Persistence in Ferromagnetic Ordering: Dependence upon initial configuration

1.3 MB·English

by  Saikat Chakraborty

Persistence in Ferromagnetic Ordering: Dependence upon initial configuration
∗
Saikat Chakraborty and Subir K. Das
Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced
5 Scientific Research, Jakkur P.O, Bangalore 560064, India
1
(Dated: January 22, 2015)
0
2 We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of the Ising
model,employingtheGlauberspin-flipmechanism,inspacedimensionsd=2and3. Resultsforthe
n
persistence probability and the domain growth are discussed for quenches to various temperatures
a
J (Tf) below the critical one (Tc), from different initial temperatures Ti ≥Tc. In long time limit, for
T >T , the persistence probability exhibits power-law decay with exponents θ ≃0.22 and ≃ 0.18
1 i c
in d = 2 and 3, respectively. For finite T , the early time behavior is a different power-law whose
2 i
life-time diverges and exponent decreases as T → T . The crossover length between the two steps
i c
] diverges as the equilibrium correlation length. Ti = Tc is expected to provide a new universality
h class for which we obtain θ ≃ 0.035 in d = 2 and ≃ 0.10 in d = 3. The time dependence of the
c average domain size ℓ, however, is observed to be rather insensitive to thechoice of T .
i
e
m
- I INTRODUCTION quencheswouldformanew universality classandwas
t
a shownthatthedecayoftheabovecorrelationwassig-
st When a homogeneous system is quenched below nificantly slower for Ti = Tc than Ti = ∞. In view
t. the critical point, the system becomes unstable to of that a slower decay of P is also expected. Apriori,
a fluctuations and approaches towards the new equilib- however, it is unclear whether there will be quantita-
m
rium via the formation and growth of particle rich tive similarity between the degree of changes in the
- and particle poor domains [1–4]. In such nonequi- two quantities. On the other hand, the behavior of
d
n librium evolutions, over several decades, aspects that P and ℓ are expected to be disconnected [34]. Nev-
o received significant attention are the domain pattern ertheless, the rate of growth of ℓ may be different for
c [3, 5–9], rate of domaingrowth[5, 10–15], persistence Ti =Tc and Ti =∞, at least during the transient pe-
[
[16–20, 22, 23] and aging [24–29]. Average size, ℓ, of riod. In this paper, we address the Ti dependence for

1 domains typically grows with time (t) as [5] persistence and domain growth in a ferromagnet, via
v Monte Carlo (MC) simulations [35] of nearest neigh-
8 ℓ∼tα. (1) bor Ising model [35]
7
1
The value of the exponent α for nonconserved order-
5
0 parameterdynamics[5,12],e.g.,duringorderinginan
1. uniaxialferromagnet,is1/2,inspacedimensiond=2. H =−J SiSj; Si =±1, J >0, (3)
0 In addition to the interesting structures exhibited by <Xij>
5 the domains of like spins (or atomic magnets) in a
1 ferromagnet, the unaffected or persistent spins also
: form beautiful fractal patterns [16–20, 22]. Typically, in d = 2 and d = 3, on square and simple cubic lat-
v
i fraction of such spins, henceforth will be referred to tices, respectively.
X
as the persistent probability, P, decays as Starting from a high value, as T approaches T [≃
r i c
a P ∼t−θ, (2) 2.27J/kB ind=2or4.51J/kB ind=3, kB beingthe
Boltzmann constant], a two-step decay in P becomes
with [20, 21] θ having a value ≃ 0.22 for the Ising prominent, except for Ti =Tc. For the latter, power-
model in space dimension d=2 and ≃0.18 in d=3. law behavior with exponents much smaller than the
ones observed for quenches from T = ∞ lives for-
The values of the exponents mentioned above are i
ever. In addition to identifying these facts, a primary
understoodto be true for the perfectly randominitial
objective of the paper is to accurately quantify these
configurations, mimicking the paramagnetic phase at
temperature T =∞. Another relevant situation is to decays. For the domain growth, on the other hand,
we do not observe a modification to time dependence
quench a system from finite initial temperature (T )
i
with the variation of T , almost from the very begin-
with a large enough equilibrium correlation length ξ. i
ning.
