Interplay of interfacial noise and curvature driven dynamics in two dimensions

1.3 MB·

by  Parna Roy

Interplay of interfacial noise and curvature driven dynamics in two dimensions
Parna Roy1 and Parongama Sen1
1Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India.
We explore the effect of interplay of interfacial noise and curvature driven dynamics in a binary
spinsystem. Anappropriatemodelisthegeneralised twodimensionalvotermodelproposedearlier
7 (J. Phys. A: Math. Gen. 26, 2317 (1993)), where the flipping probability of a spin depends
1
on the state of its neighbours and is given in terms of two parameters x and y. x = 0.5,y = 1
0
correspondstotheconventionalvotermodelwhichispurelyinterfacialnoisedrivenwhilex=1and
2
y = 1 corresponds to the Ising model, where coarsening is fully curvature driven. The coarsening

b phenomena for 0.5 < x < 1 keeping y = 1 is studied in detail. The dynamical behaviour of the
e relevant quantities show characteristic differences from both x = 0.5 and 1. The most remarkable
F result is the existence of two time scales for x ≥ x where x ≈ 0.7. On the other hand, we have
c c
studied the exit probability which shows Ising like behaviour with an universal exponent for any
8
value of x > 0.5; the effect of x appears in altering the value of the parameter occurring in the

] scaling function only.
h
c PACSnumbers: 89.75.Da,64.60.De,75.78.Fg
e
m
- Nonequilibrium phenomena associated with the zero two models. For the the IM, P(t) shows algebraic decay
at temperatureorderingprocessinclassicalIsingandVoter even in higher dimensions; in two dimensions θ ≈ 0.2
t models [1–4] have been extensively studied in the recent [11–14]. However,for the two dimensional VM, P(t) has
s
. past. Both models are two state models and the states the behaviour exp[−constant(lnt)2] [15]. The spin auto-
t
a can be represented by Ising spins. There is, however a correlationfunctionisanotherdynamicalquantitywhich
m basicdifference. TheIsingmodel(IM)isdefinedusingan again shows different behaviour for the VM and the IM
- energyfunction(H =−J σiσj; whereσ =±1andthe in two dimensions [15–19].
d sumis usuallyovernearesPtneighbours)andit hasno in- Another interesting feature of the Z models (models
n 2
trinsicdynamics. However,startingfromaconfiguration withspinupandspindownsymmetry)withtwoabsorb-
o
far from equilibrium, one can study the time dependent ing states (which are either all spins up or down) is the
c
[ behaviour of the so called kinetic Ising model. At zero exit probability E(ρ). E(ρ) is defined as the probabil-
temperature, the time evolution essentially corresponds ity that the final state has all up spins starting with a
2
to anenergyminimising scheme [5]inthe standardrules density ρ of up spins. The exit probability E(ρ) is also
v
likesinglespinflipGlauberorMetropolisdynamics. The different in the two models for d > 1. It can be eas-
7
1 Voter model (VM) on the other hand has no such en- ily argued that in all dimensions, E(ρ) = ρ for the VM.
1 ergy function associated - it is defined by the dynamical With only nearest neighbour interaction, E(ρ) is obvi-
9 rule that an agent follows the state of a randomly cho- ously linear for the IM also in one dimension. However,
0 sen neighbour at each time step. The kinetic Ising and allowingfurtherneighbourinteractionsandotherparam-
1. Votermodels areknowntobe identicalinone dimension eters governingthe dynamics, a nonlinear behaviour can
0 while in higher dimensions the dynamical schemes are be observed even in one dimension for both Ising model
7 markedlydifferent[6]. While the coarseningis curvature and Voter model [20–22]. This implies that there is a
1 driven in the Ising model, it is interfacial noise driven scope for phenomena like “minority spreading” [23] al-
:
v in the Voter model. This results in different behaviour ways in one dimension. In the two dimensional Ising
i of the relevant dynamical variables like density of active modelE(ρ)showsnon-linearbehaviour[24]whichinthe
X
bonds n(t), persistence probability P(t) and time scales. thermodynamiclimitapproachesastepfunction. Itmay
r Activebondsarethosewhichconnectneighbouringspins be mentioned here that in the zero temperature order-
a
withoppositesigns. Inone dimension,forboththe Ising ing of Ising model, one encounters the problem of frozen
−1
and Voter models, n(t) shows power law decay as t 2. states [25, 26]. Hence, the calculation of exit probability
This behaviour is true for the Ising modelevenin higher is made using only those configurations which reach the
dimensions. But for the Voter model, n(t) asymptoti- all up or all down states.
