lc, where qLc ∼ O(1). Thus,
the early evolution should be analogous to that for the XY model, both in terms of the
domain growth law and correlation function. In Figure 2(a), the solid line has a slope of 1/2
and the initial growth (at least for β ≤ 0.75) appears to be consistent with the behavior for
the XY model, i.e., L(t) ∼ (t/ ln t)1/2 for d = 2. We also expect the saturation length Ls to
scale with Lc. Figure 2(b) plots Ls vs. q
−1 for a range of β-values, and demonstrates that
our numerical data is consistent with Ls ∼ q−1. We can also obtain the scaling law for the
crossover time and the corresponding numerical results (not shown here) are in agreement
with it.
Next, we consider the correlation function for the evolution morphology shown in Figure
1. It is obviously relevant to first consider the correlation function for a single spiral of length
L, as the snapshots in Figure 1 can be thought of as consisting of disjoint spirals of size
L. (Of course, this ignores modulations of the order parameter at spiral-spiral boundaries
but we will discuss those later.) We have approximated the 1-armed single-spiral solution as
ψ(~r, t) ≃ √1− q2 exp [−iωt+ i(θ − qr)]. The correlation function is obtained by considering
the correlation between points ~r1 and ~r2 (= ~r1 + ~r12) and integrating over ~r1 as follows:
C(r12) =
1
V
∫
d~r1Re {ψ(~r1, t)ψ(~r2, t)∗} h(L− r2)
4
=
(1− q2)
V
Re
∫
d~r1 exp [i(θ1 − θ2 − qr1 + q|~r1 + ~r12|)] h(L− |~r1 + ~r12|), (4)
where V is the spiral volume; and we have introduced the step function h(x) = 1 (0) if
x ≥ 0 (x < 0). The step function ensures that we do not include points which lie outside
the defect of size L.
It is convenient to introduce variables θ1 − θ12 = θ; x = r1/L; r = r12/L, to obtain
C(r12) =
(1− q2)
π
Re
∫ 1
0
dxx
∫ 2pi
0
dθ
x+ reiθ
(x2 + r2 + 2xr cos θ)1/2
×
exp
[
−iqL
{
x− (x2 + r2 + 2xr cos θ)1/2
}]
h[1− (x2 + r2 + 2xr cos θ)1/2], (5)
where we have used V = πL2 in d = 2. Thus, the scaling form of the single-spiral correlation
function is C(r12)/C(0) ≡ g(r12/L, q2L2). In general, there is no scaling with the spiral size
because of the additional factor qL. We recover scaling only in the limit q = 0 (β = 0),
which corresponds to the case of a vortex. Essentially, spirals of different sizes are not
morphologically equivalent because there is more rotation in the phase as one goes out
further from the core.
Figure 3 plots C(r12)/C(0) vs. r12/L for the case with β = 0.75 (q ≃ 0.203). These
results are obtained by a direct numerical integration of Eq. (5). We consider 4 different
values of L. The functional form in Figure 3 exhibits near-monotonic behavior for small
values of L (i.e., in the vortex or XY limit); and pronounced oscillatory behavior for larger
values of L, as is expected from the integral expression. Notice that r12/L ≤ 2 – larger
values of r12 correspond to the point ~r2 lying outside the defect.
The asymptotic behavior of the correlation function in the limit r = r12/L→ 0 (though
r12/ξ ≫ 1) is of considerable importance as it determines the tail of the momentum-space
structure factor [5]. In particular, we are interested in the singular part of the correlation
function as r → 0. In this limit, we can discard the step function in Eq. (5) as it only
provides corrections at the edge of the defect. The asymptotic analysis of the integral in Eq.
(5) involves considerable algebra, which we will report in detail elsewhere. Here, we confine
ourselves to quoting the final result for the singular part of C(r12):
5
Csing(r12) =
1
2
∞∑
p=0
∞∑
m=0
(−1)p+m (qL)
2(p+m)
(2p)!(2m)!