However,thisproblemhasreceivedonlyoccasionalat-
tention [30–33], though experimentally very relevant. The rest of the paper is organized as follows. In
In this context, the behavior of the two-time equal section II we briefly describe the methods. Results
point correlation function, relevant in the aging phe- fromboththe dimensionsarepresentedinsectionIII.
nomena, was studied [31, 32] in d = 2 for T = T , Section IV concludes the paper with a summary and
i c
thecriticaltemperature. Itwaspointedoutthatsuch outlook.
2
II METHODS
The nonconserved order-parameter dynamics in
the MC simulations have been incorporated via the
Glauber spin-flip mechanism [36]. In this method,
a randomly chosen spin is tried for a change in sign
whichisacceptedaccordingtothestandardMetropo-
lis algorithm [35]. We apply periodic boundary con-
ditions in all directions. Time is expressed in units
of MC steps (MCS), each MCS consisting of Ld tri-
als, L being the linear dimensionofa squarebox. We
have computed ℓ from the domain size distribution,
P (ℓ ,t), as [15]
d d
ℓ(t)= ℓ P (l ,t)dℓ , (4)
Z d d d d
whereℓ iscalculatedasthedistancebetweentwosuc-
d
cessive interfaces in any direction. All lengths are ex-
pressed in units of the lattice constant a. We present
theresultsafteraveragingovermultiple initialconfig-
urations. This number ranges from 20 (for L=1024)
FIG. 1. Upper panels show snapshots during the evolu-
to 200 (for L=400)in d=2 and 10 (for L=256)to
tion of the Glauber Ising model with T = ∞, T = 0
i f
50 (for L = 64) in d = 3. The initial configurations and L = 512. The black regions represent domains of up
forTi closetoTc werecarefullypreparedviaverylong spins. Thelowerpanelsshowtheunaffectedspins,marked
runs. At T , depending upon the system size, length in black, corresponding to the evolution snapshots above
c
of such runs varied between 5×106 to 108 MCS. them. These results correspond to d=2.
III RESULTS
A. d=2
Growth of the domains have been demonstrated in
theupperframesofFig. 1forthesystemsizeL=512
in d = 2. In this figure we show snapshots from two
different times during the evolution of the Glauber
Isingmodel. Inthelowerframesofthefigure,weshow
pictures marking only the persistent spins. Beautiful
patterns are visible. These results correspond to a
quench from T =∞ to T =0.
i f
Plots of P, for T = ∞ and few different values of
i
T are shown vs t in Fig. 2. The data for T = 0
f f
and 0.25T are consistent with each other and follow
c
power-law, the exponent being θ ≃ 0.22. The flat
behavior at the end is due to the finite-size effects.
FIG. 2. Plots of persistence probability, P, vs, time, on a
This value of θ is consistent with the observation by log-log scale, for quenchesfrom T =∞, with L=512, in
i
Manoj and Ray [20]. However, for higher values of d=2. Four different values of T are included. The solid
f
T , as also previously observed [18, 19], the decay is line therehas a power-law decay with exponent 0.22.
f
not of power-law type. This is thought to be due to
thermal fluctuation. When this fluctuation is taken
care of, we observe θ ≃ 0.22 for all the values of T In Fig. 3(a) we show P vs t plots, on a log-log
f
included in Fig. 2, in agreement with Ref. [21]. It scale, for quenches to T = 0, from a few different
f
is thought that persistence and domain growth are values of T , all for the same system size L = 512. It
i
not strongly connected to each other. Interestingly, appears that, in the long time limit, for T > T , the
i c
different behavior in Fig. 2 for T >0.25T is strongly decay is power-law,with the same exponentθ ≃0.22.
