Cluster mean-field approximations with the coherent-anomaly-method analysis for the driven pair contact process with diffusion

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by  Su-Chan Park

Cluster mean-field approximations with the coherent-anomaly-method analysis for the
driven pair contact process with diffusion
Su-Chan Park and Hyunggyu Park
School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
5 (Dated: February 2, 2008)
0
Theclustermean-field approximationsareperformed, upto13 clustersizes, tostudythecritical
0
behavior of the driven pair contact process with diffusion (DPCPD) and its precedent, the PCPD
2
inonedimension. Criticalpointsareestimatedbyextrapolatingourdatatotheinfiniteclustersize

n limit, which are in good accordance with recent simulation results. Within the cluster mean-field
a approximationscheme,thePCPDandtheDPCPDsharethesamemean-fieldcriticalbehavior. The
J
application ofthecoherent anomaly method, however,shows that thetwomodels develop different

1 coherentanomalies,whichleadtodifferenttruecriticalscaling. Thevaluesofthecriticalexponents
3 for the particle density, the pair density, the correlation length, and the relaxation time are fairly
wellestimatedfortheDPCPD.Theseresultssupportandcomplementourrecentsimulationresults
]
for theDPCPD.
h
c
PACSnumbers: 05.70.Ln,02.60.-x,64.60.Ht
e
m
- The absorbing phase transition (APT) has been stud- alongwithafinite-size-scalinganalysis[16]anddirectin-
t
a ied extensively to understand many-body cooperative tegrations of corresponding Langevin equations [17] are
t
s phenomena in nonequilibrium systems [1]. Up to now, two typical examples. Another frequently used method
. two universality classeshave been firmly established: di- is the cluster mean-field (CMF) approximation [18] fol-
t
a rected percolation (DP) and parity conservation (PC) lowed by the coherent-anomaly method (CAM) analysis
m
universality classes [2]. A few other candidates for dif- [19]. This method is known to be effective to obtain a
- ferent universality classes have been reported in recent quantitative phase diagram and sometimes even explore
d
literatures. One is the DP system coupled with a static a true critical scaling behavior [20]. However, the ac-
n
conserved field [3]. Although the reported values for the curate measurement of critical indices is only limited to
o
c critical indices are rather scattered [4, 5, 6], it is widely rathersimple DP systems. More complex criticalbehav-
[ believedthatthesesystemsformauniversalityclass,dif- iors like in the PC and the PCPD classescould not have

ferentfromtheDPandthePCclass. Anothercandidate been probed with a reasonable accuracy as yet by the
2
v is the pair contact process with diffusion (PCPD) that CAM analysis [21, 22].
5 hasas yetdefiedanyconsensusonthe universalityissue. Inthis paper,motivatedby ourrecentresultsthat the
1 Various scenarios have been proposed, including a new DPCPD exhibits a distinct critical behavior from the
1 single universality class [7, 8], a marginally perturbed PCPD and also a mean-field-like behavior even in one
9
DPprocesswithcontinuouslyvaryingexponents[9],and dimension [13], we develop the CMF approximations for
0
a DP process with a huge crossover time [10, 11], which the DPCPD and the PCPD, expecting that the CAM
4
0 are summarized in a recent review [12]. analysis would produce a reasonable estimate for the
/ Recently, we studied the driven PCPD (DPCPD) mean-field-like critical indices of the DPCPD. Also, di-
t
a which is a variant of the PCPD by introducing biased rect comparison of the CMF data for two models may
m diffusion [13]. It is shown that the driving is relevant provide an independent support for different scaling be-
- and the DPCPD exhibits a “mean-field-like” critical be- haviors.
d
havior even in one dimension. Since the DP class is in- We set up dynamic CMF equations up to n = 13
n
sensitive to the driving, the DP scenario with a huge cluster size. The steady-state solutions are obtained
o
c crossovertime should be eliminated. There was a recent withinmachineaccuracyusingmathematica. Dynamic
: attempttounderstandthePCPDusingtherenormaliza- information is also extracted from the smallest eigen-
v
tion group (RG) analysis on a single-species Bosonic ac- valueofthe“linearized”transitionmatrix. Subsequently,
i
X tionderivedfromthemicroscopicmasterequation. How- throughthe CAManalysis,weestimate thevaluesofthe
r ever,itturnedouttobeimpropertodescribethecritical critical exponents for the particle density, the pair den-
a
behavior of the PCPD [14]. In our previous work [13], sity, the correlation length, and the relaxation time.