Γ
(
1
2
+m
)2
Γ
(
1
2
− p
)2
(m+ p+ 1)!2
×
(2m+ 1)(2p+ 1)r2(m+p+1) ln r. (6)
Eq. (6) is one of the central results of this paper and we would like to briefly discuss its
implications. The leading-order singularity is the same as that for the XY model (β = q = 0),
Csing(r12) =
1
2
r2 ln r [12], as expected. However, there is also a sequence of sub-dominant
singularities proportional to (qL)2r4 ln r, (qL)4r6 ln r, etc., and these become increasingly
important as the length scale L increases. These sub-dominant terms in Csing(r12) are
reminiscent of the leading-order singularities in models with O(n) symmetry, where n is
even [5,12]. Of course, in the context of O(n) models, these singularities only arise for n ≤ d
as there are no topological defects unless this condition is satisfied. In the present context,
all these terms are already present for d = 2. The implication for the structure-factor tail
is a sequence of power-law decays with S(k) ∼ (qL)2(m−1)Ld/(kL)d+2m, where m = 1, 2,
etc. Thus, though the true asymptotic behavior in d = 2 is still the generalized Porod tail,
S(k) ∼ L2(kL)−4, it may be difficult to disentangle this from other power-law decays.
Finally, Figure 4 compares our numerical data for the correlation function with the
functional form of the single-spiral correlation function. Recall that the correlation function
does not scale with the characteristic length because of the spiral nature of the defects. In
Figures 4(a)-(c), we have plotted numerical data for C(r12, t)/C(0, t) vs. r12 at t = 500, and
β = 0.75, 1.0, 1.25. For the comparison with Eq. (5), the length scale L is taken to be an
adjustable parameter. In each case, the best-fit value of L matches the length scale obtained
from the inverse defect density (see Figure 2(a)) within 10 percent. As is seen from Figure
4, the single-spiral correlation function is in good agreement with the numerical data for the
multi-spiral morphology upto (approximately) the first minimum. As a matter of fact, the
agreement is excellent (perhaps fortuitously) for β = 1.25, shown in Figure 4(c).
In the context of phase ordering dynamics, the Gaussian auxiliary field (GAF) ansatz
[5]- [8] has proven particularly useful for the characterization of multi-defect morphologies.
6
We have critically examined the utility of the GAF ansatz in the present context [13] and
find that it is only reasonable at early times – where, in any case, the ordering process is
analogous to that for the XY model. We are presently studying methods of improving the
GAF ansatz for the CGL equation and will discuss this elsewhere.
More generally, the utility of the GAF ansatz arises from the summation over phases
from many defects, which results in a near-Gaussian distribution for the auxiliary field.
However, in the present context, the shocks between spirals effectively isolate one spiral
region from the influence of other regions. As a matter of fact, the waves from other spirals
decay exponentially through the shock and the phase of a point is always dominated by the
nearest spiral. Therefore, we expect that the correlation function will be dominated by the
single-spiral result – in accordance with our numerical results.
To summarize: we have undertaken a detailed analytical and numerical study of nonequi-
librium dynamics in the CGL equation. For early times (L < Lc ∼ q−1), the domain growth
process is analogous to that for the XY model, which is well understood. At later times,
distinct effects due to spirals are seen and the evolving system freezes (in a statistical sense)
into a multi-spiral morphology. We have undertaken an asymptotic analysis of the correla-
tion function C(r12) for a single spiral. It exhibits a sequence of singularities as r12/L→ 0.
Furthermore, this correlation function is in good agreement with the numerical data for
multi-spiral morphologies, over an extended range of distances.
ACKNOWLEDGEMENTS
SP is grateful to A.J. Bray and H. Chate for useful discussions. SKD is grateful to
the University Grants Commission, India, for financial support in the form of a research
fellowship. MCC thanks the School of Physical Sciences, JNU for hospitality during the
stay in which this work was begun.