c
reflected in the domain growth also. For brevity we However, crossover to this functional form gets de-
do not present it here. layedasT approachesT . Infact,inthepre-crossover
i c
3
FIG.3. Log-logplotsofP vst,forquenchesfromdifferent
values of T (> T ), to T = 0. Continuous lines there
i c f
correspond to power-law decays with exponents 0.22 and FIG. 4. (a) Instantaneousexponentsθ areplotted vs1/ℓ
i
0.04. (b) Instantaneous exponents θi are plotted vs 1/ℓ, fortwoofthequenchesinFig. 3(a). Herewehavefocussed
for the quenches in (a), excluding Ti = Tc case. Here we on the first step of the decays, exponents for which are
haveincludedonlythelatetimebehavior. Thecontinuous obtained from the flat regions, marked by the horizontal
linesinthisfigurearelinearfits,providingthevaluesofθ solid lines. These values of θ are plotted, vs ǫ= T −T ,
i c
astheordinateintercepts. Allresultsarefromsimulations in the inset of the figure. The continuous line there is
in d=2. a power-law fit (see text). (b) Plots of P vs t, for two
different system sizes, with T = T and T = 0. The
i c f
upperinsetshowsθ vs1/ℓforthreedifferentvaluesofL.
i
regime, another power-law decay, with smaller value Thedashedlineinthisinsetisalinearextrapolationusing
ofθ,becomesprominentwiththedecreaseofT . Such data in the small ℓ region. The L-dependent exponents
i
a slower decay becomes ever-lived for T =T . obtained from flat regions of the plots (see the horizontal
i c
In Fig. 3(b), we present the instantaneous expo- solidlines)areplottedvs1/Linthelowerinset. Thesolid
line thereis a linear fit. All results correspond to d=2.
nent, θ , calculated as [14, 15]
i
dlnP
θ =− , (5)
i
dlnt
limit better. Data sets for each T are fitted to linear
i
vs 1/ℓ, with the objective of accurate quantification functions to obtain the exponents. Within statistical
of the second step of the decays for T close to, but error, for all the presented temperatures, it appears
i
greater than, T . For the abscissa variable we have that θ is consistent with the quench from T = ∞ to
c i
adopted1/ℓ,insteadof1/t,to visualize the long time T = 0. Nonzero slopes in these plots are due to the
f
4
presence of off-sets at the crossovertimes. providing θ =0.037. From all these exercises we con-
Next we move to identify the exponent for Ti =Tc clude that θ(Ti = Tc) = 0.035±0.005. This picture
and T = 0. In Fig. 3(a), it appears that the remains true for quenches from T to nonzero values
f c
Ti =Tc data are reasonably consistent with θ =0.04. of Tf, if thermal fluctuation effects are appropriately
Nevertheless,before the finalfinite-size effects appear taken care of. On this issue of thermal fluctuation,
(showingflatnatureatverylatetime),thereisafaster here,aswellasforTi =∞,ourstudieswererestricted
decay, albeit for a brief period. This can well be due to only small system sizes.
to the fact that for a finite system, ξ is not infinite The decay of the two-time correlation is also of
at T = Tc, effectively implying that the initial con- power-lawtype. ForquenchesfromTi =Tc,thevalue
figurations are prepared away from T . Thus, in this of the exponent gets reduced by a factor ≃ 10, com-
c
problemfinite-sizeeffectshavetwosources. Onecom- pared to T = ∞. In the present case the reduction
i
ing from the finiteness of the equilibrium correlation factor is ≃ 6.3. While there may be connection be-
length,otherbeingfacedwhenthenonequilibriumdo- tween the two phenomena, but a search for matching
main size is close to the system size. Thus, a quan- between the two factors may not be justified. This
tificationoftheexponentθ,forT =T ,viafinite-size is because, only spins with odd number of flips are
i c
scaling[37], becomes a challengingtask. However,we important for the decay of the two-time correlation
appropriately take care of the shortcoming below, in function.
