Table Of ContentThe double domain structure of pair contact process with diffusion
Sungchul Kwon and Yup Kim
Department of Physics and Research Institute of Basic Sciences, Kyung Hee University, Seoul 130-701, Korea
(Dated: February 6, 2008)
7 We investigate the domain structure of pair contact process with diffusion (PCPD). PCPD is a
0 stochasticreaction-diffusion modelwhichevolvesbythecompetitionoftwobinaryreactions,2A→
0 3A and 2A → 0. In addition, each particle diffuses isotropically, which leads to the bidirectional
2 couplingbetweensolitary particlesandpairs. Thecouplingfrompairstosolitaryparticlesislinear,
n while the opposite coupling is quadratic. The spreading domain formed from localized activities
a in vacuum consists of two regions, the coupled region of size Rp where pairs and solitary particles
J coexist and theuncoupled region of size R where only solitary particles exist respectively. As the
U
2 size of the whole domain R is given as R = Rp +RU, Rp and RU are the basic length scales of
2 PCPD. At criticality, Rp and RU scale as Rp ∼t1/Zp and RU ∼t1/ZU with ZU >Zp. Weestimate
Zp =1.61(1) and ZU =1.768(8). Hence, the correction to thescaling of R, Q=RU/Rp extremely
] slowly decays, which makes it practically impossible to identify the asymptotic scaling behavior of
h
R. In addition to the generic feature of the bidirectional coupling, the double domain structure is
c
anotherreason for the extremely slow approach to theasymptotic scaling regime of PCPD.
e
m
PACSnumbers: 64.60.-i,05.40.-a,82.20.-w,05.70.Ln
-
t
a
t I. INTRODUCTION complex behavior [14]. However, linearly coupled sys-
s
. tems belonging to DP or PC class exhibit mean-field or
t
a non-trivial critical behavior [9, 15, 16, 17, 18]. Linearly
The concept of universality makes possible to un-
m andunidirectionallycoupledsystemsexhibitnewcritical
derstand and classify the various and complicate crit-
- ical behavior of a number of equilibrium models [1]. behavior atmulticritical point where all sub-systems are
d critical [19, 20, 21, 22, 23, 24, 25, 26].
n Fornonequilibriumcriticalphenomena,extensivestudies
o duringpastdecadesrevealedthatvariousnonequilibrium Among single species systems, pair contact process
c systems also exhibit universal behavior characterized by with diffusion (PCPD) can be regarded as a two species
[ severalfeatures[2]. Hence,ithasbeenanimportantissue system. PCPD has been extensively studied during last
3 to identify nonequilibrim universality classes by finding yearsduetoitsnontrivialcriticalbehavior(see[27]forre-
v common physical features. view). However,in spite of extensive theoreticaland nu-
5 Among nonequilibrium critical phenomena, absorbing mericalstudies,thecriticalbehaviorisnotclearlyuncov-
1 phase transitions (APT’s) from fluctuating active states ered yet. PCPD is a stochastic reaction-diffusion model,
3
into absorbingstates in which the system is trapped for- which evolves by the competition of two processes, fis-
1
0 ever have been a field of growing interest during last sion (2A → 3A) and annihilation (2A → ∅). In addi-
7 decades [2, 3]. Recent theoretical and numerical stud- tion, each particle performs isotropic diffusion. With-
0 ies show that APT’s exhibit universality and it can be out diffusion, the model is so-called pair contact process
/ classified according to conservation laws, dimensionality (PCP) belonging to DP class [28]. Since the reactions
t
a of systems and symmetries of absorbing states [2, 3, 4]. involve pairs, diffusing solitary particles are not engaged
m However only a few universality classes have been iden- in the binary reactions. However when two solitary par-
- tified so far. Directed percolation (DP) [3, 4, 5, 6] and ticles form a pair,the reactionstake place. Onthe other
d parity conserving (PC) [7, 8, 9, 10, 11, 12, 13] class are hand, solitary particles are created from pairs by diffu-
n
wellstudiedclassesamongothers. DPclassincludessys- sion. Hence, PCPD can be regarded as a bidirection-
o
c tems with no special attributes except the time reversal allycoupledtwospeciessysteminwhichthe orderofthe
: symmetry, so that mostsystems studied so far belong to coupling is linear in the direction from pairs to solitary
v
this class. particles and quadratic in the opposite direction. This
i
X observation leads to the cyclically coupled DP and pair
As a research direction to search for further unknown
annihilation which exhibits the similar type of critical
r universality classes, coupled systems have been studied
a behavior to that of PCPD [29].
