Asymmetrically coupled directed percolation systems

0.13 MB·English

by  Jae Dong Noh

Asymmetricallycoupled directed percolationsystems
Jae Dong Noh
Department of Physics, Chungnam National University, Daejeon 305-764, Korea
5 Hyunggyu Park
0 School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
0 (Dated:February2,2008)
2
Weintroduceadynamicalmodelofcoupleddirectedpercolationsystemswithtwoparticlespecies. Thetwo
n speciesAandBarecoupledasymmetricallyinthatAparticlesbranchBparticleswhereasBparticlespreyonA
a
particles. Thismodelmaydescribeepidemicspreadingcontrolledbyreactiveimmunizationagents. Westudy
J
nonequilibriumphasetransitionswithfocusedattentiontothemulticriticalpoint wherebothspeciesundergo

2 theabsorbingphasetransitionsimultaneously. Inonedimension, wefindthattheinhibitorycouplingfromB
2 toAisirrelevantandthemodelbelongstotheunidirectionallycoupleddirectedpercolationuniversalityclass.

Onthecontrary, ameanfieldanalysispredictsthattheinhibitorycoupling isrelevant andanew universality
]
h appearswithavariabledynamicexponent. Extensivenumericalsimulationsonsmall-worldnetworksconfirm
c ourpredictions.
e
m PACSnumbers:64.60.Ht,05.70.Ln,05.40.-a,89.75.Da
-
t
a It is a challengingproblemto establish the conceptof the phenomenabelong to the DP class. The other species X ,
t k>0
s universalityclassinnonequilibriumphasetransitions.Incon- however,aredressedbythecriticalfluctuationsofparticlesof
.
at trast to equilibrium systems, nonequilibrium systems lack a species Xk′<k at the multicritical point where all species un-
generalframeworktobeginwithanddisplaytoodiversecriti- dergotheabsorbingphasetransitionsimultaneously.Itturned
m
calphenomenatobeclassifiedintoafewuniversalityclasses. out that their critical behaviors are described by the critical
-
d Recently the absorbing phase transition has been attracting exponents distinct from those of the DP class [7, 8]. This
n much theoreticalinterest since the universalityclass concept novelcriticalbehavioris destroyedandthe DP scalingis re-
o canbeapplied.Itisaphasetransitionfromaninactive(dead) coveredas soon as we turn on the excitatorycouplingin the
c
phaseintoanactive(live)phase,whichmayoccurinsystems other(downward)direction[6,8].
[
withmicroscopictrapped(absorbing)stateswhichthesystem,
It is noteworthy that the UCDP process was proposed for
1 oncetrapped,cannotexcapeoutof.
describing an interesting roughening transition in an one-
v
2 Absorbing critical phenomena are categorized into a few dimensional growth process [9], where the monomer-type
4 universality classes depending on conservation in dynam- restricted-solid-on-solid(RSOS)modelisconsideredwiththe
5 ics and symmetry between absorbing states (see for review constraint that evaporation is allowed only at the edges (not
1
Ref. [1]). The directed percolation (DP) class is the most terraces). It can be easily shown that the growth model is
0
5 well-established one, which encompasses a vast number of mappedtotheUCDPprocess,exceptthattheRSOScondition
0 systems[2,3]. Thecontactprocess[4]isanarchetypeofthe generatesanextrainhibitorycouplinginthedownwarddirec-
t/ DP class. In this model, each lattice site is either empty (0/) tion, i.e. the offspringproductionof Xk is suppressed by the
a or occupied by a particle (X). Each particle may annihilate presenceofXk′ fork′ >k (see Sec.VA in [8]). The coupling
m
spontaneously(X →0/)orbranchanoffspringtoaneighbor- should be at least bilinear, so the simplest process one may
d- ing site (X →XX). The contactprocess was originallysug- considerisXkXk+1→Xk+1. Numericalstudies[8,9]revealed
n gestedasamodelforepidemicspreadingwithoutimmuniza- thattheone-dimensionalgrowthmodeldisplaysthesamecrit-
o tion [4], wherethe particlerepresentsan infectedindividual, icalbehaviorasthe UCDP, whichmayimplytheirrelevance
c who may recover spontaneously or infect its healthy neigh- oftheinhibitorycouplingincoupledDPprocesses.
