A Monte Carlo investigation of the critical behavior of Stavskaya's probabilistic cellular automaton

0.49 MB·English

by  J. Ricardo G. Mendonça

AMonteCarloinvestigationofthecriticalbehaviorofStavskaya’sprobabilisticcellularautomaton
J. Ricardo G. Mendonc¸a∗
InstitutodeF´ısica,UniversidadedeSa˜oPaulo–CaixaPostal66318,05314-970Sa˜oPaulo,SP,Brazil
Stavskaya’smodelisaone-dimensionalprobabilisticcellularautomaton(PCA)introducedintheendofthe
1960’s as an example of a model displaying a nonequilibrium phase transition. Although its absorbing state
phasetransitioniswellunderstoodnowadays,themodelneverreceivedafullnumericaltreatmenttoinvestigate
itscriticalbehavior. InthisbriefreportwecharacterizethecriticalbehaviorofStavskaya’sPCAbymeansof
Monte Carlo simulations and finite-size scaling analysis. The critical exponents of the model are calculated
andindicatethatitsphasetransitionbelongstothedirectedpercolationuniversalityclassofcriticalbehavior,
as it would be expected on the basis of the directed percolation conjecture. We also explicitly establish the
relationshipofthemodelwiththeDomany-KinzelPCAonitsdirectedsitepercolationline,aconnectionthat
seemstohavegoneunnoticedintheliteraturesofar.
PACSnumbers:05.70.Fh,64.60.Ht,64.60.De
1 Keywords:Stavskayamodel,probabilisticcellularautomata,phasetransition,Domany-Kinzel,directedpercolation
1
0
2 I. INTRODUCTION verysimplerule: withprobabilityε ∈ [0, 1], ηi(t+1) = 1,
otherwiseη (t+1)=η (t)·η (t).
n i i−1 i
a Stavskaya’s model is a one-dimensional probabilistic cel- Clearly, 1 = (1,1,...,1) is an absorbing state of the
J lular automaton (PCA) proposed in the end of the 1960’s by model. Itcanbeproventhatthereexistsacriticalε∗suchthat
2 theRussianschoolofMarkovprocessesasanexampleofan forε > ε∗ theonlyinvariantmeasureofStavskaya’sPCAis
2 interactingparticlesystempresentinganonequilibriumphase δ1, the measure concentrated in 1, and that for ε < ε∗ the
transition [1–3]. The model is related with the directed site invariant measures are translation-invariant convex combina-
]
h percolation(DP)process,ofwhichitcanbeviewedasaone- tions of the form αµε + (1 − α)δ1, with 0 < α < 1 and
c sidedversion,aswellaswiththeDomany-KinzelPCAinone µε themeasurethatputsmassonconfigurationswithdensity
e ofitsmanifolds(cf.Sec.II)[4]. Roughlyspeaking,thephase 0<µε(1)<1[2,3,6]. Earlyboundsonthecriticalpointes-
m transitioninStavskaya’smodelfollowsfromitsattractiveness timate0.09<ε∗ <0.323[3,6]. Theupperboundwaseven-
- (itstendencyforformingclusters)andtheexistenceofanab- tually confirmed, but not improved, by different techniques
t
a sorbingstate,andiswellunderstoodnowadays. [8], while the lower bound never received a reassessment; it
t However, while many rigorous results exist for this model should be remarked that lower bounds on critical values of
s
. [1–3, 5–11], it has never received a full numerical treatment interactingparticlesystemsarenotoriouslydifficulttoobtain.
t
a to estimate its critical point and critical exponents. In this Stavskaya’sPCAisrelatedwiththeDomany-Kinzel(DK)
m
briefreportweproceedtosuchaninvestigationofStavskaya’s PCA [4] by taking the complementary (negated) variables
- model by Monte Carlo simulations and finite-size scaling η¯ = 1−η . ItcanthenbeseenthatStavskaya’sPCAcorre-
d i i
techniques. Besides closing a gap in the characterization of sponds to the DK PCA on the line p = p = 1 − ε, i.e.,
n 1 2
themodel,ourresultsaddanotherbitofevidenceinfavorof over the directed site percolation (site DP) line of the DK
o
c the DP conjecture, according to which phase transitions into PCA parameter space. Notice, however, that the dynamics
[ anabsorbingstateinshort-rangedsinglecomponentsystems in the DK PCA is defined for each of its two sublattices in

intheabsenceofconservedquantitiesallbelongtothesame the time direction (even and odd time steps), while the dy-
2
universality class of critical behavior [12, 13]. Remind that namicsinStavskaya’sPCAisdirect. OnthesiteDPline,the
v
9 although the DP conjecture is grounded on solid theoretical DK PCA displays an inactive-active phase transition at the
8 argumentsandhasbeenverifiedinahostofmodelsystems,it critical point p∗ = p∗ = 0.705489(4) [17], corresponding
1 2
4 (i) could not be proved rigorously yet, and (ii) has only very toε∗ = 0.294511(4),withintherigorousboundsmentioned
1 thinexperimentalevidence[14,15],sothatitcontinuestorely before (the numbers between parentheses indicate the uncer-
.