cally vanishes as ln1t in two dimensions and for dimen- Sincetheinterfacialnoiseandsurfacetensiongoverned
sions d > 2, n(t) ∼ a−bt−d/2. The dynamics in VM dynamicsdefinitelyleadtohighlydifferentdynamicalbe-
is slower and the consensus time (by consensus we mean haviour of several important quantities, it is worthwhile
the all up and all down absorbing states of the system) to study models in which both are present in a tunable
typicallybehavesasL2logLincontrasttoL2 fortheIM manner. One such model had been proposed in [27],
intwodimensions(Listhesystemsize). Thepersistence namely the generalisedvoter model. We have, therefore,
probability P(t), defined as the probability that a spin considered this particular model to study the interplay
does not change sign till time t, shows algebraic decay of the interfacial noise and curvature driven dynamics.
as t−θ with θ = 0.375 in one dimension [7–10] for the Inthe generalisedvotermodel,the dynamicalrule has
2
been parameterised so that one can recover a number of t=100 t=500
models for specific values of the parameters. We have (a) (b)
investigatedthe behaviourofthepersistenceprobability,
decayof activebonds, exitprobability andtime to reach
consensus. These quantities are not related in general
and hence the effect of changing the parameter values
may be different for each of them. (c) (d)
Let us briefly review the generalised Voter model
(GVMhenceforth)proposedin[27]. Here,ateachsiteof
the square lattice there is a spin variable σ = ±1. The
i
configurationevolvesintime accordingto singlespinflip t=2000 t=12000
stochastic dynamics. The spin flip probability w (σ) for
i
the ith spin is given by, FIG.2. Certainconfigurationsshowveryslowrelaxationasx
1 isincreased. Snapshotsshowsuchaconfigurationfordifferent
wi(σ)= [1−σifi(σ)], (1) times for x=0.9.
2
where f (σ) = f( σ ), a function of the sum of
i δ i+δ
the nearest neighbPor spin variables. The model is de-
configurations show the existence of nearly striped pat-
fined taking f(0) = 0, f(2) = −f(−2) = x and f(4) =
terns which do reach consensus but very slowly as the
−f(−4) = y, where x and y are restricted to the con-
interfaces take long time to vanish. In Figs 1 and 2, we
ditions x ≤ 1 and y ≤ 1. The original VM is recovered
have shown snapshots for two values of x; for x = 0.6,
for x = 0.5 and y = 1 whereas the IM corresponds to
thecurvaturedrivencoarseningisseentodominatewhile
x=1,y =1. Along the line y =1,there aretwo absorb-
for x = 0.9, a case is shown where coarsening has led to
ing states: all spins up and all spins down for x ≥ 0.5
domains with nearly straight edges prevailing over large
(apartfrompossiblefrozenstates). Thesestatesarehow-
timedurations. Snapshotsforothervaluesofxaregiven
ever unstable for x<0.5. As the limiting values x=0.5
in the [28].
and x=1.0 correspond to the two different models, one
The variation of the density of active bonds n(t)
can expect either a sharp transition or a crossover be-
against time is plotted in Fig. 3 for different values of
haviour at an intermediate value of x.
x. As x increases from 0.5, we find that n(t) goes to
We have studied the non-equilibrium behaviour and
zero involving larger time scales. However, the variation
exit probability E(ρ) of the GVM keeping y = 1 and
is faster than the 1/logt behaviour known for the Voter
varyingxusingMonteCarlosimulationsonL×Lsquare
model. It is found that for any 0.5 ≤ x < 1 the system
lattices with L≤80. Periodic boundary conditions have
always reaches the equilibrium ground state since n(t)
been used and at least 2500 different initial configura-
vanishes which implies the freezing probability is zero.
tionshavebeensimulated. Persistenceprobability,active
Only at x = 1 a frozen state may be reached. As x ap-
bonds dynamics and consensus times are the quantities
proaches unity (but not equal to it), the initial decay of
estimated. These quantities are unrelated and therefore
n(t) can be fitted quite accurately by a power law, while
itisusefultostudyallofthesetocheckhoweachofthem
there is a clear crossover to a much slower evolution at
is effected by tuning of the parameter x.