7
REFERENCES
[1] M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[2] For example, see Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag,
Berlin (1984); W. van Saarloos, in Spatiotemporal Patterns in Nonequilibrium Systems (ed. P.E.
Cladis and P. Palffy-Muhoray), Addison-Wesley, Reading, MA (1994).
[3] For example, see H. Chate, Nonlinearity 7, 185 (1994) for the d = 1 CGL equation; H. Chate
and P. Manneville, Physica A 224, 348 (1996) for the d = 2 CGL equation.
[4] K. Binder, in Materials Science and Technology, Vol. 5 : Phase Transformations of Materials
(ed. R.W. Cahn, P. Haasen and E.J. Kramer), p. 405, VCH, Weinheim (1991).
[5] A.J. Bray, Adv. Phys. 43, 357 (1994).
[6] T. Ohta, D. Jasnow and K. Kawasaki, Phys. Rev. Lett. 49, 1223 (1982).
[7] A.J. Bray and S. Puri, Phys. Rev. Lett. 67, 2670 (1991); H.Toyoki, Phys. Rev. B 45, 1965
(1992).
[8] G.F. Mazenko Phys. Rev. Lett. 63, 1605 (1989), Phys. Rev. B 42, 4487 (1990), Phys. Rev. B
43, 5747 (1991).
[9] P.S. Hagan, SIAM J. Appl. Math 42, 762 (1982).
[10] I.S. Aranson, L.B. Aranson, L. Kramer and A. Weber, Phys. Rev. A 46, R2992 (1992); A.
Weber, L. Kramer, I.S. Aranson and L.B. Aranson, Physica D 61, 279 (1992).
[11] A.N. Pargellis, P. Finn, J.W. Goodby, P. Pannizza, B. Yurke and P.E. Cladis, Phys. Rev. A
46, 7765 (1992); B. Yurke, A.N. Pargellis, T. Kovacs and D.A. Huse, Phys. Rev. E 47, 1525
(1993).
8
[12] A.J. Bray and K. Humayun, Phys. Rev. E 47, R9 (1993).
[13] S.K. Das, S. Puri and M.C. Cross, in preparation.
9
FIGURE CAPTIONS
Figure 1: Evolution of the CGL equation from a small-amplitude random initial condi-
tion. The evolution pictures were obtained from an Euler-discretized version of Eq. (1)
with α = 0, β = 0.75, implemented on an N2-lattice (N = 256). The discretization mesh
sizes were ∆t = 0.01, ∆x = 1.0; and periodic boundary conditions were imposed in both
directions. The snapshots show regions of constant phase θψ = tan
−1(Imψ/Reψ), measured
in radians, with the following color coding: θψ ∈ [1.85, 2.15] (black); θψ ∈ [3.85, 4.15] (red);
θψ ∈ [5.85, 6.15] (green). The snapshots are labeled by the appropriate evolution times.
Figure 2: (a) Plot of ln[L(t)] vs. ln t for α = 0 and β = 0.25, 0.5, 0.75, 1.0, 1.25 – de-
noted by the specified symbols. The characteristic length scale, L(t), is obtained from the
square root of the inverse defect density – measured directly from snapshots as shown in
Figure 1. The numerical data shown here was obtained as an average over 5 independent
runs for N2-lattices (with N = 1024). The solid line has a slope of 1/2.
(b) Plot of saturation length Ls vs. q
−1 for a range of β-values. The corresponding values
of q (as a function of β) are obtained from Hagan’s solution, cf. Figure 5 of Ref. [9]. The
solid line denotes the best linear fit to the numerical data.
Figure 3: Correlation function for the 1-armed spiral solution when β = 0.75 (q ≃ 0.203).
We plot C(r12)/C(0) vs. r12/L for different spiral sizes, L = 15, 25, 50, 100 – denoted by the
specified line-types. These results are obtained from a direct numerical integration of Eq.
(5).