variousdifferentwayswhichprovideresultsconsistent It is certainly important to ask, if, like the decay
with each other. of the persistence probability and the two-time cor-
In Fig. 4(a) we show the instantaneous exponents relation [31], the growth of the average domain size
θ , vs 1/ℓ, with the objective of quantifying the first also exhibits initial temperature dependence, at least
i
stepofthedecay,fortwovaluesofT ,closeenoughto atthetransientlevel. Hereweonlyexaminethecases
i
Tc. Asdemonstrated,fromtheflatregionsweidentify Ti = ∞ and Ti = Tc, for quenches to Tf = 0. In Fig.
the exponents for the early time or the first step of 5 we present the ℓ vs t plots for these two cases using
the decay for various values of T . These numbers a log-log scale. Both the data sets appear to grow
i
are plotted in the inset of this figure as a function of slowerthant1/2. Thiscanwellbeduetothe presence
ǫ = Ti−Tc. The continuous line there is a fit to the ofsignificantlybiginitiallengthℓ0,whichweexamine
form below. While from this figure it is difficult to identify
anydifferenceinthegrowthexponentbetweenthetwo
θ(T )=θ(T )+Aǫx, (6) cases,therecertainlyexistsdifferenceinthefinite-size
i c
effects, noting that L=512 in both the cases.
providing θ(T =T )=0.034, A=0.15 and x=0.54. To learn better about the exponents, in the inset
i c
Recall that this value of θ is the only decay exponent we present the instantaneous exponents [14, 15]
for T =T .
i c
dlnℓ
To verify the above value of θ further, in Fig. 4(b) α = , (7)
i
dlnt
we present an exercise with different system sizes. In
the main frame of this figure, we present P vs t data, with the variation of t. Here, while calculating α ,
i
for Ti = Tc, from two different values of L. It is seen we have subtracted ℓ0 which are ≃ 2 and ≃ 6.65, re-
that with the increase of the system size, there is a spectively, for T = ∞ and T . This subtraction is
i c
tendencyofthedatatosettledowntoapower-lawfor meaningful, considering the fact that the pure scaling
a longer period of time, following a marginally faster withrespecttotimeiscontainedinℓ−ℓ0. Calculation
decay at early time. In the upper inset of this figure of α , without such subtraction, provided early time
i
weshowθ vs1/ℓforthreedifferentsystemsizeswith exponents muchsmallerthanthe theoreticalexpecta-
i
T = T . The early time behavior appears linear, ex- tion for the conserved dynamics [14]. This has previ-
i c
trapolation of which leads to θ ≃ 0.029. However, ously been understood to be due to the curvature de-
if the data in the main frame is closely examined, pendent correction. Such confusion has recently been
as already mentioned above, this part corresponds corrected [15]. Note that, there may be a delay time
to the preasymptotic behavior, thus, should be dis- for a system to become unstable following a quench.
carded from the analysis. Actual asymptotic expo- Thus, for an appropriate understanding of a time de-
nents should be extracted from the flat regions of the pendent exponent, the value of ℓ0 need not be the
plots. The numbers obtained from these flat parts, length at t = 0. Via finite-size scaling analysis, this
as discussed, differs due to the finite-size effects and was demonstrated in a recent work [15]. Here, how-
thus,shouldbe extrapolatedto L=∞appropriately. ever, we do not undertake such a task.
These values are plotted in the lower inset as a func- It may be argued that the value of ℓ0 should be of
tionof1/L. Averyreasonablelinearfit (see the solid theorderofsystemsizeforT =T ,sinceξ ∼LatT .
i c c
line) is obtained, providing θ = 0.035. On the other Note here that at criticality fluctuations exist at all
hand, a nonlinear fit appears to be nearly quadratic lengthscales. AtTi =Tc,smallvalueofℓ0,compared
5
to L, is due to the fact that our calculations did not C being a constant. Such full forms are useful for a
ignore such fluctuations. finite-size scaling analysis to accurately quantify the
FirstimportantobservationfromtheinsetofFig. 5 exponent α. It appears that even for a power-law
is that the value of α approaches1/2 fromthe upper behavior of f(1/ℓ), the asymptotic behavior in the
i
side. This is in contradiction with the corresponding growth law is reached exponentially fast. Of course,
behavior for the conservedorder parameter dynamics from least square fitting exercise of the ℓ vs t data
with T very close to zero [38]. In the latter case, also one can aim to obtain the early time corrections.