latertimes asshowninthe insetofFig. 3. This suggests
t=100 t=200 that there are two different time regimes. There is an
(a) (b) initial time scale up to which the behaviour is similar to
the Ising model, i.e., n(t) shows a power law decay with
exponentcloseto1/2. Beyondthisscale,anonalgebraic
slow decay is observed. However, we have checked that
the behaviour in the later regime is not like 1/logt as in
the Voter model but may be even slower than that. For
general values of x, we conjecture that at initial times a
powerlawbehaviouroccurswithsomecorrectiontoscal-
ing as indicated by the plots in Fig. 3; such corrections
(c) (d)
becomeweakerasxdeviatesfrom0.5. Thesecondregime
t=400 t=800
with the slower decay exists only for x > x , x ≈ 0.7.
c c
For exactly x = 1, the power law behaviour is exact be-
FIG.1. Typicalsnapshotsatdifferenttimesforx=0.6shows
that coarsening is curvaturedriven. fore n(t) saturates to a time independent non-zero value
due to the frozen stable states.
Snapshots taken during the evolution help in under- In order to gain more insight in the dynamical be-
standing the process quite well. We find that in general haviour, we have estimated the time τ required to reach
for x > 0.5 , the pictures looks very similar to the cur- the consensus state and its distribution D(τ). In a de-
vature driven case. However for x close to 1, certain tailedstudy made forL=32,we findthatD(τ) changes
3
its nature remarkably as x is increased (Fig. 4). For 1
x = 0.5, D(τ) shows a conventional behaviour; it in- 1
creases for small τ, has a broad peak and a long ex- 0.1 0.94
0.01 0.96
ponential tail. This behaviour continues till xc ≈ 0.70 0.1 0.001 0.98
bsmeyaollndanwdhliachrgweevafilnudesthoaftτD(s(eτe) Fdiiffge,r8s cinonSsMide)r.aAbplypaforr- n(t) 1 10 100 10000
ently it is an overlap of a symmetric function of finite 0.01 xx==00..76 L=80
width peaked about a small value of τ and a slow ex- x=0.8
x=0.9
ponentially decaying function extending to large values x=1.0
0.001
of τ (a magnified figure is shown in Fig. 7 in the SM). 1 10 100 1000 10000 100000 1e+06
Exactly at x = 1, the width of the symmetric function t
is minimum and the exponential partexists over a much
shorterrange. Weconjecturethatinthethermodynamic FIG.3. PlotofdensityofactivebondswithtimetforL=80
limit, the exponential part of D(τ) for x = 1 will van- for x = 0.6,0.7,0.8,0.9,1. Inset shows variation of active
ish altogether; this is supported by the data for L = 64 bondsfor x=0.94,0.96,0.98.
(shown in [28]). As the tail of the distribution for any x
may be fit by anexponential function exp(−ωτ) one can
0.01
define a time scale τ =1/ω for each x. x=0.5
eff
x=0.8
In Fig 5, we plot the three time scales: τ , hτi and
eff 0.001 x=1.0
τ where τ denotes the most probable value. In
mp mp
general, the average values of τ are different from the
) 0.0001
τ
most probable values. The difference becomes consider- (
D
able for x > x due to the long exponential tails. From
c 1e-05
Fig. 5 we can conclude that τ is very weakly de-
mp
pendent on x; in fact for x ≥ 0.55, it is almost a con-
1e-06
stant. On the other hand the two other timescales are
strongly dependent on x; τ and hτi initially decrease 0 2000 4000 6000 8000 10000
eff
andthenincreaserapidlywithx. Evidently,τ denotes τ
mp
the time to reachconsensus in absence of any intermedi-
ate metastable state and is presumably constant for any
FIG. 4. Plot of distribution of consensus time for x =
x > 0.5. On the other hand, τeff corresponds to the 0.5,0.8,1.
time scaleassociatedwiththe configurationswithnearly
frozenintermediatestates whichincreaseinnumber asx
increases. hτi shows the increasingtrend simply because 5000
it is an overallaverage.
