Table Of ContentA Non-Equilibrium Defect-Unbinding Transition: Defect Trajectories and Loop
Statistics
∗
Glen D. Granzow and Hermann Riecke
Department of Engineering Sciences and Applied Mathematics,
Northwestern University, 2145 Sheridan Rd, Evanston, IL, 60208, USA
system non-variational.
Various results have been obtained for the dynam-
In a Ginzburg-Landau model for parametrically driven
ics of defects in spatio-temporally chaotic systems. In
waves a transition between astate of ordered and one of dis-
binary-mixtureconvectionprogresshasbeenmadetore-
1 ordered spatio-temporal defect chaos is found. To character-
construct the full wave pattern from the locations of the
0 ize the two different chaotic states and to get insight into
dislocations[6]. Theprobabilitydistributionfunctionfor
0 the break-down of the order, the trajectories of the defects
thenumberofdislocationshasbeenmeasuredinelectro-
2 are tracked in detail. Since the defects are always created
n and annihilated in pairs the trajectories form loops in space convection [9] and found to agree quite well with results
basedonthecomplexGinzburg-Landauequationandon
a time. Theprobabilitydistributionfunctionsforthesizeofthe
a simple diffusive model [10]. The dynamical relevance
J loops and the number of defects involved in them undergo a
ofdislocationshasbeendemonstratedbestsofarinwork
9 transition from exponential decay in the ordered regime to a
power-law decay in the disordered regime. These power laws thatsucceededinextractingthecontributionofeachdis-
] are also found in a simple lattice model of randomly created locationtothetotalfractaldimensionofthe(extensively)
D
defect pairs that diffuseand annihilate upon collision. chaotic attractor of the CGLE [11].
C Very early in the investigationof pattern-forming sys- Thespatio-temporallychaoticstatesfoundinpattern-
. tems far from thermodynamic equilibrium it has been forming systems typically arise from ordered states
n
recognized that dislocations in stripe patterns near on- through some transition when a control parameter is
i
l set are mathematically closely related to defects in the changed. An interesting question is what actually hap-
n
[ equilibrium xy-model since both systems are described penswhentheorderofthepatternbreaksdown. Inanal-
byasinglecomplexorderparameterAintermsofwhich ogy with the melting of two-dimensional crystals [12,13]
1 the dislocations are givenby locations of vanishing mag- one may expect that defects in the pattern may play an
v
nitude |A| = 0. One of the fascinating phase-transition important role. In this Letter we present results for a
9
phenomena studied extensively in the xy-model is the transition between two spatio-temporally chaotic states
1
0 Kosterlitz-Thouless transition, which is associated with in a Ginzburg-Landau model for parametrically excited
1 the unbinding of defect pairs [1]. This motivated early waves [14]. While one state is disordered in space, the
0 efforts to identify related phenomena also in pattern- other retains a stripe-like order despite the chaotic cre-
1 forming systems like Rayleigh-B´enard convection [2]. A ation and annihilation of defect pairs. We characterize
0 directanalogybetweenthesesystemsdoesnothold,how- the break-down of order in terms of the defect dynam-
/
n ever, for a number of reasons. Most significantly, the ics and find that the transition to the disorderedstate is
i phase transitions occur at finite temperature and are in- associated with what one may call an unbinding of the
l
n timatelyrelatedtothe relevanceoffluctuations,whereas pairwise created defects. Tentative results for this un-
v: in macroscopic systems like Rayleigh-B´enardconvection binding transition have been presented earlier in [15,14].