f
the early time dynamics provides a growth exponent However,thismethodismorearbitrary. Oftenderiva-
muchsmallerthantheexpectedasymptoticvalue1/3. tives help guessing the functional forms better. We
Second, after t = 5, both the data sets practically leave the exercise of a finite-size scaling analysis to
followeachother,implyingnodifferenceinthegrowth accurately estimate α and the finite-size effects for a
of ℓ almost from the beginning! future work.
Before moving on to presenting results in d = 3,
we discuss the issue of persistence again. The essen-
tial feature in the initial configurations prepared at
different temperatures is the difference in the equilib-
rium correlation length ξ. The basic question, prior
to the study, one asks, how the size of ξ affects the
decay of persistence probability? For each value of ξ,
do we have a unique exponent θ describing the full
decay? The answer, as we have observed, is certainly
notinaffirmative. Essentially,the decayexponentfor
T = ∞ is recovered for all ξ (< ∞) in the long time
i
limit. The crossoverto this asymptotic behavior gets
delayed with the increase of ξ. It is then natural to
askifthis crossoveroccurswhenℓcrossesξ. Thiscan
beansweredbyappropriatelyestimatingthecrossover
length ℓ .
c
FIG. 5. Average domain sizes, ℓ(t), are plotted vs t, for
quenches to T = 0 from T = ∞ and T , in d = 2. The
f i c
solid line represents t1/2 behavior. Corresponding instan-
taneous exponents, vs 1/t, are shown in the inset. All
these results are for L=512.
From the length (or time) dependence of α , one
i
can write
α =α+f(1/ℓ), (8)
i
to obtain
dℓ
=lnt. (9)
Z αl 1+ 1f(1/ℓ)
α
(cid:2) (cid:3)
If f(1/ℓ) can be quantified accurately from the simu-
lation data, a full time dependence of ℓ is obtainable.
E.g.,if f(1/ℓ)is a powerlaw, A /ℓβ, A being a con-
β β
stant, by taking αlβ >A , one finds
β
FIG. 6. Scaling plot of persistence probability versus t/t
ℓ1/α 1 c
ln ∼ . (10) where the crossover time (to the asymptotic behavior) tc
t α2βℓβ has been used as an adjustable parameter to obtain opti-
mum data collapse. Inset: Plots of ℓ −1 versus ǫ. The
c
Assuming thata correctiondisappearsfast, suchthat circles correspond to estimates of ℓ from t , the squares
c c
ℓ≃tα, we obtain are directly obtained from the scaling plots of P vs ℓ/ℓc.
The solid line has d=2 Ising critical divergence of corre-
C lation length. All results were obtained using L= 512 in
ℓ∼tαexp(− ), (11)
αβtαβ d=2.
6
In the main frame of Fig. 6 we show plots of per-
sistence from different values of T , for quenches to
i
T = 0. Here the time axis is scaled by appropri-
f
ate factors (expected to be proportional to cross over
time t ) to obtain collapse of data in the asymptotic
c
regime. Qualityofcollapse,ontopoftheT =∞data
i
set, again confirms that θ ≃ 0.22 in the t → ∞ limit
for all T (> T ). From the square roots of these T
i c i
dependent scaling factors, one can obtain ℓ which is
c
expected to scale as ℓ ∼ ξ ∼ ǫ−ν. Note that for the
c
Ising model ν =1 in d=2 and ≃0.63 in d=3. con-
sidering that the T = ∞ data have been used as the
i
reference case, it will be appropriate to fit the data
set for ℓ to the form
c
ℓ =1+A ǫ−ν, (12)
c c
since ℓ → 1 for T = ∞. Unless we are very close to
c i
T such additional term cannot be neglected. In the
c
inset of Fig. 6 we ahve plotted ℓ −1 as a function FIG. 8. Instantaneous exponents θi are plotted vs 1/ℓ,
c for quenches from T = T to T = 0 in d = 3. Results
of ǫ, on a log-log scale. The data set (circles) appear i c f
from different values of L are included. The horizontal
consistent with ν = 1. When ℓ is extracted from
c solid lines are related to theestimation of L-dependentθ.