4000
The existenceofthe twodifferenttime scalesis clearly
shown by the above result. The non-monotonicity in hτi 3000 τ
and τeff are to be noted, both time scales drop imme- <τe>ff
diately as x deviates from 0.5 and again at x = 1. This 2000 τ
mp
suggests there are discontinuities at the two well known
1000
limiting points.
The average consensus times as a function of the dif-
0
ferent system sizes apparently show a behaviour faster 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
than L2logL for x > 0.5. hτi for other values of L are
x
shown in the [28]. The possibility of discontinuities ex-
isting at x = 0.5 and x = 1 is more strongly supported
FIG. 5. Plot of different time scales τ , hτi and τ as a
by the data as system size increases. A more conclusive eff mp
function of x.
statement about the scaling of the timescales with the
system sizes can only be made after a detailed study of
the other time scales which is to be reported later. tained numerically in [15]. For 0.5 < x < 1, P(t) also
We next discuss some other results in context of the approaches a non-zero saturation value, however there
nonequilibrium phenomena. The persistence probability is no clear power law behaviour. The saturation value
P(t)asafunctionoftimetisplottedinFig. 6. Forx=1, increases with increase in x in a non-linear manner.
P(t)showspowerlawdecayast−θ withθ ≃0.2agreeing Lastly we discuss the results for the exit probability
with the known result [11–14]. For x = 0.5, persistence (see Fig. 7). The plot of E(ρ) as a function of ρ shows
decays to a very small fraction (≈ 0) following the be- that it is nonlinear except for x=0.5 having strong sys-
haviour exp[−0.31(lnt)2]. This behaviour is obtained by tem size dependence. Different curves intersect at a sin-
fitting the data and agrees very well with the form ob- gle point ρ = ρ ≃ 0.5 (ρ should be equal to 0.5 from
c c
4
1
1 1.2
0.1
1
0.8
λ 0.8
P(t) 0 0.0.0011 1100--42 x=0.5 x=0.6 ρE() 00..46 00..46 0.6 0.7 x0.8 0.9 1 ρE() 0000 ....12468
1 5 10 15 20 25 30 x=0.7 0
x=0.8 0.2 0.4 0.5 0.6
(log(t))2 x=1.0 ρ
0.0001
0
1 10 100 1000 10000 -6 -4 -2((ρ-ρ ) /0ρ )L1/ν 2 4 6
t c c
FIG.6. PlotofpersistenceprobabilitywithtimetforL=80 FIG. 7. Data collapse of E(ρ) is plotted against (ρ−ρc)L1/ν
ρc
for x = 0.6,0.7,0.8,1. Inset shows variation of P(t) as a for system sizes L = 32,48,64,80 for x = 0.6. Bottom inset
function of (lnt)2 for x=0.5. shows plot of unscaled data against ρ. Top inset shows the
plot of λ as a function of x.
symmetry argument). The curve becomes steeper as the
systemsizeis increased. Finite size scalinganalysisasin that while for smaller values of x the model is closer to
[29] can be made using the scaling form the voter model, the average consensus time increases
as x approaches unity, the Ising limit. This is appar-
(ρ−ρ )
E(ρ,L)=f c L1/ν , (2) ently counter intuitive as it is known that the evolution
(cid:20) ρc (cid:21) in the Voter model is slower compared to that in the
Ising model in two dimensions. Actually the metastable
wheref(y)→0fory <<0andequalto1fory >>0,so
states increase in number as x is increased (which is not
that the data for different system sizes L collapse when
surprising knowing the result for x = 1) enabling longer
E(ρ) is plotted against (ρ−ρc)L1/ν. The data collapse
ρc time scales for the system. However average consensus
takes place with ν = 1.3 ± 0.01 (the unscaled data is times are still less than that at x = 0.5 up to a certain
shown in the bottom inset) for all values of x>0.5. We value of x. Results for different system sizes indicate
conclude that like the Ising model, E(ρ) becomes a step that this value is very close to x (shown in [28]) in the
c
function in the thermodynamic limit. The scaling form thermodynamic limit which is consistent with the other
given by eq. (2) is found to fit very well with a general results.
form [29]
The exit probability shows a nonlinear behaviour for
any x > 0.5 with a universal exponent ν ≈ 1.3 and a
1+tanh(λy)
f(y)= , (3) non-universalparameter λ entering the scaling function.