Figure 4: Numerical data for the correlation function C(r12, t)/C(0, t) vs. r12 at t = 500
for the cases α = 0 and (a) β = 0.75; (b) β = 1.0; (c) β = 1.25. The numerical data
was obtained as an average over 5 independent runs for N2-lattices (with N = 1024). The
10
solid line refers to the numerical integration of Eq. (5) with L as an adjustable parameter.
Subsequently, the r12-axis is scaled so that the point C(r12, t)/C(0, t) = 1/2 is matched for
the numerical data and the analytical expression.
11
This figure "fig1.gif" is available in "gif"
 format from:
http://arXiv.org/ps/cond-mat/0103527v1


Non equilibrium dynamics in the complex Ginzburg-Landau equation

Crossref

Nonequilibrium dynamics in the complex Ginzburg-Landau equation

Results from a comprehensive analytical and numerical study of nonequilibrium dynamics in the two-dimensional complex Ginzburg-Landau equation have been presented. In particular, spiral defects have been used to characterize the domain growth law and the evolution morphology. An asymptotic analysis of the single-spiral correlation function shows a sequence of singularities—analogous to those seen for time-dependent Ginzburg-Landau models with O(n) symmetry, where n is even

Caltech Authors - Main

PHYSICAL REVIEW E, VOLUME 64, 056140Nonequilibrium dynamics in the complex Ginzburg-Landau equation
Sanjay Puri,1 Subir K. Das,1 and M. C. Cross2
1School of Physical Sciences, Jawaharlal Nehru University, New Delhi–110067, India
2Department of Physics, California Institute of Technology, Pasadena, California 91125
~Received 18 June 2001; published 30 October 2001!
Results from a comprehensive analytical and numerical study of nonequilibrium dynamics in the two-
dimensional complex Ginzburg-Landau equation have been presented. In particular, spiral defects have been
used to characterize the domain growth law and the evolution morphology. An asymptotic analysis of the
single-spiral correlation function shows a sequence of singularities—analogous to those seen for time-
dependent Ginzburg-Landau models with O(n) symmetry, where n is even.
DOI: 10.1103/PhysRevE.64.056140 PACS number~s!: 64.60.Cn, 05.70.Ln, 74.20.DeMuch recent interest has focused on pattern formation in
the complex Ginzburg-Landau ~CGL! equation
]c~rW ,t !
]t
5c~rW ,t !1~11ia!„2c~rW ,t !
2~11ib!uc~rW ,t !u2c~rW ,t !, ~1!
where c(rW ,t) is a complex order-parameter field that de-
pends on space (rW) and time (t). In Eq. ~1!, a and b are real
parameters. The CGL equation arises in a range of diverse
contexts, as reviewed by Cross and Hohenberg @1#. This uni-
versality arises from the fact that the CGL equation provides
a generic description of oscillations in a spatially-extended
system near a Hopf bifurcation @2#.
The CGL equation exhibits rich dynamical behavior with
variation of the parameters a and b , and the ‘‘phase dia-
gram’’ has been investigated ~mostly numerically! in various
studies @3#. In a large range of parameter space, the emer-
gence and interaction of spiral defect structures characterizes
the morphology. In this paper, we study the nonequilibrium
dynamics of the CGL equation resulting from a small-
amplitude random initial condition. In general, this nonequi-
librium evolution is referred to as ‘‘phase ordering dynam-
ics’’ or ‘‘domain growth,’’ and constitutes a well-studied
example of far-from-equilibrium statistical physics @4,5#.
Our analytical understanding of phase ordering systems has
depended critically upon modeling the dynamics of defects
in these systems ~e.g., interfaces, vortices, monopoles, etc.!
@5–8#. In this communication, we use spiral defect structures
to characterize the evolution morphology in the CGL equa-
tion. Many important features emerge in our study, which
should be of great relevance for both experiments and sub-
sequent numerical simulations.