tc, a better exercise requires incorporation of ℓ0 and Inset: System size dependent θ are plotted as a function
growthamplitude for each T . To avoidthis problem, of 1/L. The continuous line there is a linear fitting (see
i
wehavealsoobtainedℓ directlyfromthescalingplots text for details).
c
of persistence data vs ℓ/ℓ (see the squares). Both
c
data setsappear nicely consistentwith eachother. In
fact, fitting these data to the form in Eq. (12) we Next we present results in d = 3. All the impor-
obtain ν ≃0.95. tant facts being discussed in the previous subsection,
here we straightway present the results. Noting that
nothing remarkable happened for domain growth in
B. d=3 the lower dimension, we stick only to the persistence
probability.
In Fig. 7 we show the P vs t plots for quenches
from T = ∞ and T = T , keeping T = 0 in both
i i c f
the cases. For each value of T , results from multiple
i
system sizes are presented. The data for T = ∞ are
i
consistent with θ = 0.18, previously reported in [21].
Thus, here we aim to accuratelyquantify the value of
θ for T =T .
i c
Even though, for T = T , data from both the sys-
i c
tem sizes in Fig. 7 look consistent with each other,
finite-size effects are detectable from a closer look. In
the main frame of Fig. 8 we plot θ versus 1/ℓ for a
i
few different values of L. Like in d=2, from the flat
regions we identify system size dependent θ, a plot of
which is shown in the inset of this figure. Again, the
θ vs 1/L data exhibits a reasonable linear trend and
an extrapolation to L=∞ provides θ ≃0.106.
Similar to d=2, for T <T <∞, two step decays
c i
exist in d = 3 as well. In the main frame of Fig.
FIG. 7. Plot of P vs t, for quenches from T = ∞ and 9(a)wehavedemonstratedtheestimationofθ forthe
i
Ti = Tc, to Tf = 0. In each of the cases results from first step, for two representative values of Ti. In the
two different system sizes are included. The solid lines inset of Fig. 9(a) we have plotted these exponents
have power-law decays with exponents 0.1 and 0.18, as as a function of ǫ. A fit of this data set to the form
indicated on the figure. All results corresponds tod=3. in Eq. (6) provides θ(T = T ) = 0.096, A = 0.07
i c
and x = 0.34. This value of θ is in good agreement
7
this, as well as to get rid of the influence of T depen-
i
dent ℓ0 and growth amplitude, we have obtained ℓc
from these plots only.
Finally in the inset of Fig. 9(b) we plot ℓ −1 as a
c
functionofǫ,onalog-logscale. Thedivergenceofthe
length scale is consistent with a power law exponent
0.63 which is the critical exponent for ξ in d = 3. In
fact, a fitting to the form in Eq. (12), excluding the
point closest to T , provides ν ≃0.67. The reasonfor
c
theexclusionofthelastpointisfinite-sizeeffectwhich
is clearly visible. Data below ǫ = 0.1 are, however,
included in the nonlinear fitting in the inset of Fig.
9(a). Thisisthe reasonforquotinglargererrorbarin
the final value of θ there.
IV CONCLUSIONS
In conclusion, we have studied phase ordering dy-
namicsinIsingferromagnetsforvariouscombinations
of initial (T ) and final (T ) temperatures in d = 2
i f
and 3. In this work, the primary focus has been on
thepersistenceprobability,P,anditsconnectionwith
the growth of average domain size, ℓ, as well as the
equilibrium initial correlation length ξ.