1 onmodelsystemstosustainitself. taintyinthelastdigitordigitsofthedata). Curiously,there-
1 lationshipbetweenStavskaya’sPCAandtheDKPCAseems
0 tohavegoneunnoticedinpreviousinvestigations[6–11], al-
1 II. STAVSKAYA’SMODEL thoughacouplingschemewithan“independentorientedper-
:
v colation” process equivalent with site DP was used in [8]. It
i Stavskaya’s model is a two-state PCA defined on a one- isworthmentioningthatStavskaya’smodel,togetherwithan-
X
dimensional periodic lattice of L cells specified by the con- other PCA introduced by the same epoch, Vasil’ev’s model
r
a figuration η(t) = (η1(t), η2(t), ..., ηL(t)) ∈ {0,1}L and [6,18]—whichcorrespondstothep2 =0lineintheDKPCA
evolving in discrete time t ∈ N according to the following or, equivalently, to a probabilistic version of CA rule 18 in
Wolfram’sclassificationscheme[19]—predatestheDKPCA
andrelatedmodelsbyalmosttwodecades,butdidnotreceive
much attention, not even when CA and PCA reentered the
∗Email:[email protected] mainstreamscientificagendainthe1980’s.
2
11
00..55
LL== 88000000 ((aa))
00..88
00..66 00..3322
ρρ
−− 00..2244
11 ρρ
00..44 00..1166 −−
11
00..0088
00..22
00
0.288 0.291 0.294 0.297
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05
ε
2000 20000 200000
FIG.1. StationarydensityofinactivecellsinanautomatonofL = t
8000cellsaveragedover1000samples. Theinsetshowsthecurve
00..2255
close to the critical point ε∗ = 0.29450(5) (value obtained from ((bb))
time-dependentsimulations).
00..2200
III. THECRITICALBEHAVIOR
δδ 00..1155
Our Monte Carlo (MC) simulations of Stavskaya’s model
ranasfollows. Foragivenε,thePCAisinitializedwitheach 0.10
η (0) = 1,1 (cid:54) i (cid:54) L,drawnindependentlywithprobability
i
1/2. Stationarystatequantities,e.g. thedensityofactivecells
ρ =L−1(cid:80) η ,arethensampledafterthesystemisrelaxed 0.05
L i i 0.000005 0.00005 0.0005
through100LMCsteps(MCS),withoneMCSequivalentto
1/t
a synchronous update of the states of all L cells of the au-
tomaton. This amount of relaxation proved enough for our
FIG.2. (a)Logarithmicplotof1−ρ (t)withL = 20000cells.
L
purposes. Moreover, except for the data shown in Figure 1, (b) Instantaneous values of δ obtained from the curves in (a). In
our results were obtained from time-dependent simulations, both graphs we have, from the uppermost curve downwards, ε =
so that estimates on the stationary state did not concern us 0.2943,0.2944,0.2945,0.2946,and0.2947.Fromthesecurveswe
much. Wereferthereaderto[16]foraniceexpositionofthe estimatedε∗ = 0.29450(5)andδ = 0.155(5). Thedashedlinein
time-dependenttechniquesemployedinwhatfollows. panel(b)indicatesthebestvalueavailableforδDP.
The critical behavior of the model can be determined by
assumingthescalingrelation
Noticethatthisestimateofε∗completelyagreeswiththecrit-
1−ρL(t;∆)∼t−β/ν(cid:107)Φ(∆t1/ν(cid:107), tν⊥/ν(cid:107)/L) (1) icalpointp∗1 =p∗2 =0.705489(4)=1−ε∗foundforthesite
DPtransitionintheDKPCA[17].
close to the critical point ε∗, with ∆ = ε − ε∗ (cid:62) 0. We Theexponentν canbeobtainedbyplottingtδ(1−ρ (t))
L L (cid:107) L
do not put a subscript ‘L’ on ∆ or the critical exponents to versus t∆ν(cid:107) and tuning ν(cid:107) to achieve data collapse with dif-
lightenthenotation. Foraverylargesystem,relation(1)be- ferent ∆. The collapsed curves shown in Figure 3 were ob-
comes1−ρL(t;∆) ∼ t−β/ν(cid:107)Φ(∆t1/ν(cid:107)),withΦ(x (cid:28) 1) ∼ tained with a combination of central values ε∗ = 0.29451,
constantandΦ(x (cid:29) 1) ∼ xβ. Theinvestigationofthetime- δ =0.157,andν =1.73. Wefoundithardtodiscernvalues
(cid:107)
dependent profiles ρ (t;∆) then allow for the simultaneous ofν bylessthan±0.02. Otherwise, wefoundthedatacol-
L (cid:107)
determination of ε∗ and δ = β/ν , and judicious perusal of lapseverysensitivetothechoiceofε∗; infact,itcouldhave
L (cid:107)
(1)andderivedrelationsfurnishtheotherexponents. beenusedtolocateε∗ withinquitetightbounds. Combining
Figure 1 displays the density profile ρ for an automaton δandν furnishesβ =δν =0.27(1)(orβ =0.268(10)).