For simplicity, we will focus on the CGL equation with
a50 and dimensionality d52. However, the results pre-
sented here are also relevant for the cases with aÞ0 and d
.2, as the underlying paradigm does not change, i.e., spirals
continue to determine the morphology in large regions of
parameter space. Following the work of Hagan @9#, Aranson
et al. @10#, and Chate and Manneville @3#, let us briefly dis-
cuss the phase diagram of the d52 CGL equation with a
50. The limiting case b50 corresponds to the dynamical
XY model, which is well understood. The appropriate ~point!1063-651X/2001/64~5!/056140~5!/$20.00 64 0561defects are vortices, and domain growth is driven by the
attraction and annihilation of vortex-antivortex pairs. The
relevant growth law for the characteristic length scale is
L(t);(t/ln t)1/2 @11,5#; and the analytic form of the time-
dependent correlation function ~which characterizes the
evolving morphology! has been obtained by Bray and Puri,
and ~independently! Toyoki @7#. Without loss of generality,
we focus on the case with b>0. For 0<b<b1 (b1
.1.397 @9#!, spirals ~which are asymptotically plane waves!
are linearly stable to fluctuations. For b1,b<b2 (b2
.1.82 @10,3#!, spirals are linearly unstable to fluctuations,
but the growing fluctuations are advected away, i.e., the spi-
ral structure is globally stable. Finally, for b2,b , the spirals
are globally unstable and cannot exist for extended times
@10#. Our results are relevant for the parameter regime with
b<b2, where spiral defects are an important feature of the
morphology.
Figure 1 shows the typical evolution from a small-
amplitude random initial condition for the case with b
50.75. Our numerical simulations were performed by imple-
menting an isotropic Euler-discretization of Eq. ~1! on
N2-lattices (N5256 for Fig. 1!, with periodic boundary con-
ditions in both directions. The discretization mesh sizes were
Dt50.01 and Dx51.0. In Fig. 1, we plot constant-phase
regions and the relevant color coding is provided in the fig-
ure caption. The evolving morphology is characterized by
spirals and antispirals, and there is a typical length scale L,
e.g., inter spiral spacing or the square root of inverse defect
density, which is the definition we will use subsequently.
Figure 2~a! plots ln@L(t)# vs ln t for five representative
values of b . The length-scale data was obtained from five
independent runs on N2 lattices with N51024. After an ini-
tial transient period, the length scale L(t) should saturate to
an equilibrium value (Ls) because of an effective spiral-
antispiral repulsive potential @1#. However, we stress that the
local order parameter continues to be time dependent—only
the morphology of the system undergoes statistical ‘‘freez-
ing.’’ This should be contrasted with the b50 case, where
vortices continue to anneal ~at zero temperature! as t→‘ . As
a matter of fact, the data for b50.25,0.5 in Fig. 2~a! does not
exhibit this morphological freezing on the time scales of our
simulation, though signs of the crossover are evident for b
50.5.
To understand this crossover behavior, we recall the ana-
lytical solution for an m-armed spiral due to Hagan @9#©2001 The American Physical Society40-1
SANJAY PURI, SUBIR K. DAS, AND M. C. CROSS PHYSICAL REVIEW E 64 056140c~rW ,t !5r~r !exp@2ivt1imu2if~r !# , ~2!
where rW[(r ,u); and v5b(12q2), where q is a constant
that is determined by b @9#. The limiting forms of the func-
tions r(r) and f(r) are
r~r !→A12q2, f8~r !→q , as r→‘ ,
r~r !→arm, f8~r !→r , as r→0, ~3!
where a is a constant that is determined by finiteness condi-
tions. We will focus on the case with m561, as only one-
armed spirals are stable in the evolution @9#. Furthermore, we
are only interested in distances r@j , where j is the defect
core size. Our numerical results show that the order-
parameter amplitude saturates to its maximum value on a
length scale j;O(1) dimensionless unit. In Table I, we
present typical values of j for the different values of b con-
sidered here. We define j as the radial distance from the
spiral center where the order-parameter amplitude reaches
half its maximum value. On the other hand, the maximum
defect length scale for ~say! b51.25 in Fig 2~a! is Ls
.13.0. Thus, we consider the spiral form in Eq. ~2! with
r(r)5A12q2 and f(r)5qr ~appropriate for r→‘).