we pointed outa possible reasonforthis failure andsug-
The model is defined on a one-dimensional lattice of
gested that the PCPD may be described properly by a
L sites with periodic boundary conditions. At each site,
fieldtheorywithtwoindependentfields. Still, thesearch
there is at most one particle and no multiple occupancy
for the coarse-grainedaction adequate for the PCPD re-
is allowed. Hence the configuration is specified by the
mains a challenge.
occupation number which is either 1 or 0 at every site.
Besides the RG technique on the proper action [15], Eachparticle hopsto the right(left) with transitionrate
there are a few other efficient methods to investigate D (D ). The total number of particles in the system
R L
theabsorbingcriticalphenomena. Numericalsimulations variesby branchingand annihilating events mediatedby
2
a particle pair (2A → 3A and 2A → ∅). The transition 10-4
10-8
rate is p (1−p) for the annihilating (branching) event
with 0≤ p ≤ 1. These three dynamics can be described
by the master equation which takes the form 10-9
p
(cid:26)
∂
|P;ti=−Hˆ|P;ti, (1)
∂t s 10-10
(cid:26)
10-6 10-5
where|P;tiisthe state vectorattime t andthe “Hamil-
12
tonian” is written as Hˆ =PL Hˆ with 10-5 p
(cid:0)p
i=1 i
Hˆ =D (ρˆvˆ −aˆ aˆ† )+D (vˆρˆ −aˆ†aˆ )
i R i i+1 i i+1 L i i+1 PSfiragi+r1epla
ements
−p(aˆ aˆ −ρˆρˆ )
i i+1 i i+1 (2)
− 1−p(aˆ† +aˆ† −vˆ −vˆ )ρˆρˆ , 10-7 10-6 10-5
2 i−1 i+2 i−1 i+2 i i+1
12
p
(cid:0)p
where aˆ (aˆ†) is the annihilation (creation) operator of
hard corie piarticles, satisfying {aˆ ,aˆ†} = 1 and [aˆ ,aˆ ] = FIG. 1: Log-log plot of ρs vs p1c2−p obtained from the 12-
i i i j clusterCMFapproximationfortheDPCPD.Theslopeofthe
[aˆi,aˆ†j] = 0 for i 6= j, ρˆi = aˆ†iaˆi is the number operator, straightlineis1. Intheinset,ρp’sareplottedagainstp1c2−p.
and vˆi =1−ρˆi. The slope of thestraight line is 2.
Three different cases arise depending on the values of
D and D . The case of D = D = 0 represents the
R L R L
paircontactprocess(PCP)whichhasinfinitelymanyab- ThestationaryprobabilitydistributionfunctionPs(ρ)
n
sorbingstates andis knownto belong to the DP class at can be obtained by solving the set of coupled equa-
least for static situations [23]. Since the cluster approxi- tions Fρ = 0. For given n, the number of equations
mations alongwith the CAM analysis havebeen already and the number of variables are both 2n, but not all
performedforthePCPbyseveralauthorspreviously[24], are independent. The translational invariance and the
we skip the analysis of the PCP here. The PCPD corre- normalization condition guarantee that all Pn(ρ) with
sponds to DR = DL 6= 0 and the DPCPD corresponds ρ1 = 0 can be expressed in a linear combination of
to DR 6= DL. In what follows, we set DR + DL = 1 Pn(ρ)’s with ρ1 = 1. For example, in case of n = 4, one
for convenience and DR is chosen to be 1/2 (1) for the can easily show that P4(0011) = P3(011)−P4(1011) =
PCPD (DPCPD). P2(11) − P3(111)− P4(1011) = P3(110)− P4(1011) =
We consider an n-site probability function Pn(ρ;t). It P4(1101)+ P4(1100)− P4(1011). The absorbing (vac-
is defined as the probability at time t to find an n-site uum)probabilityP4(0000)canbedeterminedbythenor-
cluster of the configurational state ρ = (ρ ,ρ ,...,ρ ), malization condition. Hence the DPCPD case has 2n−1
1 2 n
where an occupational state ρ at site i takes either 0 or independent variables. In case of the PCPD, the left-
i
1. Tracing out Eq. (1) over occupational states outside right symmetry further reduces the number of indepen-
the cluster ({ρi} with i ≤ 0 or i ≥ n+1), one may find dent variables, for example, P4(1101)=P4(1011) and so
a formal exact expression on.