:
v bors. Thevacuumstatewithallsitesemptyistheuniqueab- InthisLetter,westudyacoupledDPsystemwithtwopar-
i sorbing state, which is one of the key ingredients in the DP ticle species, which has a linear excitatory coupling in one
X
class. direction and a bilinear inhibitory coupling in the other di-
r
a rection. We show that the inhibitory coupling is irrelevant
Recently, coupled DP systems have been studied exten-
only in low dimensions (UCDP class) but becomes relevant
sively [5, 6, 7, 8]. In particular, the unidirectionally cou-
in high dimensions, at least where the mean field theory be-
pledDP(UCDP)processintroducedbyTa¨uberetal.[7]was
comesvalid. Ourmodelmayalsoserveasaprototypemodel
foundto displaya seriesofnewmulticriticalphenomena. In
forepidemicspreadingprocessinthepresenceofreactiveim-
the UCDP process, there are an infinite number of particle
munizationagents,whichwillbeexplainedshortly.
species X with k=0,1,2,··· with dynamicrules character-
k
izedbyX →0/,X →X X ,andX →X X . Thecoupling WeconsideracoupledDPsystemwithtwoparticlespecies
k k k k k k k+1
isunidirectional(upward),linear,andexcitatory. Thespecies A and B on lattices or networks in general. Specifically we
X is not affected by the others, hence its absorbing critical choose the contact process dynamics for both species in the
0
2
p Q AB process for large p region. Therefore, one may easily
B
B
expect that the three critical lines should belong to the DP
I class. That was confirmed by numerical simulations, which
III
wedonotpresenthere.Interestingcriticalphenomenacanbe
M R observedalonga linethroughthe multicriticalpointM from
the phase III to the phase I. In this Letter, we determine the
locationofMwithhighprecisionandmainlyfocusondensity
II
decay dynamics of both species at M, which may display a
P novelcriticalbehaviordistinctfromtheDPandalsofromthe
p
A UCDP.
First,weinvestigatethemodelusingthemeanfieldtheory
FIG. 1: A schematic phase diagram of the ACDP process in the which assumes that local densities of A and B particles are
(pA,pB)planewithfixedl andµ. uniform and uncorrelated spatially with the mean values r
A
and r , respectively. The mean field rate equations can be
B
writtenas
noninteracting limit; each particle annihilates spontaneously
or branchesone offspringof the same kind to a neighboring r˙ = a r −r 2−eµr r ,
A A A A A B
site. The two species are coupled asymmetrically through e
r˙ = a r −r 2+rl , (1)
branching and predation processes, that is, an A particle B B B B A
branches a B particle to a neighboring site (excitatory cou-
where the densities are rescaled to fix the coefficient of the
pling), while a B particle preys on an A particle at the same
r 2 terms to be unity and the parametersare functionsof the
site(inhibitorycoupling).
modelparameters.Higherordertermsareneglected.
Thedynamicrulesaresummarizedasfollows:(i)A(B)→
The stationary state densities are obtained by solving
0/ withprobability pA(pB),(ii)A→AAwithprobability(1− Eq.(1)withr˙ =r˙ =0andhencethephasediagramisde-
p )(1−l ),(iii)B→BBwithprobability1−p ,(iv)A→AB A B
A B termined. It has the same structure as shown in Fig. 1. The
withprobability(1−p )l ,and(v)AB→Bwithprobabilityµ.
A criticallinesMR,MQ,andMParegivenbya =0(a ≤0),
B A
Theorderparametersforthephasetransitionaretheparticle a =0 (a ≤0), and a =eµa (a ≥0), respectively. The
densities, r and r . The model will be referred to as the A B A B B
A B multicriticalpointMlocatesat(a ,a )=(0,0).
A B
asymmetrically coupled DP (ACDP) process. When µ=0,
Approaching the multicritical point along a straight line
thecouplingbecomesunidirectionalandthemodelreducesto (a ,a )=(e ,re )withfixedrinsidethephaseIII,wefindthat
A B
thetwo-speciescaseoftheUCDPprocess. thedensityvanishesasr A,B∼e b A,B withtheorderparameter
In the context of epidemiology, A particles correspond to exponentsb =2andb =1. Spatialfluctuationsintheorder
A B
antigens (or viruses) and B particles correspond to antibod-
parameter are easily incorporated into the mean field theory
ies (orvaccines). Both antigensand antibodiescan replicate by including diffusion terms (cid:209) 2r in the mean field equa-
A,B
themselvesandmaydieoutbythemselves(processes(i),(ii),
tions. Withthose,onecanshowthatthecharacteristiclength
and (iii)). In addition, an antigen promotesan antibody (iv) scaledivergesasx A,B∼e −n A,B withthecorrelationlengthex-
andcanbedestroyedbytheantibody(v). ponentn =n =1/2. Thecharacteristictimescalediverges
A B
AschematicphasediagramoftheACDPprocessisshown with the length scale as t ∼x zA,B with the dynamicexpo-
A,B A,B
inFig.1. InthephaseIwherebothannihilationprobabilities nentz =z =2.