Our generalobservationhas been that, irrespective
of the value of T , the decay ofP becomes faster with
i
the increase of T , after a certain critical number for
f
the latter. This is understood to be due to spins af-
fected by thermal fluctuations. When this effect is
taken care of [19], the long time decay appears to be
power law with exponent [20, 21] consistent with the
one for quench to T =0.
f
As T approaches T , two-steppower-law decay be-
i c
comes prominent, the second part having exponent
θ ≃ 0.22 in d = 2 and ≃ 0.18 in d = 3, same as
FIG. 9. (a) Estimation of θ for the first step of decay is T = ∞ and T = 0 case. For T = T , thought to
demonstratedind=3. Inset: Exponentθforthefirststep i f i c
provide a new universality class, the first part of the
of the decay is plotted as a function of ǫ, in d = 3. The
continuous line is a non-linear fitting. Further details are two-stepprocesslivesforever. Thecorrespondingval-
providedinthetext. (b)ScalingplotofP,versusℓ/ℓ ,in uesoftheexponenthavebeenidentifiedtobe≃0.035
c
d=3. Thesolidlinehasapower-lawdecaywithexponent in d=2 and ≃0.10 in d=3. Thus, even though, the
0.54. Inset: Plot of ℓc−1, in d = 3, versus ǫ. The solid decay of persistence probability is disconnected with
line there has d = 3 Ising critical divergence. All results
the growth of domain length, its behavior is strongly
were obtained using L=128.
connected with the initial correlation length. It has
been shown that the crossover length to the second
step of decays diverges as the equilibrium correlation
with the one obtained from Fig. 8. Thus, in d = 3, lengthinboththedimensions. Thisleadstotheques-
for Ti = Tc, we quote θ = 0.10±0.02. The effect of tion of difference in the fractal dimensions in the pre
growingcorrelationlengthintheinitialconfigurations and post crossover regimes which needs to be looked
certainly appears weaker in this space dimension. at carefully.
InFig. 9(b)weshowscalingplots ofP,vsℓ/ℓ ,us- We have not observed any initial configuration de-
c
ing data from different values of T . Collapse of data pendence of the growth of the average domain size.
i
is good. The late time behavior is power-law and is Thisisconsistentwithapreviousstudy[34]butmore
consistent with a decay exponent 0.54. Considering explicitly demonstrated here. Essentially, even the
that θ ≃ 0.18 in d = 3, this implies α = 1/3 in d = 3 transients are not affected due to change in initial
[39]. As stated in Ref. [39], deviation of α, in this temperature. However, stronger finite-size effects are
dimension, from 1/2, is not yet understood. To avoid detected for lower values of T . For domain growth,
i
8
a striking observationis that the early time exponent [16] MAJUMDARS.N.,SIREC.,BRAYA.J.andCOR-
is much higher than the asymptotic value, despite T NELL S.J., Phys. Rev. Lett. 77, 2867 (1996).
f
being zero. This is at variance with the conservedor- [17] MAJUMDAR S. N., BRAY A. J., CORNELL S. J.
and SIREC., Phys. Rev. Lett. 77, 3704 (1996).
derparameterdynamics. Theseareallinterestingnew
[18] DERRIDA B., HAKIM V. and ZEITAK R., Phys.
results, requiring appropriate theoretical attention.
Rev. Lett. 77, 2871 (1996).
In future we will focus on persistence for the con- [19] DERRIDAB., Phys. Rev. E 55, 3705 (1997).
served order parameter dynamics. For the conserved [20] MANOJ G. and RAY P., Phys. Rev. E 62, 7755
dynamics, initial temperature dependence of aging (2000).
[21] STAUFFER D.,Int. J. Mod. Phys. C 8, 361 (1997).
anddomaingrowtharealsoimportantopenproblems.
[22] CHAKRABORTYD.andBHATTACHARJEEJ.K.,
Phys. Rev. E 76, 031117 (2007).
[23] SAHARAY M. and SEN P., Physica A 318, 243
(2003).
ACKNOWLEDGEMENT [24] FISHER D. S. and HUSE D. A., Phys. Rev. B 38,
373 (1988).