recently [9, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26]. Acoupledsystemisa multi-species systeminwhich In this paper, we investigate the domain structure of
eachspeciesiscoupledtothe othersincertainwayssuch PCPD. When a spreading domain is formed from local-
as bidirectional and unidirectional coupling in linear or izedinitialactivities,thequadraticcouplingfromsolitary
quadratic ways. However the coupled systems do not al- particlestopairsallowsthepair-freeregioninwhichonly
ways exhibit new critical behavior. For bidirectionally solitaryparticlesarepresent. Wecallthepair-freeregion
coupled systems, the critical behavior depends on the so-called uncoupled region. On the other hand, the lin-
manner of the coupling. For instance, quadraticallycou- earcouplingfrompairstosolitaryparticlesresultsinthe
pled DP systems still belong to DP class despite their so-called coupled region in which pairs and solitary par-
2
tttt
that the double domain structure of PCPD also makes
RRRR
UUUU itdifficulttoidentify the criticalspreadingbehaviorpre-
RRRR cisely. Theaimofthispaperistoinvestigatetheeffectof
pppp
RRRR the double domain structure on the critical spreading of
RRRR UUUU
x PCPDandto estimate criticalexponentsmoreprecisely.
We consider the PCPD of Ref. [30]. In this model, a
randomly selectedparticle attempts to hop to one of the
FIG.1: (Coloronline)ThedoubledomainstructureofPCPD.
nearest neighbor sites with an equal probability. If the
The region with size R is the coupled region in which pairs
p
target site is empty, the attempt is accepted. However
(blue) and solitary particles (red) coexist. The region with
if the target site is occupied, (i) two particles annihilate
sizeR istheuncoupledregioninwhichonlysolitaryparticles
U
exist. withprobabilityp or(ii) thehopping attemptisrejected
and the pair (the chosen particle and one at the target
site) tries to create a particle at the randomly chosen
ticles coexist. The pair-free region encloses the coupled nearestneighborsite of the pair. When the targetsite is
region as shown in Fig. 1. Hence, the spreading domain occupied,the branchingattempt isrejected. The critical
is divided into two regions, coupled and uncoupled re- pointofthismodelispc =0.133519(3)inonedimension
gion. This kind of double domain structure was found [30].
in unidirectionally coupled two level hierarchies [25, 26]. PCPDhasthreesectorsinconfigurationspacesaccord-
As showninpreviousstudies onunidirectionallycoupled ingtothe existenceofpairs(P)andsolitaryparticles(S).
systems [22, 23, 24, 25, 26], the measurements of criti- Theoneistheconfigurationsinwhichbothpairsandsoli-
cal exponents are very difficult due to long-time drift of tary particles are present (PS-ensemble). We call con-
the exponents. The one reasonof the drift is the generic figurations with at least one pair (two solitary particles)
feature of the unidirectional coupling [22, 23], the other P-ensemble (S-ensemble). In P-ensemble (S-ensemble),
is the double domain structure [25, 26]. In measuring solitary particles (pairs) may be present or not. The P-
critical exponents, one can overcome the latter effect by ensembleisthe reactivesubspaceofRef. [31]. Aconven-
measuring quantities in each region separately. tionalensemble includes configurationswith at leasttwo
To explain the effect of the domain structure on the particleswhichcanbeeithertwosolitaryparticlesorone
criticalbehavior,letusconsidertheunidirectionallycou- pair. We call the conventional ensemble All-ensemble.