We expect that spirals behave similarly to vortices for L
,Lc , where qLc;O(1). This is because the distinction be-
FIG. 1. Evolution of the CGL equation from a small-amplitude
random initial condition. The evolution pictures were obtained from
an Euler-discretized version of Eq. ~1! with a50, b50.75, imple-
mented on an N2 lattice (N5256). The discretization mesh sizes
were Dt50.01, Dx51.0; and periodic boundary conditions were
imposed in both directions. The snapshots show regions of constant
phase uc5tan21(Im c/Re c), measured in radians, with the fol-
lowing color coding: ucP@1.85,2.15# ~black!; @3.85,4.15# ~dark
gray!; @5.85,6.15# ~gray!. The snapshots are labeled by the appro-
priate evolution times.05614tween a spiral and a vortex is only apparent on length scales
qr.1. Thus, the early evolution should be analogous to that
for the XY model, both in terms of the domain growth law
and correlation function. In Fig. 2~a!, the solid line has a
FIG. 2. ~a! Plot of ln@L(t)# vs ln t for a50 and b
50.25,0.5,0.75,1.0,1.25—denoted by the specified symbols. The
characteristic length scale L(t) is obtained from the square root of
the inverse defect density—measured directly from snapshots as
shown in Fig. 1. The numerical data shown here were obtained as
an average over five independent runs for N2 lattices ~with N
51024). The solid line has a slope of 1/2. ~b! Plot of saturation
length Ls vs q21 for b values ranging from 0.75 to 1.20. The
corresponding values of q ~as a function of b) are obtained from
Hagan’s solution, cf. Fig. 5 of Ref. @9#. The solid line denotes the
best linear fit to the numerical data.0-2
NONEQUILIBRIUM DYNAMICS IN THE COMPLEX . . . PHYSICAL REVIEW E 64 056140slope of 1/2 and the initial growth ~at least for b<0.75)
appears to be comparable with the behavior for the XY
model, i.e., L(t);(t/ln t)1/2 for d52. Over extended time
intervals, this growth law is similar to power-law growth
with an ‘‘effective exponent’’ less than 1/2 @12#. From scal-
ing arguments, we also expect the saturation length Ls to
scale with Lc . Figure 2~b! plots Ls vs q21 for a range of b
values, and demonstrates that our numerical data is consis-
tent with Ls;q21. We can also obtain the scaling law for the
crossover time and the corresponding numerical results ~not
shown here! are in agreement with it.
Next, we consider the correlation function for the evolu-
tion morphology shown in Fig. 1. It is obviously relevant to
first consider the correlation function for a single spiral of
length L, as the snapshots in Fig. 1 can be thought of as
consisting of disjoint spirals of size L. ~Of course, this ig-
nores modulations of the order parameter at spiral-spiral
boundaries but we will discuss those later!. We have ap-
proximated the one-armed single-spiral solution as c(rW ,t)
.A12q2 exp@2ivt1i(u2qr)#. The correlation function is
obtained by considering the correlation between points rW1
and rW2 (5rW11rW12) and integrating over rW1 as follows:
C~r12!5
1
VE drW1 Re$c~rW1 ,t !c~rW2 ,t !*%h~L2r2!
5
~12q2!
V ReE drW1 exp@ i~u12u22qr1
1qurW11rW12u!#h~L2urW11rW12u!, ~4!
where V is the spiral volume; and we have introduced the
step function h(x)51 (0) if x>0 (x,0). The step func-
tion ensures that we do not include points which lie outside
the defect of size L.