We use mathematica to find the stationarysolutions
dPnd(tρ;t) =F˜ρ(Pn,Pn+1,Pn+2), (3) Pginsv(eρn)vuapluetsoonf p=ar1a3mwetitehrs,mpacahnidneDacc[u2r5a].cyW(i1t0h−P20s),fwoer
R n
calculate the particle density ρ and the pair density ρ
where the function F˜ involves the sets of n-, (n+1)-, s p
ρ in the steady state as
and (n+2)-site probability. Notice that P and P
n+1 n+2
terms show up due to the boundary dynamics of the n- ρ = Ps(1)=Tr Ps(ρ)δ ,
s 1 ρ n ρk,1
site cluster. ρ = Ps(11)=Tr Ps(ρ)δ δ , (5)
As the infinite hierarchy appearing in Eq. (3) is the p 2 ρ n ρk,1 ρk+1,1
major obstacle towards analytic treatment, we need an where k (and k+1) denotes an arbitrary site inside the
approximationschemetotruncate the hierarchyatfinite cluster. At a fixed value of D , the order parameters,
R
n. In this paper, we take the so-called (n + 1,n) ap- ρ andρ ,simultaneouslyvanishforlargep(pairannihi-
s p
proximations [18], where P and P are expressed lation rate) and the system exhibits an absorbing phase
n+2 n+1
in terms of products of P ’s. Then, the rate equations transition into vacuum at p=pn.
n c
for the n-site cluster probability function become Near the transition point pn, the order parameters
c
scale as
dP (ρ;t)
n
dt =Fρ({Pn}), (4) ρs ≃An(pnc −p)β1MF, ρp ≃Bn(pnc −p)β2MF, (6)
where P is now the approximate (mean-field) probabil- wherewe findthe mean-fieldvalues for the orderparam-
n
ity function. eter exponents: βMF = 1 and βMF = 2. Figure 1 shows
1 2
3
0.1
TABLE I: Cluster approximation results for the PCPD
model. The errors are in the last digits.
n pnc An Bn Cn
4 0.209 692 7263 4.473 51.855 17.59
5 0.194 357 9912 4.720 72.928 19.15
n
6 0.184 167 8676 4.859 93.789 19.93 (cid:1)
7 0.177 119 7696 4.963 116.26 20.66
8 0.171 815 3824 5.039 139.91 21.22
9 0.167 700 6591 5.100 165.04 21.72
10 0.164 396 9333 5.151 191.63 22.17 DPCPD
PCPD
11 0.161 685 1815 5.194 219.71 22.58
12 0.159 416 2244 5.232 249.28 22.96 0.01
13 0.157 488 7140 5.265 280.35 PSfragrepla
ements 0.08 0.1 0.2 0.3
1=n
then=12clusterapproximationresultsfortheDPCPD. FIG. 2: Log-log plots of ∆n vs 1/n for the PCPD and
We estimate the critical point pn and the critical ampli- the DPCPD. In this figure, the value of pc for the PCPD
c (DPCPD) is set tobe 0.1335 (0.153).
tudes A and B by fitting five data near the transition
n n
(|p−pn|≤5×10−6),linearlyforρ andquadraticallyfor
c s
ρp. Our results are tabulated in Table I for the PCPD exponential form. However, this method does not pro-
modelandinTableIIfortheDPCPDmodel. Noticethat duce high-precision data. In this paper, we propose a
the relative errors for pnc are extremely small (∼ 10−9), differentmethodtocalculateτ withmachineaccuracyin
buttheamplitudesAnandBnstillhaveasizablerelative the CMF approximationscheme.