pled DP-PC coupling [26]. In the DP-PC coupling, PC Since solitary particles are effectively linearly coupled to
process (B) is unidirectionally coupled with DP process pairs, the existence of pairs implies the existence of soli-
(A) via A→A+2B (linear). Since DP process spreads taryparticles. Hence,PS-ensembleshouldcoincide with
fasterthanPCprocess,thecouplingchangesthe spread- P-ensemble asymptotically. However, solitary particles
ing behavior of PC process. There are no interactions transformintopairsbycollisions,thecouplinginthisdi-
between A and B species. Starting with a single A par- rectionisquadratic. Hence,theexistenceofsolitarypar-
ticle on A-level, a B-domain is form and spreads faster ticlesdoesnotalwaysguaranteethepresenceofpairsdue
than a A-domain does due to the coupling. As a result, to the long life time of solitary particles. So S-ensemble
theuncoupledregionwherenoAparticlesexistisformed coincide with All-ensemble. As a result, there are two
[25]. The coupled region is just the region of B-domain distinct ensembles in PCPD.
overlapped with the A-domain where the coupling ex- We define the size of each domain as follows (See
ists. Hence, the size of the B-domain (R ) is the sum Fig. 1). The size of the whole domain at time t (R(t))
B
of the size of coupled (R ) and uncoupled region (R ), is defined as the distance between the leftmost and the
C U
R = R +R . At the multicritical point where both rightmost particle. When both pairs and solitary parti-
B C U
species are critical, the size of each domain increases in cles exist simultaneously, we can define the size of the
power-law such as RA ∼ tzA, RB ∼ tzB, RC ∼ tzC and coupled region(RC(t)) and the size of the uncoupled re-
RU ∼ tzU. Since zC = zA by definition, the scaling of gion(RU(t)). Sincesolitaryparticlesarelinearlycoupled
RB is given as RB ∼atzA +btzU, a and b are constants. to pairs, we define the size of the coupled region(RC(t))
Arecentstudy showedz ≥z forthe fastspreadingA- as the spreading distance of pairs (R ) defined as the
A U p
domain(source)[26]. Hence,RB scalesastzA asymptoti- distance between the leftmost and the rightmost pair.
callyforfasterspreadingsourceA,andR playsthecor- Then, R (t) is given as R = R−R . Hence, we have
U U U p
rectiontothescalingofR . Sincethedifferencez −z three different lengths, R, R andR in PCPD.To take
B A U p U
issufficientlysmallintheDP-PCcoupling,the existence intoaccountthethreelengthscalesatthesametime,one
of the uncoupled region makes precise estimates of z should use PS-ensemble in which only two lengths, R
B p
difficult within moderate simulation time. For the DP- and R are the fundamental length scales of PCPD due
U
PC coupling, the estimate of 2z is 1.24(1) which is less to R=R +R .
B p U
than the expectation 2zB = 2zDP = 1.265 [26]. On the At pc = 0.133 519, we perform defect Monte Carlo
other hand, the estimate of 2zU is 2zU = 1.20(2) which simulations with a pair on one dimensional empty lat-
is compatible with the leading scaling. tice. We run simulations up to t = 107 time steps using
As in unidirectionally coupled systems, it is expected 3.6 × 106 independent runs. We measure the squared
3
0.42
4.8
1.02
2/ZU 4.4 0.40
2R / tU 34..60 ht 0.38 N / Nsp 1.01
Z, 3.2 / p 0.36 1.00
2/ N
2R / t 22..48 (a) 0.34 103 104 1t05 106 107
2.0 0.32
103 104 105 106 107
0.30
3.50
3.48 103 104 105 106 107
3.46 t
Zp
2/t 3.44
/
2 p 3.42 FIG. 4: The scaling plot of Np. The main plot shows Np/tη
R
with η =0.275. The inset shows the semilogarithmic plot of
3.40
theratio N /N .