It is convenient to introduce variables u12u125u; x
5r1 /L; r5r12 /L , to obtain
C~r12!5
~12q2!
p
ReE
0
1
dxxE
0
2p
du
x1reiu
~x21r212xr cos u!1/2
3exp@2iqL$x2~x21r212xr cos u!1/2%#
3h@12~x21r212xr cos u!1/2# , ~5!
where we have used V5pL2 in d52. Thus, the scaling form
of the single-spiral correlation function is C(r12)/C(0)
[g(r12 /L ,q2L2). In general, there is no scaling with the
spiral size because of the additional factor qL . We recover
scaling only in the limit q50 (b50), which corresponds to
the case of a vortex. Essentially, spirals of different sizes are
TABLE I. Spiral core size (j) for different values of b . We
define j as the radial distance from the spiral center where the
order-parameter amplitude reaches half its maximum value.
b 0.0 0.25 0.50 0.75 1.0 1.25
j 0.997 1.002 1.014 1.025 1.031 1.03205614not morphologically equivalent because there is more rota-
tion in the phase as one goes out further from the core.
Figure 3 plots C(r12)/C(0) vs r12 /L for the case with
b50.75 (q.0.203). These results are obtained by a direct
numerical integration of Eq. ~5!. We consider four different
values of L. The functional form in Fig. 3 exhibits near-
monotonic behavior for small values of L ~i.e., in the vortex
or XY limit!; and pronounced oscillatory behavior for larger
values of L, as is expected from the integral expression. No-
tice that r12 /L<2—larger values of r12 correspond to the
point rW2 lying outside the defect.
The asymptotic behavior of the correlation function in the
limit r5r12 /L→0 ~though r12 /j@1) is of considerable im-
portance as it determines the tail of the momentum-space
structure factor @5#. In particular, we are interested in the
singular part of the correlation function as r→0. In this
limit, we can discard the step function in Eq. ~5! as it only
provides corrections at the edge of the defect. The
asymptotic analysis of the integral in Eq. ~5! involves con-
siderable algebra, which we will report in detail elsewhere.
Here, we confine ourselves to quoting the final result for the
singular part of C(r12),
Csing~r12!5
1
2 (p50
‘
(
m50
‘
~21 !p1m
~qL !2(p1m)
~2p !!~2m !!
3
G~ 12 1m !
2
G~ 12 2p !2~m1p11 !!2
3~2m11 !~2p11 !r2(m1p11) ln r . ~6!
FIG. 3. Correlation function for the one-armed spiral solution
when b50.75 (q.0.203). We plot C(r12)/C(0) vs r12 /L for dif-
ferent spiral sizes, L515,25,50,100—denoted by the specified line
types. These results are obtained from a direct numerical integration
of Eq. ~5!.0-3
SANJAY PURI, SUBIR K. DAS, AND M. C. CROSS PHYSICAL REVIEW E 64 056140Equation ~6! is one of the central results of this paper and
we would like to briefly discuss its implications. The
leading-order singularity is the same as that for the XY
model (b5q50), Csing(r12)5 12 r2 ln r @13#, as expected.
However, there is also a sequence of subdominant singulari-
ties proportional to (qL)2r4 ln r, (qL)4r6 ln r, etc., and these
become increasingly important as the length scale L
increases. These subdominant terms in Csing(r12) are
reminiscent of the leading-order singularities in models
with O(n) symmetry, where n is even @5,13#. Of course,
in the context of O(n) models, these singularities only arise
for n<d as there are no topological defects unless this
condition is satisfied. In the present context, all these terms
are already present for d52. The implication for the
structure-factor tail is a sequence of power-law decays
with S(k);(qL)2(m21)Ld/(kL)d12m, where m51,2, etc.
Thus, though the true asymptotic behavior in d52 is
still the generalized Porod tail, S(k);L2(kL)24, it
may be difficult to disentangle this from other power-law
decays.