[25] LIUF.andMAZENKOG.F.,Phys.Rev.B 44,9185
Theauthorsarethankfultothe DepartmentofSci- (1991).
ence and Technology, India, for financial support via [26] CORBERI F., LIPPIELLO E., ZANNETTI M.,
Phys. Rev. E 74, 041106 (2006).
grant number SR/S2/RJN-13/2009.
[27] AHMAD S., CORBERI F., DAS S. K., LIPPIELLO
∗ [email protected] E., PURI S. and ZANNETTI M., Phys. Rev. E 86,
061129 (2012).
[28] MAJUMDERS.andDASS.K.,Phys.Rev.Lett.111,
055503 (2013).
[29] MIDYAJ.,MAJUMDERS.andDASS.K.,J. Phys.
[1] ONUKIA.,Phase Transition Dynamics,(Cambridge : Condens. Matter. 26, 452202 (2014).
UniversityPress, Cambridge, 2002). [30] DASGUPTA C. and PANDIT R., Phys. Rev. B 33,
[2] PURI S. and WADHAWAN V. , Kinetics of Phase 4752 (1986).
Transitions, (Boca Raton, CRCPress, 2009). [31] HUMAYUNK. and BRAYA.J., J. Phys. A : Math.
[3] BRAYA.J., Adv. Phys. 51, 481 (2002). Gen. 24, 1915 (1991).
[4] BINDER K., in Phase transformation of materials, [32] BRAY A. J., HUMAYUN K. and NEWMAN T. J.,
edited by CAHN R. W., HAASEN P., KRAMER Phys. Rev. B 43, 3699 (1991).
E. J., Mater. Sci. Technol. Vol. 5 (VCH, Wein- [33] BLANCHARD T., CUGLIANDOLO L. F. and
heim,1991), p. 405. PICCO M., J. Stat. Mech: Theory and experiment
[5] CROSS M. C. and HOHENBERG P. C., Rev. Mod. P12021 (2014). This very recent work was brought
Phys. 65, 851 (1993). to our notice by an anonymous referee. The value
[6] OHTA T., JASNOW D. and KAWASAKI K., Phys. θ = 0.035 obtained by us is extremely close to the
Rev. Lett. 49, 1223 (1982). conclusion from this study [33] for triangular lattice.
[7] BRAY A. J. and PURI S., Phys. Rev. Lett. 67, 2670 This implies, the lattice structure plays insignificant
(1991). role.
[8] TOYOKIH., Phys. Rev. B 45, 1965 (1992). [34] SICILIA A., ARENZON J. A., BRAY A. J. and
[9] DAS S. K., PURI S. and CROSS M. C., Phys. Rev. CUGLIANDOLO L. F., Phys. Rev. E 76, 061116
E 64, 46206 (2001). (2007).
[10] LIFSHITZ I. M. and SLYOZOV V. V., J. Phys. [35] LANDAUD. P. and BINDER K.,A Guide to Monte
Chem. Solids 19, 35 (1961). Carlo Simulations in Statistical Physics (Cambridge
[11] BINDERK.andSTAUFFERD.,Phys.Rev.Lett.33, University Press, Cambridge, 2009).
1006 (1974). [36] GLAUBER R. J., J. Math. Phys. 4, 294 (1963).
[12] ALLEN S. M. and CAHN J. W., Acta Metall. 27, [37] FISHER M. E., in Critical Phenomena, edited by
1085 (1979). GREEN M. S. (AcademicPress, London, 1971).
[13] SIGGIAE. D.,Phys. Rev. A 20, 595 (1979). [38] MAJUMDERS.andDASS.K.,Phys. Chem. Chem.
[14] HUSED. A., Phys. Rev. B 34, 7845 (1986). Phys. 15, 13209 (2013).
[15] MAJUMDER S. and DAS S. K., Phys. Rev. E 81, [39] CUEILLE S. and SIRE C., J. Phys. A: Math. Gen.
050102 (2010). 30, L791 (1997).