L (cid:107) (cid:107)
ofL=8000cellsinthestationarystate. Weactuallyplotthe Thethirdindependentexponentcanbeobtainedbyplotting
densityofinactivecells1−ρ instead,becauseitisthisquan- tδ(1−ρ (t)) versus t/Lz for different L and tuning z until
L L
tity that enters the scaling relation (1). The steep transition data collapse for some z. Since z = ν /ν by definition,
(cid:107) ⊥
aboutε∗ anticipatesasmallvaluefortheexponentβ. Toesti- thisprocedurealsogivesν onceν isknown. Thefinite-size
L ⊥ (cid:107)
mateε∗ moreprecisely,weplot1−ρ (t)closetoε(cid:39)0.294 curvesappearinFigure4. Wefoundz =1.6(1). Thesethree
L
forsomelargeL. Onthecriticalpoint,1−ρ (t) ∼ t−δ and exponents, δ, ν , and z, suffice to determine the universality
L (cid:107)
wecanestimateδbyplottinglog [(1−ρ (t/b))/(1−ρ (t))] class of critical behavior of the model, the other exponents
b L L
against 1/t for some small b. Our data for L = 20000 and followingfromwellknownhyperscalingrelations[16].
b = 10appearinFigure2. Fromthesedatawecouldextract The best values available for δ, ν , β, and z for the
(cid:107)
the estimates ε∗ = 0.29450(5) and δ = 0.155(5). Similar DP process on the square lattice are δ = 0.159464(6),
DP
estimates using 4000 (cid:54) L (cid:54) 16000 confirm these values. ν = 1.733847(6), β = 0.276486(8), and z =
(cid:107)DP DP DP
3
00..9900 1.580745(10)[20]. Thus,withintheerrorbarsourestimates
for these exponents put the critical behavior of Stavskaya’s
modelphasetransitionintheDPuniversalityclass,asitwould
00..8800 beexpectedonthebasisoftheDPconjecture.
δδ
tt
))
ρρ
−−
11
(( 00..7700 IV. SUMMARYANDCONCLUSIONS
We estimated the critical point of Stavskaya’s model at
0.60 ε∗ = 0.29450(5) and found that the model belongs to the
0.0003 0.003 0.03 DPuniversalityclassofcriticalbehavior. Thevalueof1−ε∗
t∆ν is in excellent agreement with the critical point p∗ = p∗ =
(cid:1) 1 2
0.705489(4)forthesiteDPtransitionintheDKPCA[17].
FIG.3. Datacollapseofthescaledtime-dependentdensityprofiles
The estimates of the critical point as well as of the criti-
forε−ε∗ =±0.0001,±0.0002.Theupper(lower)branchescorre-
cal exponents could be improved by larger simulations, but
spondtoε<ε∗(ε>ε∗). Thebestdatacollapsewasobtainedwith
thecentralvaluesε∗ =0.29451,δ=0.157,andν =1.73. we believe that this would be superfluous, since both the lo-
(cid:107)
cationofthecriticalpointofthemodelwithinbetterbounds
thanthoseprovidedbyrigorousandmean-fieldanalysesand
00..7777
thedeterminationofitsuniversalityclassofcriticalbehavior
couldbeestablishedwithinthecomputationaleffortsreported
00..7766
here, namely, afewthousandhoursofCPUtimeonInteli7-
δδtt 00..7755 860processorsrunningGCC/Linuxat2.8GHz.
ρρ)) Takentogether,ournumericalresultsforStavskaya’sPCA
−−
11 00..7744 and the rigorous results existent on its relationship with the
((
generaltheoryofcellularautomataandpercolationprocesses
0.73 provideareasonablycompletecharacterizationofthemodel.
The establishment of its relationship with the DK PCA on
0.72 its site DP line allows the translation of results between the
0.0001 0.001 0.01 0.1 1 two models, with potential benefits for future developments
t/Lz involving either model per se or as approximations (e.g., on
couplingschemes)tomodelsofgreatercomplexity.
FIG.4.Finite-sizedatacollapseofthescaledtime-dependentdensity
profilesonthecriticalpointε∗ =0.29450for4000(cid:54)L(cid:54)20000.
Bestdatacollapsewasobtainedwithδ=0.157(thesamevalueasin
Fig.3)andz =1.6.Noticethatthedataisspreadover(cid:38)3decades ACKNOWLEDGMENTS
andthattherangeinthey-axisisreasonablytight.
The author thanks Prof. Ma´rio J. de Oliveira (IF/USP) for
severalhelpfulconversations.
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