i the effect of (thermal) noise is negligible in most situa- Toillustratethepossibleroleofthedefectdynamicsin
X
tions [3]. Moreover, in the absence of noise there are no thebreak-downoforder,Fig.1presentsaspace-timedia-
r persistent dynamics in the Ginzburg-Landau with real gramofthedefectdynamicsintheregimewithpersistent
a
coefficients that describes the equilibrium system. spatialorder. They-locationofeachdefectwithpositive
More recently, spatio-temporal chaos in pattern- topological charge is shown as a solid circle while the y-
forming systems has found considerable attention. A location of each defect with negative charge is shown as
number of variations of convection experiments have re- a dot. The trajectories of almost all of the defect pairs
vealed various different types of spatio-temporal chaos formsimpleloopsinspace-time. Thus,whileinanygiven
like spiral-defect chaos (e.g. [4]), domain chaos triggered system defects are always annihilated in pairs of oppo-
by the Ku¨ppers-Lortz instability (e.g. [5]), dislocation- sitecharge,intheorderedregimetheannihilatingdefects
dominated chaos in binary-mixture convection [6] and havealsobeencreatedtogether. Itisinthissensethatwe
in electroconvection [7]. Extensive theoretical investi- considerthemtobebound pairs. Thepreservationofthe
gations have focussed on the complex Ginzburg-Landau stripe-like order in this regime can then be understood
equation (CGLE) for the complex amplitude of oscilla- intuitively, since the dynamics of defects in such simple
tionsarisinginaHopfbifurcation(e.g.[8]). Incontrastto defect loops affect only a very small portion of the sys-
the Ginzburg-Landau equation describing the xy-model temand,moreover,renderthe systemalmostunchanged
the coefficients in the CGLE are complex and make this after their disappearance. In some cases the space-time
1
loopsinvolvetwo(seearrowinfig.1)orpossiblythreede- plex. The traveling waves arise in a Hopf bifurcation at
fectpairs. Thentheareaofthe systemthatisperturbed a ≡ Re(a) = 0. For c < 0 the bifurcation is super-
r r
by the defects is larger, but after their annihilation the critical and the waves exist for a > 0. The parametric
r
systemis still left in essentially the same state as before. forcing is applied at close to twice the Hopf frequency.
It therefore induces a resonant interaction between the
15250 counter-propagating waves at linear order [16,17]. For
a <0andb2 >a2+a2 standingwavesareexcitedpara-
r r i
metrically that are phase-locked to the forcing. These
15200 waves correspond qualitatively to the waves excited in
Faradayexperiments. Inallofthesimulationsweusepe-
e riodic boundary conditions in both directions and solve
m
(1,2)pseudo-spectrallywithafourth-orderRunge-Kutta
Ti 15150
scheme using an integrating factor for the linear deriva-
tive terms.
Numerical simulations of (1,2) show two distinct
15100
regimes of spatio-temporal chaos for the parametrically
excited waves: a conventional regime in which the spa-
tial correlation function decays rapidly in an essentially
0 10 20 30 40 50 60
isotropic way and a regime of spatio-temporal chaos
y-Location whichexhibitsastrikinglyorderedstateinwhichthecor-
FIG. 1. Space-time diagram of defect trajectories. for relationfunctionrevealsastrongstripe-likeorderthatis
a=0.25,c=−1+4i,d=1+0.5i,s=0.2,g=−1−12i,and alsoapparentinindividualsnapshots[15,14]. Thetransi-
b=2. Circles anddotsdenotedefectsofoppositetopological tionbetweenthe twostates is discontinuousasindicated
charge. by a jump in the averagenumber of defects.
In order to get insight into the role of the defect dy-
Theorientationandpositionofthestripepatternisaf- namicsinthisorder-disordertransitionwetrackeachde-
fected significantly only between the defects. Therefore, fect from its creation to its annihilation. In particular,
one may expect that the destruction of order requires we identify which defect was annihilated by which de-
that the defects in a pair have to unbind and separate fect. From these data we extract the distribution func-
arbitrarily far from each other. However, it may also be tions for the number of defect pairs that are participat-
sufficient if the defects in a given pair are annihilated by ing in a space-time loop and for the spatial and tempo-
defects from two other pairs, which in turn are annihi- ral extent of the loops. Since the relevant information
latedbyfurtherdefectpairs,generatingachainofevents turns out to be in the rare, large loops care has to be
whose trajectoryin space-time is a largeloop that spans taken to identify annihilation and creation processes re-
the whole system. Simulations show that in fact even liably and distinguish them from situations in which a
in the disordered regime most defects are annihilated by defect simply moved relatively far in one time step. In
their‘own’partnerformingasimpleloopandmostloops our defect-tracking scheme we recursively check the dis-
are small compared to the system size. Thus, if a dis- tances between defects of equal and of opposite charge.
tinction between the two regimes can be made based on If in consecutive time steps two defects of equal charge
defect trajectories it must involve more subtle aspects are closer than some threshold value δ(n) = n·δ they
like the distribution of defect loops as a function oftheir 1 1
qualifyas asingledefect thathasmovedfromoneto the
size. Using a detailed statistical analysis of the defect
other position. If more than two defects fall into this
dynamics we show in the following that indeed the sta-
category the closest defects are taken to be ‘continuing’
tistical properties of the defect loops in space-time are
defects. Defectsthatarenotcontinuingdefectsarecandi-
qualitatively different in the two regimes.