2
Since we omit the frozen states for x = 1 and the time
whereλdependsonx. Thedependenceofλonxisshown scales are irrelevant for this measure, it is not surprising
in the top left inset of Fig. 7. λ which increases with x that the behaviour is Ising like. However, the fact that
quantifies the steepness of the E(ρ) curve. λ increases λ shows a discontinuity at x = 1 again shows that the
continuously as x is increased from 0.5, consistent with x=1 point has a distinctive feature with respect to the
the factthat the E(ρ) shoulddeviate fromthe linearbe- exit probability as well.
haviour. However,exactlyatx=1,λincreasesabruptly In conclusion, quite a few interesting results due to
toacomparativelylargervalueindicatingadiscontinuity. the interplay of the two types of dynamics are obtained
Let us now discuss the results obtained in this Rapid in the genaralised Voter model. The main result is the
Communication. First of all it appears that the cur- existenceoftwotimescalesinthesystemforx>x .One
c
vature driven coarsening governs the dynamics for any of them, τ is nearly independent of x while τ , the
mp eff
x>0.5atleastintheinitialstages. Thisisevidentfrom other timescale is a highly nonlinear function of x. Both
Fig. 3,whichshowsthatatinitialtimesthe orderingbe- x = 0.5 and x = 1 with completely different kind of
comes much faster compared to the Voter model as x is dynamical rules have unique features. At x = 0.5, P(t)
increasedfrom0.5(seeFig. 5in[28]). Atthe sametime, andE(ρ)behavedifferentlycomparedtoanyothervalue
the interfacialnoisedrivencoarseningpresentinthe sys- of x > 0.5, hτi is discontinuous. On the other hand,
tem, however small, is crucial for leading the system to x = 1 is also unique in the sense freezing occurs only at
consensus for larger values of x when the dominant cur- this point, hτi and λ show discontinuity here while well
vature driven process tends to generate nearly straight known power law behaviour in the relevant quantities
interfaces. It is only because of its presence the freez- exist. The intermediate region 0.5 < x < 1 does not
ing probability is zero in the system for any x ≥ 0.5 show any freezing phenomena. Although not algebraic,
(but not equal to unity). Thus it is interesting to note here P(t) reaches saturation for 0.5 < x < 1 in contrast
5
to that at x = 0.5. No sharp transition is observed for Acknowledgement: PR acknowledges financial sup-
any value of 0.5 < x < 1, but a crossover behaviour at portfromUGC.PSacknowledgesfinancialsupportfrom
x=x ≈0.7 is seen to exist. CSIR project.
c
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Interplay of interfacial noise and curvature driven coarsening: Supplementary material
Parna Roy1 and Parongama Sen1
1Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India.
7 I. SNAPSHOTS AND THE DENSITY OF InFigs3and4,twodifferentcasesofcoarseningforx=1
1 INTERFACES are shown with or without freezing.
0
2
The snapshots for different values of x are plotted in t=100 t=500
b the figures below for system size 64×64. For x = 0.5
e
(Fig. 1) we can see that there is no clear domain forma-
F
tion, only rough interfaces exist as is well known. Dy-

8 namical evolution in this case is interfacial noise driven;

in a finite system, a random fluctuation of large size ul-
]
h timately leads to a consensusstate. For values of x close
c to unity, we note that there are two possibilities. Either
e
the system reaches a consensus state in a short time or
m
it may take a much larger time as the system evolves
t- through metastable states with minimum curvature. t=2000 t=10000
a
t t=100 t=1000
s FIG. 3. Typical snapshots at different times for x = 1
.
t shows that coarsening is curvature driven. This configura-
a tion reached striped frozen state.
m
-
d
n t=100 t=200
o
c
[

2
v t=5000 t=10000
7
1 FIG.1. Typicalsnapshotsatdifferenttimesforx=0.5shows
1 that coarsening is interfacial noise driven.
9
0
. t=400 t=800
1 t=100 t=500
0
7 FIG. 4. Typical snapshots at different times for x = 1
1 shows that coarsening is curvature driven. This configura-
: tion reached the consensus state.
v
i
X
r
a 1
0.1
t=800 t=1000
n(t)
FIG. 2. Certain configurations show fast relaxation. Snap- 0.01 x=0.5
shotsshowsuchaconfigurationfordifferenttimesforx=0.9. x=0.55
x=0.6
x=0.7
In the main text, we have shown the latter case for 0.001
1 10 100 1000 10000 100000
x=0.9. Hereweshowanexampleforthefirstcase(Fig.