A B
are large, both species are inactive with r A =r B =0 in the We note that the critical exponents for the B species are
steadystate.WhenpBissmaller,theBspeciesbecomesactive identicaltothemeanfieldDPexponentswithb =1,n =1/2,
withtheAspeciesremaininginactive. Hence,inthephaseII, and z=2 [1]. On the otherhand, the A species dynamicsis
r A=0andr B6=0inthesteadystate. WhenpAissmall,theA dressedbythecriticalDPfluctuationsandfallsintothenon-
speciesisactive,soistheBspeciesregardlessofthevalueof DPclass. IntheUCDPlimitwitheµ=0,wehavetheopposite
pBduetotheprocessA→AB.So,bothspeciesareactivewith situation; A species falls into the DP class with b A =1 and
r A6=0andr B6=0inthephaseIII.Sincer A=0throughout Bspeciesintothenon-DPclasswithb B=1/2[7]. Fromthe
thephasesIandII,thephaseboundarylineMRshouldbeflat meanfieldanalysisweconcludethattheinhibitorycouplingis
with pB = pcB which is a constantindependentof pA, l , and relevantandthattheACDPformsanindependentuniversality
µ,andisidenticaltothecriticalprobabilityoftheuncoupled classdistinctfromtheUCDPclass.
system. More interesting is the critical density decay dynamics at
Thenatureofthephasetransitionsiseasilyinferred.Across the multicritical point. Putting a =a =0 in Eq. (1), after
A B
thelineMR,theBspeciesundergoestheabsorbingtransition straightforwardcalculations,weobtainthatthedensitydecays
with r A =0. Across the line MP, the A species undergoes algebraically in time as r A ∼t−a A and r B ∼t−a B with the
the transition while r B remains finite. Across the line MQ, exponenta B=1andthevariableexponent
bothspeciesundergoesthetransitionwithr (cid:181)r sincetheB
B A
particlesaremostlyslavedbytheAparticlesthroughtheA→ a =max{2,eµ}. (2)
A
3
At eµ = 2, logarithmic corrections appear as r A ≃ 100 104
e
t−2(2l lnt)−1andr B≃t−1(1+1/(2lnt)). Inthenormalscal- 10-1 0.49 0.52
ing, we expect that the density decay exponent is given by 0.98 103
aTh=isbis/a(znno)th[e1r],nwovheilchfedatouersenooftthheolAdCinDoPuurnmivoedrseallfiotyrceµla>ss2.. r 10-2 rr AB ( a ) t ( b )0.51 tt AB
The mean field characteristics may be realized on regu- 10-1300 101 102 103 102 104 105 106
larlatticesinhigherspacedimensionsthantheuppercritical 100 t 104 N
dimension and also on various networks including complete 10-1 1.02 0.50
graphs, random networks, and small-world networks [10]. 10-2 103
Refeficceinentltyi,nthsetusdmyianlgl-wmoeralndfineeltdwcorrikticisalfpohuenndotmoebneau[1se1f]ualnadnidt r 1100--43 rr AB ( c ) 2.2 t102 0.33 ( d ) tt AB
reflectssomeofnetworkaspectsinrealbiologicalandsocial 10-5 101
systems. InthisLetter,weadoptthesmall-worldnetworksto 100 101 102 103 104 105 106
t N
test our mean-field result and also to get the information on
theuppercriticaldimensionofourmodel. FIG.2: (a)Densitydecay(N=2×106)and(b)characteristictime
Thesmall-worldnetworkischaracterizedbythetotalnum- scaleatµ=0andpA=0.3202.(c)Densitydecay(N=2×106)and
ber of sites N, the average coordination number 2Z, and (d)characteristictimescaleatµ=1.0andpA=0.2902.