datesforannihilationandcreationevents. Amongthose,
We consider a Ginzburg-Landau model for paramet-
two defects of opposite charge and closer than a second
rically excited waves in an axially anisotropic system in
threshold δ(n) = n·δ are identified as a pair that was
whichatthebifurcationpointthewavestravelonlyalong 2 2
annihilated(orcreated)inthis time step. After one step
the x-axis,
of this analysis the same analysis is repeated with new,
∂ A+s∂ A=d∇2A+aA+bB+ increased values for the thresholds, δ(n+1) =(n+1)·δ ,
t x i i
c|A|2A+g|B|2A, (1) until all defects have been assigned.
∂ B−s∂ B =d∗∇2B+a∗B+bA+ Fig.2 shows the relative frequency of loops consisting
t x x of at least n defect pairs. The results in figs.2,3,4 are
c∗|B|2B+g∗|A|2B. (2) basedon8000timesteps(dt=0.5)withanaveragenum-
ber of 7000 defects at any given time in the disordered
Here A and B are the complex amplitudes of right- and
regime. In the ordered regime (b ≥ 0.7) very few loops
left-traveling waves. All the coefficients except for the
containmorethan5defectpairsandthe distributionde-
group velocity s and the parametric forcing b are com-
2
cays essentially exponentially. In the disordered regime is of the order of one wavelength. This may indicate
(b≤0.625),however,thenumberofloopswithmanyde- somepinningofthedefectsbythepatterninthatitmay
fectsisgreatlyincreasedandthedecayofthedistribution restrict their motion to a predominantly climbing mo-
function is only algebraic with an exponent of α≈1.5. tion. In the disordered pattern, on the other hand, a
noticeable number of loops is of the order of the system
0 size L = 1088 and even larger (i.e. wrapping around
10
b=0.4 the system due to the periodic boundary conditions). In
10-1 b=0.5 a simple interpretation these events would be associated
cy b=0.625 withapersistentchangeintheaveragewavevectorofthe
en 10-2 b=0.7 pattern. Again, the distribution functions are exponen-
u
q b=1.0 tialinthe orderedregimeandexhibita powerlawin the
e
Fr 10-3 disordered regime with exponent β ≈3. The distinction
e betweentheregimesisnotquiteasstrikinginthespatial
v
ati 10-4 n-1.5 extent∆y inthey-direction. Evenintheorderedregime
el theloopscanreachasizeof∆y ≈100,whiletheloopsin
R
-5
10 the disorderedregimeextendto sizesof∆y ≈1000. The
distributionisstillexponentialintheorderedregimeand
-6
10 appears to be power-law in the disordered regime. How-
0 1 2 3
10 10 10 10 ever,the measuredexponentincreasesfrom about2.8 to
Number n of Defects in Loop
4 as b is increased from 0.4 to 0.625.
FIG.2. Relative frequency of loops made up of at least n
defect pairs. Parameters as in fig.1 except for b. System size
0
L=1088 in the disordered regime; L=272 and L=136 for 10
b=0.4
b=0.7 and b=1.0, respectively.
-1 b=0.5
10
cy b=0.625
Since the defect motion essentially affects only the en 10-2 b=0.7
(vaguely defined) part of the system between the defects u
q b=1.0
a better indicator for the expected loss of order are the re -3
F 10
spatialextents∆x≡xmax−xmin and∆y ≡ymax−ymin e
oHfetrheexdmefinec,mtaloxoapnsdinytmhienx,m-aaxndreyfe-drirteocttihoen,mreinspimecatlivaenlyd. elativ 10-4
maximal values of x and y in any given loop. Note R -5 ∆t-2.7
10
thatthese valuesneednotbe obtainedatthe sametime.
Thus, in principle a small loop could still yield large ∆x -6
10
or ∆y if it traveled. 101 102 103
t-span ∆t
100 FIG. 4. Relative frequency of loops with lifetime at least
b=0.4 ∆t. Parameters as in fig.2.
-1 b=0.5
10
cy b=0.625 The duration ∆t of the loops is also of interest and
en 10-2 b=0.7 the correspondingrelativefrequencies areshownin fig.4.
u
q b=1.0 The exponential and power-law character of the distri-
e
r -3
F 10 butionsarequite clearinthe respectiveregimeswiththe
ve exponent of the power-lawγ ≈2.7.