s p
3.38
(b)
3.36
103 104 105 106 107
t cially Z =1.70(5)[10]. Since ZU >Zp, the total spread-
ing distance R(= Rp +RU) should scale as R ∼ t1/Zp,
FIG. 2: (Color online) The scaling plots of various sizes. (a) and RU plays the role of the correction to the scaling as
Thescaling plotofR2 (black)andR2 (blue)withZ =1.663 in unidirectionally coupled systems. Hence, we conclude
U
and ZU =1.768. (b) thescaling plot of Rp2 with Zp =1.61. Z =Zp =1.61(1) which is the smallest value among the
estimates of previous studies [27]. For reference, we also
2.1 measure R2 using All-ensemble. The difference of All-
1.2 ensemble from PS-ensemble or P-ensemble is that All-
2.0 ensemble includes configurations without pairs. Since
1
1.9 Q(t) 0.8 asovleirtaagryedpoavrteircAlesll-sepnrseeamdbdlieffsucsaivleeslyd,iffiteirsenetxlpyefcrtoemd tthhaattoRf
PS-ensemble. From the scaling plot of R2/t2/Z, we esti-
fQ t 1.8 mate Z =1.676(3)for All-ensemble which is larger than
0.6
103 104 105 106 107 thatofPS-ensemble(notshown). Theslowspreadingof
1.7
t the whole domainin All-ensembleresults fromthe diffu-
sivemotionsofsolitaryparticlesinconfigurationwithout
1.6
pairs.
1.5 Since R scales as R = R (1+R /R ), the correction
p U p
103 104 105 106 107 to the R is Q = R /R which decays as Q ∼ t−φ with
U p
t φ=(Z −Z )/Z Z . To see how the correction decays
U p p U
slowly in time, we plot Q in Fig. 3. When Q ≪ 1, the
FIG. 3: The scaling plot of the ratio Q=R /R . The main correction is negligible. However, as shown in the inset,
U p
plot shows Qtφ with φ = 0.054. The inset shows the double Q is still comparable to one even at t = 107. We obtain
logarithmic plot of Q(t). the best scaling plot of Qtφ with φ = 0.054(4). As Q
decays with very small φ, it is practically impossible to
reach the asymptotic scaling regime of R ∼ R . As a
p
sizes, R2, R2 and R2 for surviving PS-ensemble. The result, one should take the double domain structure into
p U
PCPDhastwoabsorbingstates,vacuumandstateswith accountforthemoreprecisemeasurementofthedynamic
one diffusing solitary particle. Hence, we stop the sim- exponent Z.
ulations when the total number of particles N is less In addition to the sizes, we also measure the number
then two. At the criticality, the squared sizes scale ofpairs(N )andofsolitaryparticles(N )averagedover
as R2 ∼ t2/Z, Rp2 ∼ t2/Zp and RU2 ∼ t2/ZU. Fig. 2 allsamplesp, andthe survivalprobabilitysofPS-ensemble
shows the scaling plots of the squared sizes, Rx2/t2/Zx. (Pps) and All-ensemble (Pall). As solitary particles are
We obtain the best scaling plot with Z = 1.663(5), linearly coupled to pairs, N is proportional to N . At
s p
Z = 1.61(1) and Z = 1.768(8) respectively. The er- criticality, N scales as N ∼ tη. Fig. 4 shows the scal-
p U p p
rors of our estimates should be larger due to the error ing plot N /tη. We obtain the best scaling plot with
p
of p . Within the numerical errors at the criticality, our η = 0.275(5). The inset show the ratio of N /N , which
c s p
estimate of Z agree with the previous studies [27], espe- converges to one as expected. P (t) and P (t) de-
ps all
4
0.9
(R ). Hence,R andR arethebasiclengthscaleschar-
U p U
acterizing the scaling behavior of the spreading domain
'dP taall 00..78 P , Ppsall0.1 RiHnepPn∼cCetP,1D/RZ.psWhaoneudnlduRmaUsey∼rmictap1l/tlZoyUtificnwadliltyhthsZactUalRe>paZsaCnRda∼Rt cUtr1i/stZcicaalweliittayhs.
, Z = Zp and RU plays the role of the correction to the
'dtps 0.6 102 103 104 t105 106 107 scaling. However, the direct measurement of R leads to
the underestimate of the asymptotic value of Z because
s
Pp 0.5 the correction Q = RU/Rp decays with very small ex-
ponent. Since it is practically impossible to reach the
asymptotic scaling region of R ∼ t1/Zp, it is important
to take the domainstructure into accountin simulations
102 103 104 105 106 107
for more precise estimate of the dynamic exponent Z of
t
PCPD.