the so-called rewiring probability p . It is constructed as
r
follows [10]: First, N sites are arranged along an one-
bystraightlines in the plot. Note thatthe resultsare consis-
dimensionalringandeachsiteisconnecteduptoZneighbor-
tentwiththemeanfieldresultsoftheUCDP[7]. InFig.2(b),
ingsites. Second,eachbondisrewiredrandomlywithproba-
wepresentthescalingofthecharacteristictimet atwhichthe
bility p . Itinterpolatesthenormalone-dimensionallatticeat
r
survivalprobabilitybecomes1/2. Itgrowsalgebraicallywith
p =0andtherandomnetworkat p =1. Insimulations,we
r r
chooseZ=10, p =0.5,andN≤2×106. the totalnumberof sites N with theeffectivedynamicexpo-
r
nent z¯ =0.51(2) and z¯ =0.52(2) as indicated by straight
In networks, the concepts of dimensionality as well as A B
lines. Using Eq. (3) and the mean field dynamic exponents
lengthscalearemeaningless. Botetetal.[12]suggestedthat
acorrelationvolume,insteadoflength,x beaproperscaling zA=zB=2[7], we obtainthattheuppercriticaldimension-
V alityis givenbydu =4 forbothspecies, whichisconsistent
variableandthecriticalitybedescribedbythecorrelationvol-
umeexponentn¯ andtheeffectivedynamicexponentz¯. More- withthepredictionfortheUCDPclass[7].
over,theyconjecturedthattheexponentsaregivenby Turning on the inhibitory coupling (µ 6= 0), we find dif-
ferent scaling exponents. In Fig. 2 (c), the density decay is
n¯ =n MFdu and z¯=zMF/du (3) shownatthemulticriticalpoint pcA=0.2902(2)atµ=1. As
predictedinourmeanfieldtheory,theestimateddecayexpo-
where n MF and zMF are the mean-field exponents and du is nenta B=1.02(3)isconsistentwiththemean-fieldDPvalue
theuppercriticaldimension. Althoughtheconjecturehasnot anda A=2.20(5)representsthenon-DPclass. We alsoesti-
beenprovedyet,itisshowntobevalidforsomeequilibrium matetheeffectivedynamicexponentsz¯A=0.33(1)≃1/3and
systems [11, 12]. We will use this relation to find the upper z¯B =0.50(1) in Fig. 2 (d). Since zMF =2 for both species,
criticaldimensionofoursystem. Eq.(3)suggeststhatthetwospecieshavedifferentuppercrit-
ThephaseboundaryMRisgivenbyaflatlineof p = pc, ical dimensions, i.e. du =6 and du =4. It is remarkable to
B B A B
which is the critical probability of the uncoupled system. see that the two interacting species begin to show the mean
We perform Monte Carlo simulations starting with the fully field critical behaviors at different dimensionality. A sim-
occupied configuration of B particles only. All numerical plepowercountingforcoupledLangevinequationsobtained
data shown below are obtained after average over, at least, from Eq. (1) by adding diffusion terms and DP-type multi-
several thousands of samples. As the particle density de- plicativenoisesreproducestheaboveresults,butmoreefforts
cays algebraically in time at criticality, we can easily locate are neededto understandthis peculiarpropertybothanalyti-
pcB =0.48505(5)where r B ∼t−a B with a B =0.98(5). This callyandnumerically.
confirms that the critical line MR belongs to the mean-field We performedthe similaranalysisforvariousvaluesofµ.
DPclass. Theresultsforthedecayexponentsandtheeffectivedynamic
Once pc being found,the multicriticalpointM can be lo- exponentsarepresentedinFig.3. Thenumericalresultsarein
B
cated by examiningthe decay dynamicsof r , starting from goodagreementwiththemeanfieldresults. Nearµ=0,e.g.,
A
the fully occupiedconfigurationof A particles, as one varies atµ=0.25,theexponentsseemtodeviatefromthemeanfield
p with p = pc fixed. For the coupling constants, we fix values. These are attributed to crossoversdue to the nearby
A B B
l =0.5andvaryµ.WefirststudytheUCDPlimitatµ=0.In UCDP fixedpoint. Themean field theoryandthe numerical
Fig.2(a),theparticledensitydecayispresentedatthemulti- resultssuggestthattheACDPprocesswithµ6=0formadis-
criticalpointatpc =0.3202(1).Thedensitydecayexponents tinctuniversalityclassdifferentfromtheUCDPclass.