Relati 10-4 ∆x-3 inTthoegdivisetrfuibruthtieornsfuupnpcotriotnfsorinththeeexdiisstoerndceereodfrpeogwimerelaawnds
-5
10 to getmore insightinto their originwe haveinvestigated
a simple lattice model of the defect dynamics, which is
-6
10 based on the observation that the single-defect statis-
0 1 2 3
10 10 10 10
tics in the disorderedregime show the same signature as
x-span ∆x
those of the defect chaos in the CGLE. This type of dis-
FIG.3. Relative frequency of loops with spatial extent in
tributionhasbeenshowntoariseifthedefectsbehaveas
thex-direction at least ∆x . Parameters as in fig.2.
randomwalkersthat are annihilated upon colliding with
any other defect of opposite charge [10].
Similar to fig.2, fig.3 shows the relative frequency of
Theresultsofourimplementationofthesimplelattice
loops with size in the x-direction larger than ∆x. In the
model are shown in fig.5. With a probability p, ran-
ordered regime the decay is again very rapid and there
dom walkers of opposite charge are created pairwise at
are essentially no loops with ∆x larger than 10, which
the same randomly chosen site of a square lattice. They
3
interact only with walkers of opposite charge and an- experiments in electroconvection[9] and in thermal con-
nihilate upon contact. Fig.5 shows the loop statistics vection in an inclined layer [18]. It would be exciting to
for these walkers in a system of size L = 1600 and a study also the loop distribution functions in these sys-
probability of creation p = 0.000016 (thick lines) and tems. The amount of detailed data necessary to deter-
p = 0.0001 (thin lines). All measured quantities, i.e. mine the loop statistics represents presumably a chal-
number of defects in a loop and the spatial as well as lenging task in the current set-ups.
the temporalextentofthe loops,showpower-lawbehav- The simplicity of the lattice model suggests that the
ior for large loops. The simplicity of the lattice model power laws may also be amenable to an analytic ap-
suggests that the power laws are expected to arise in a proach. A further interesting question is whether the
much wider class of spatio-temporally chaotic systems lattice modelcanbe extended to obtainalsoa transition
including the CGLE. The exponents are measured to be frompower-lawto exponentialdecay. Introducing anat-
α = 1.6, β = 2.9, and γ = 2.4, respectively. Thus, even tractiveinteractionbetweenthedefectswouldseemtobe
the values of exponents agree quite well with those ob- a natural choice. It is not clear whether the interaction
tained in the simulations of (1,2). should be short- or long-range.
HRgratefullyacknowledgesdiscussionswithH.Chat´e,
p=0.000016 & p=0.0001 L=1600 M. Cross, L. Sander, J. Vin˜als, and thanks the Aspen
0 Center for Physics, where parts of this work were per-
10
n formed. This work was supported by the Engineering
10-1 ∆x ResearchProgramofthe Officeof BasicEnergySciences
cy -2 ∆y attheDepartmentofEnergy(DE-FG02-92ER14303)and
n 10
e ∆t by a grant from NSF (DMS-9804673).
u
q -3
re 10 ∆t-2.4
F
e 10-4
v -1.6
elati 10-5 n ∆x-2.9
R
-6
10
[1] J.KosterlitzandD.Thouless,J.Phys.C6,1181(1973).
-7
10
0 1 2 3 4 5 [2] R.Occelli, E. Guazzelli, and J. Pantaloni, J. Phys.Lett.
10 10 10 10 10 10
44, 567 (1983).
FIG. 5. Simulations of the lattice model. Thick lines de-
[3] I. Rehberget al.,Phys.Rev.Lett. 67, 596 (1991).
note p=0.000016, thin lines p=0.0001. L=1600.
[4] S. Morris, E. Bodenschatz, D. Cannell, and G. Ahlers,
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[5] Y. Hu, W. Pesch, G. Ahlers, and R. Ecke, Phys. Rev. E
pled complex Ginzburg-Landau equations we have in-
58, 5821 (1998).
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(1996).
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a significant increase in the number of loops that extend (1990).
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withthosefoundintheGinzburg-Landauequations. The [16] H. Riecke, J. Crawford, and E. Knobloch, Phys. Rev.
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[18] K. Daniel and E. Bodenschatz, priv. comm. .
found in a wider class of defect-chaotic systems and in
particular also in the single complex Ginzburg-Landau
equation.
The single-defect statistics obtained in the disordered
state and in the lattice model have also been found in
4