error (∼10−4). Sincethe stationarysolutionsofEq.(4)wereobtained
It is interesting to note that, for n ≤ 3, the diffusion with machine accuracy,we canlinearize Eq.(4) near the
bias does not enter the CMF rate equations at all. The stationarysolutionsveryaccurately. Thelinearizedequa-
functionalformofFρ({Pn})inEq.(4)isidenticalforthe tion takes the form
PCPDandthe DPCPD.Theleft-rightsymmetryamong
d|P ;ti
Pn(ρ)’sisautomaticallyenforcedduetothetranslational n =−M|Pn;ti, (7)
invariance,regardlessof the details of the dynamics. For dt
example, P (110) = P (11)−P (111) = P (011) and so
3 2 3 3 where|P ;tiisthe(n-cluster)statevectorwiththecom-
n
on. However,forn≥4,thetranslationalinvariancedoes
ponentsP (ρ;t)andM isasquarematrix. Itistrivialto
n
notguaranteetheleft-rightsymmetry,whichmaybebro-
show that the eigenvalues of M are equal to the inverse
ken by the dynamics with a broken left-right symmetry.
ofvariouscharacteristictimescalesofthedynamics. The
Innonequilibriumsystems,dynamicrelaxationbehav-
most dominant slow mode is determined by the smallest
ior provides one of the key pieces of information on the eigenvalue Λ , i.e., the relaxation time τ =Λ−1.
s s
system. Offcriticality,theorderparametersareexpected
Weanalyzethelinearizedequationupton=12. Near
to approach their stationary values exponentially with a
criticality, we find
characteristicrelaxationtimeτ. Atcriticality,τ diverges
and the order parameters decay algebraically. One may τ−1 ≃Cn(pnc −p)νkMF, (8)
roughly estimate τ by numerically integrating the rate
equations (4) and fitting time-dependent data into an where we find again the mean-field value for the relax-
ationexponent,νMF =2. Weestimatethecriticalpoints
k
pn independently, which are found to be consistent with
c
previousestimates fromthe density data in Tables I and
TABLE II: Cluster approximation results for the DPCPD
model. The errors are in the last digits. II, where we also tabulate the estimated values for the
amplitude C for both the PCPD and the DPCPD.
n pnc An Bn Cn Now,weemnploythecoherent-anomalymethod(CAM)
4 0.216 140 3513 4.254 44.37 16.69
introduced by Suzuki and co-workers [19] and estimate
5 0.202 800 9465 4.356 56.08 17.31
the values of the true critical exponents. Following the
6 0.194 381 7410 4.405 66.48 17.55
CAM analysis, the n dependence of the critical point pn
7 0.188 503 1907 4.423 76.39 17.74 c
is predicted in the large n limit as
8 0.184 102 4774 4.424 85.76 17.83
9 0.180 689 8311 4.420 94.89 17.90
∆ν⊥ ∼n−1, (9)
10 0.177 954 3360 4.410 103.7 17.94 n
11 0.175 711 7674 4.397 112.3 17.97
where ∆ = pn−p is the distance of pn from the true
12 0.173 837 8803 4.383 120.8 17.99 n c c c
criticalpoint p =lim pn andν is the true correla-
13 0.172 247 8976 4.368 128.9 c n→∞ c ⊥
tion length exponent.
4
100
100
7
n n 4.4
B 5 B
n n
A A 4.3
3 4.2
0.03 0.050.07 0.03 0.05 0.07
PSfragrepla
ements (cid:1)n PSfragrepla
ements (cid:1)n
10 10
0.01 0.1 0.01 0.1
(cid:1)n (cid:1)n
FIG. 3: The CAM analysis for the order parameters for the FIG. 4: The CAM analysis for the order parameters for the
PCPD model. The slope of the straight line is −1.51, which DPCPDmodel. Theslopeofthestraightlineis−0.92,which
leads to β2 ≈0.49. Inthe inset, theslope of the straight line leads to β2 ≈1.08. In the inset, An vs ∆n is drawn without
is −0.084, which leads to β1 ≈0.92. a log scale. An remains nearly constant which implies β1 ≈
β1MF =1.