Weclassifyparticleconfigurationsintofourensembles,
FIG. 5: (Color online) The scaling plot of P and P . The
ps all which are finally reduced to two distinct ensembles, P-
main plot shows the scaling plot of Ppstδp′s (red) and Palltδa′ ensemble and All-ensemble respectively. All-ensemble
(black) with δ′ = δ′ = 0.13. We add a constant −0.4 to
ps a includes configurations without pairs, while P-ensemble
thescaling plot of P for thebetter presentation. The inset
all doesnot. Thesurvivalprobabilitiesoftwoensemblesde-
shows thedoublelogarithmic plots.
caywiththesameexponent. However,thewholedomain
appears to spread more slowly in All-ensemble than in
cthaye cinritpicoawlietyr-.lawFiga.s5Psphsow∼s tt−hδep′sscaanlidngPpalllot∼s otf−Pδa′xtaδx′t Pttioc-letenhsseeindmocbmolenafiidnguusetrratuoticottnhuserewd,iittffhhueosudivtieffpumasiiorvste.iomHnsoentoicofens,soilnoitfaasdroydliitptaiaorrny-
and the double logarithmic plots of P and P . With
ps all particles in All-ensemble raise another correction to the
′ ′
δ = δ = 0.130(3), we obtain the best scaling plots.
ps a scaling of the total spreading distance R which does not
Thesystemescapesfromthenon-reactivesectorinwhich
appear in P-ensemble.
onlydiffusing solitaryparticlesarepresentviapairanni-
As the linear-quadratic bidirectional coupling is the
hilations of solitary particles. As an naive argument for
′ ′ commonfeatureofvariousPCPDstudiedsofar,thedou-
the equality of δ = δ , the escaping probability from
ps a ble domainstructureshouldappearinotherPCPDvari-
non-reactivesector within time τ may scale as 1−τ−1/2
ants. Among PCPD variants, we investigate the domain
which is the death probability of two diffusing particles
structure of the bosonic PCPD with soft-constraint of
undergoing the reaction S+S →0 within τ. Hence, the
Ref. [10] which is known to exhibit the clear power-law
contribution of non-reactive sector to P is negligible
all decays at the criticality. For this model, we also confirm
due to the fast escaping probability, which leads to the
theexistenceofthedoubledomainstructureandthecrit-
same scaling behavior of P as P .
all p ical spreading behavior similar to that of PCPD studied
In summary, we investigate the domain structure of
in this paper. Hence, the double domain structure is a
PCPD.Wenumericallyconfirmthe existenceofthedou-
common feature of PCPD variants.
ble domain structure in PCPD. This double domain
Asshowninrecentstudies,asPCPDextremelyslowly
structure comes from the linear and quadratic bidirec-
approachestoitsasymptoticscalingregion,itisverydif-
tional couplings. The structure intrinsically makes the
ficult to identify the critical behavior. The bidirectional
serious correction to the scaling of the critical spreading
couplingshouldbethemainreasoninitself. Ontheother
of a domain.
hand,thedoubledomainstructurenaturallyappearsdue
Starting with a pair, a domain grows and spreads in
tothelinear-quadraticbidirectionalcouplings,whichen-
vacuum via fission(A+A→3A) andspontaneous anni-
hance the long-time drift of dynamic exponent Z. One
hilationofpairs(2A→0)inPCPD.Inadditiontothebi-
canovercomesthe lattereffectbyconsideringthescaling
nary reactions, each particle diffuses isotropically, which
behaviorofsub-domainsseparatelyasinunidirectionally
leads to the bidirectional coupling between solitary par-
coupled systems.
ticles and pairs. The couplingfrom pairsto solitary par-
ticles is linear, while the opposite coupling is quadratic.
The difference of the coupling waysresults in the double
Acknowledgments
domain structure of the whole domain, the coupled and
the uncoupled region respectively. As a result, the size
of the whole domain (R) is given as the sum of the size ThisworkwassupportedbyGrantNo. R01-2004-000-
of the coupled region (R ) and of the uncoupled region 10148-0from the Basic Research Programof KOSEF.
p
5
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