Finally, Fig. 4 compares our numerical data for the
correlation function with the functional form of the single-
spiral correlation function. Recall that the correlation
function does not scale with the characteristic length because
of the spiral nature of the defects. In Figs. 4~a!–~c!, we
have plotted numerical data for C(r12 ,t)/C(0,t) vs r12 at t
5500, and b50.75,1.0,1.25. For the comparison with
Eq. ~5!, the length scale L is taken to be an adjustable
parameter. In each case, the best-fit value of L matches the
length scale obtained from the inverse defect density @see
Fig. 2~a!# within 10%. As is seen from Fig. 4, the single-
spiral correlation function is in good agreement with the
numerical data for the multispiral morphology up to
~approximately! the first minimum. As a matter of fact, the
agreement is excellent ~perhaps fortuitously! for b51.25,
shown in Fig. 4~c!. We should remark that the correlation
function data shown in Figs. 4~b! and ~c! ~corresponding to
b51.0,1.25) does not change at later times, because the de-
fect length scales have already frozen by t5500 for these b
values @see Fig. 2~a!#.
In the context of phase ordering dynamics, the Gaussian
auxiliary field ~GAF! ansatz @5–8# has proven particularly
useful for the characterization of multidefect morphologies.
We have critically examined the utility of the GAF ansatz in
the present context @14# and find that it is only reasonable at
early times—where, in any case, the ordering process is
analogous to that for the XY model. We are presently study-
ing methods of improving the GAF ansatz for the CGL equa-
tion and will discuss this elsewhere.
More generally, the utility of the GAF ansatz arises from
the summation over phases from many defects, which results
in a near-Gaussian distribution for the auxiliary field. How-
ever, in the present context, the shocks between spirals ef-
fectively isolate one spiral region from the influence of other
regions. As a matter of fact, the waves from other spirals
decay exponentially through the shock and the phase of a
point is always dominated by the nearest spiral. Therefore,05614we expect that the correlation function will be dominated by
the single-spiral result—in accordance with our numerical
results.
To summarize, we have undertaken a detailed analytical
and numerical study of nonequilibrium dynamics in the
CGL equation. For early times (L,Lc;q21), the domain
growth process is analogous to that for the XY model,
which is well understood. At later times, distinct effects
due to spirals are seen and the evolving system freezes
~in a statistical sense! into a multispiral morphology.
We have undertaken an asymptotic analysis of the
correlation function C(r12) for a single spiral. It exhibits a
sequence of singularities as r12 /L→0. Furthermore, this
correlation function is in good agreement with the numerical
data for multispiral morphologies, over an extended range of
distances.
S.P. is grateful to A.J. Bray and H. Chate for useful dis-
cussions. S.K.D. is grateful to the University Grants Com-
mission, India, for financial support. M.C.C. thanks the
School of Physical Sciences, JNU, for hospitality during the
stay in which this work was begun.
FIG. 4. Numerical data for the correlation function
C(r12 ,t)/C(0,t) vs r12 at t5500 for the cases a50 and ~a! b
50.75; ~b! b51.0; ~c! b51.25. The numerical data were obtained
as an average over five independent runs for N2 lattices ~with N
51024). The solid line refers to the numerical integration of Eq. ~5!
with L as an adjustable parameter. Subsequently, the r12 axis is
scaled so that the point C(r12 ,t)/C(0,t)51/2 is matched for the
numerical data and the analytical expression.0-4
NONEQUILIBRIUM DYNAMICS IN THE COMPLEX . . . PHYSICAL REVIEW E 64 056140@1# M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65, 851
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Wesley, Reading, MA, 1994!.
@3# For example, see H. Chate, Nonlinearity 7, 185 ~1994! for the
d51 CGL equation; H. Chate and P. Manneville, Physica A
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@4# K. Binder, in Materials Science and Technology, Vol. 5: Phase
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Haasen, and E.J. Kramer ~VCH, Weinheim, 1991!, p. 405.
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Nonequilibrium Dynamics in the Complex Ginzburg-Landau Equation

Nonequilibrium Dynamics in the Complex Ginzburg-Landau Equation

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