We estimate p by applying the Bulirsch and Stoer
c
(or BST) algorithm [26] to the series of {pn} and find
c Onthe other hand, the CAM analysisfor the DPCPD
that p = 0.134(2) for the PCPD and p = 0.154(3) for
c c looks consistent with the simulation results. In Fig. 4,
the DPCPD, which are in good agreement with simula-
we use p = 0.151 032 [13]. First, A seems not diverg-
c n
tion results of 0.133 522(2) and 0.151 032(1) [13]. Al-
ing as ∆ → 0 and reaching a nonzero constant, which
n
ternatively, we estimate pc and ν⊥ simultaneously using implies β1 = β1MF = 1. Second, Bn behaves very differ-
Eq. (9). In Fig. 2, we plot ∆ vs 1/n in a log-log plot,
n ently from A and diverges with the exponent ∼ 0.92,
n
varyingpc to findthe bestpower-lawfit. Forthe PCPD, which implies that β = βMF −0.92 ≈ 1.08. Numerical
2 2
the choice of p = 0.1335 yields the smallest fitting er-
c simulation results [13] are in complete agreement with
ror with ν = 1.04, where the data from n = 8 to 13
⊥ our CAM results. One should notice that β does not
2
are used. For the DPCPD, the best choice is p =0.153
c assume the MF value, but seems to be equal to β ex-
1
with ν = 1.01. The relative error for p is ∼ 2%, and
⊥ c cept a probable multiplicative logarithmic correction as
the error for ν is ∼10%. The best simulation result of
⊥ found in the exponent β/ν by numerical simulations
||
ν =1.09(2) for the PCPD is within the errors, but the
⊥ [13]. This mean-field-like behavior is expected for the
accuratemeasurementseemstobeoutofreachwithdata
two-dimensional PCPD [28], of which the upper critical
upton=13. Ourestimateofν =1.01fortheDPCPD
⊥ dimensionisbelievedtobe2. OurCAMresultsindepen-
is in very good agreement with the expected mean-field
dently supportthe conclusiondrawnfrom our numerical
value νMF =1.
⊥
The amplitudes A and B are expected to scale as
n n
A ∼∆−(β1MF−β1), B ∼∆−(β2MF−β2), (10) 25 DPCPD
n n n n
PCPD
where β and β are the true critical exponents for the
1 2
order parameters. In Fig. 3, the log-log plots of A and
n
B vs ∆ for the PCPD are presented. Here we use
n n
20
p =0.133522(thebestestimatefromMonteCarlosim-
c n
C
ulations)[13]. The CAM analysisleads to β ≈0.92 and
1
β ≈ 0.49 [27], both of which are far from the simula-
2
tion results of β ≈ β ≈ 0.36(2) [8, 9]. In particular,
1 2
thereisahugediscrepancybetweentheestimatedvalues
of β and β by the CAM analysis, which warns us that
1 2
the CAM estimates for the order parameter exponents 15
should be interpreted with greatcaution. This huPSgfreagdriespl-a
ements 0.02 0.05 0.08
crepancy also implies that the cluster sizes up to n=13
(cid:1)n
are still too small for the PCPD to reachthe asymptotic
regimewherethesystemisdominatedbylongspatialcor- FIG. 5: The CAM analysis for the relaxation time for the
PCPDandtheDPCPD.Theslopeofthestraightlineforthe
relations,inducedbythelong-termmemorymediatedby
PCPD(DPCP)is−0.2(−0.02)whichleadstoν =1.8(1.98).
solitary particles [9]. k
5
simulations results [13] that the DPCPD critical behav- to the mean-field value of νMF = 2, consistent with the
k
ior is distinct from the PCPD behavior and the upper simulation results.
critical dimension for the DPCPD is 1 rather than 2.
Finally, we estimate ν from the relation
k In summary, we estimated the critical exponents for
the PCPDandthe DPCPD, using CMF approximations
C ∼∆−(νkMF−νk), (11) along with the CAM analysis. For the PCPD, the val-
n n
ues of the order parameter exponents are poorly esti-
where ν is the true critical exponent for the relaxation mated, while the estimates for the correlation and the
k
time. InFig. 5, the log-logplots ofC vs ∆ areshown. relaxation exponents are consistent with simulation re-
n n
We estimate that ν ≈ 1.8 for the PCPD and ν ≈ 1.98 sults within error bars. In contrast, the CAM estimates
k k
for the DPCPD. Rather surprisingly, the PCPD result for the DPCPD are in excellent accord with simulation
is consistent with the simulation result of ν = 1.85(10) results,supportingourconjecturethatthe upper critical
k
[8, 9]. For the DPCPD, the value of ν is quite close dimension of the DPCPD is 1.
k
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