2). In this case no domains occur with nearly straight t
edges in the intermediate times. For x = 1.0 (Fig. 3)
certainconfigurationsreachfrozenstripedstatewhichdo FIG. 5. Plot of density of interfaces n(t) for x =
0.5,0.55,0.6,0.7.
not evolve further as the coarsening is curvature driven.
2
0.01 known.
x=0.5
z=0.8 0.01
x=0.65
0.001 x=1.0
x=0.7
0.001 x=0.75
) 0.0001
τ
(
D ) 0.0001
0.002 τ
1e-05 x=1.0 D(
0.001 64 1e-05
1e-06
0.0001
1e-06
0 200 400 600 800 1000
τ 10 100 1000 10000
τ
FIG. 6. Plot of D(τ) for initial time scale. Inset shows plot
of thedistribution of theconsensus time D(τ) for L=64 for FIG. 7. Plot of D(τ) for x=0.65,0.7,0.75.
x=1.0.
InFig. 5wehaveplottedthe densityofinterfacesn(t)
35000
for values of x = 0.5 and a few other values. In this 10000 x=0.5 48
figure,itcanbeseenclearlythatthecoarseningbecomes 64
τ
muchfastercomparedtox=0.5evenasthedeviationof 25000 1000 80
xfrom0.5issmall. Italsoshowsthataclearexistenceof
τ 100
a kink for x=0.7 and not smaller values supporting the
15000 20 50 100 200
conjecture thata crossoverbehaviouroccursfor x≈0.7.
L
5000
II. CONSENSUS TIME
0.5 0.75 1
Inthemaintext,theconsensustimesdistributionD(τ)
x
forL=32hasbeenreported. InFig. 6 weshowthe dis-
tributions for L = 32 magnifying the region τ ≤ 1000.
FIG. 8. Plot of consensus time τ as a function of x for L =
small values of τ only. In the inset we have shown D(τ)
48,64,80. Inset shows variation of τ with system size for
for x=1 for L=64. From this figure we can argue that
x=0.5,0.6,0.7,0.8.
for largersystem sizes,in the Ising limit x=1.0,the ex-
ponentialdecayatlargerτ isnotpresent,thedistribution
only contains a sharply peaked symmetric function of fi-
III. RESULTS FOR x<0.5
nite width. In the main text we have reported that the
conventional behaviour of D(τ) continues till xc ≈ 0.7
beyond which D(τ) changes its behaviour considerably. The focus of the paper has been on x ≥ 0.5 for which
Fig. 7 supports the statement. absorbing states can be reached. However, the region
For other values of x, only the average value of τ has x<0.5 also yields some interesting results. As the state
been estimated so far for L>32. In fig. 8 we have plot- undergoes continuous evolution, the persistence proba-
ted hτi as a function of x for L=48,64,80. The results bility goes to zero and density of active bonds remain
are qualitatively similar for L=32, and the results sup- finite. In Fig. 9 we have plotted density of active bonds
port the conjecture that discontinuities occur at x=0.5 n(t) and in the inset we have plotted the persistence
and 1. In the inset we have plotted hτi as a function probability P(t) as a function of time for L = 32 for
of system size L for different values of x. Although for x = 0.4. The persistence probability has an exponential
x = 0.5 one gets the behaviour hτi ∝ L2logL, it is dif- decay(P(t)∼exp(−bt)),i.e.,itisfasterthanthatinthe
ficult to conclude about the exact dependence for other voter model. The parameter b has nonlinear dependence
values of x. For x=1 the dependence is simply L2 as is on x (see inset of Fig. 9).
3
1
x=0.4
1
0.1
n(t) P(t) 0 0. 00.00.000111 b 0.3
1e-05 0.2
1e-06
1e-07
1 10 1000 0.1 0.2 0.3 0.4
t x
0.1
1 10 100 1000 10000
t
FIG. 9. Plot of n(t) as a function of time for L = 32 for
x = 0.4. Left inset shows variation of P(t) with t and right
inset shows variation of the persistence exponent b with x.