A
areestimatedasa =0.98(3)anda =0.49(2)asindicated Thepreviousstudyposesaquestionontherelevanceofthe
A B
4
2.5 0.6 Insummary,wehaveintroducedtheACDPprocessoftwo
particlespeciesandinvestigateditsmulticriticalbehaviors.In
2 the mean field theory, we show that the inhibitory coupling
0.5
isrelevantandtheACDPprocessformsthedifferentuniver-
sality class distinct from the UCDP process: The B species
1.5
displaystheDPtypecriticalbehaviorwhiletheAspeciesthe
a z| 0.4 dressednon-DPtypecriticalbehavior. Furthermore,thecriti-
1 caldynamicsoftheAspeciesisanomalouswiththevariable
criticalexponent.Themeanfieldresultsareconfirmedbythe
a 0.3 -
0.5 ( a ) a AB ( b ) -zz AB nnuetmweorrikcsa.lsUimsiunlgattihoensctoundjieecstuorfethineAECq.D(P3)o,nwtehealssmoafilln-dwothrladt
theuppercriticaldimensionalityforeachspeciesisgivenby
0 0.2
0 0.25 0.5 0.750.91 0 0.25 0.5 0.750.91 du =6 and du =4. On the other hand, the numericalsimu-
m m A B
lationstudiesinonedimensionshowthattheinhibitorycou-
plingisirrelevantandtheACDPprocessbelongstotheUCDP
FIG.3:(a)Thedensitydecayexponentand(b)theeffectivedynamic
exponents at the multicritical points. The solid (dashed) lines are class: The A species displays the DP type critical behavior
guidestoeyesandindicatethemeanfieldresultsfortheA(B)species whiletheBspeciesthedressednon-DPtypecriticalbehavior.
of the ACDP process. The solid (dashed) arrows point toward the
correspondingexponentvaluesofthemeanfieldUCDPclass. For future works, we suggest that it would be interesting
to study the model in higher dimensional lattices to find the
100 107 dimensionalityatwhichtheinhibitorycouplingbecomesrel-
( a ) ( b ) evant and to cofirm the upper critical dimensionalities. The
multi-species generalization would be also interesting [13].
0.10 6
10 As an application, the phase transition of the ACDP process
ongeneralcomplexnetworksmayyieldrichercriticaldynam-
1.59
ics.
5
10
r 0.159 t 1.59
This work was supported by Korea Research Foundation
Grant(KRF-2004-041-C00139).
10-1 r A 104 t A
r t
B B
3
10
2 3 4 5 6 2 3 4
10 10 10 10 10 10 10 10
t N [1] H.Hinrichsen,Adv.Phys.49,815(2000).
[2] H.K.Janssen,Z.Phys.B42,151(1981).
FIG. 4: (a) Density decay at the multicritical point of the ACDP [3] P.Grassberger,Z.Phys.B47,365(1982).
modelwithl =0.2andµ=0.2ontheone-dimensionallatticewith [4] T.E.Harris,Ann.Prob.2,969(1974).
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103,801(2001).
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[7] U.C.Ta¨uber,M.J.Howard,andH.Hinrichsen,Phys.Rev.Lett.
multicritical behavior of our model on the one-dimensional
80,2165,(1998);Y.Y.Goldschmidt,ibid.81,2178(1998).
lattice. We fix the coupling constants l =0.2 and µ=0.2.
[8] Y. Y. Goldschmidt, M. J. Howard, H. Hinrichsen, and U. C.
Numericalsimulationsshowthatthemulticriticalpointislo-
Ta¨uber,Phys.Rev.E59,6381(1999).
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cayandthecharacteristictimescalesarepresentedinFig.3. Rev.Lett.76,2746(1996);Phys.Rev.E57,4997(1998).
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A
a =0.10(2), which are consistent with the results for the (1998).
B
[11] B.J.Kim,H.Hong,P.Holme,G.S.Jeon, P.Minnhagen, and
UCDP class in one dimension [7]. The dynamic exponents
M. Y. Choi, Phys. Rev. E 64, 056135 (2001); H. Hong, B. J.
areesimatedasz ≃z ≃1.59(2),whichareconsistentwith
A B Kim,andM.Y.Choi,Phys.Rev.E66,018101(2002).
the DP value. These results suggest that the ACDP process
[12] R. Botet, R. Jullien, and P. Pfeuty, Phys. Rev. Lett. 49, 478
belongstotheUCDPclassinonedimension. Thatis,thein- (1982).
hibitorycouplingisirrelevantinonedimensioninthecontrary [13] Themeanfieldexponentsforthemulti-speciesACDPdisplay
tothehigherdimensionalcases. aninterestingFibonaccisequence.