Nonequilibrium phase transition on a randomly diluted lattice

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by  Thomas Vojta

Nonequilibrium phase transition on a randomly diluted lattice
Thomas Vojta and Man Young Lee
Department of Physics, University of Missouri-Rolla, Rolla, MO 65409
(Dated: February 2, 2008)
6 We show that the interplay between geometric criticality and dynamical fluctuations leads to a
0 novel universality class of the contact process on a randomly diluted lattice. The nonequilibrium
0
phase transition across the percolation threshold of the lattice is characterized by unconventional
2
activated (exponential) dynamical scaling and strong Griffiths effects. We calculate the critical

n behavior in two and three space dimensions, and we also relate our results to the recently found
a infinite-randomnessfixed point in thedisordered one-dimensional contact process.
J

7 PACSnumbers: 05.70.Ln,05.50.+q,64.60.Ak,02.50.Ey
2

] Nonequilibriumsystemscanundergocontinuousphase vives on the infinite percolation cluster for λ > λ∗. The
h
transitions between different steady states. These tran- (multi)criticalbirthrateλ mustbesmallerthanthecrit-
c ∗
e sitions are characterized by collective fluctuations over icalbirthrateoftheone-dimensional(1D)contactprocess
m large distances and long times similar to the behavior of because the critical percolation cluster is connected, in-
equilibrium critical points. Examples can be found in finitely extended, and its fractal dimension is D > 1.
- f
t population dynamics and epidemics, chemical reactions, For p>p , no active phase can exist because the lattice
a c
t growing surfaces, and in granular flow and traffic jams consistsofdisconnectedclustersoffinite sizethatdonot
s
. (for recent reviews see, e.g., Refs. [1, 2, 3, 4, 5, 6]). support a steady state density of active sites.
t
a If a nonequilibrium process is defined on a randomly The contact process on a site-diluted lattice therefore
m diluted spatial lattice, its dynamical fluctuations coex- hastwononequilibriumphasetransitions,separatedbya
- ist with geometric fluctuations. Site or bond dilution multicriticalpoint. Forp<pc,thetransition(markedby
d
defines a percolation problem for the lattice with a geo- “a”inFig.1)isexpectedtobeintheuniversalityclassof
n
o metric phase transition at the percolation threshold [7]. the generic disordered contact process [12, 13, 14] which
c In this Letter we address the question of how the inter- has reattracted considerable attention recently [15, 16].
[ playbetweengeometriccriticalityduetopercolationand In contrast, the phase transition across the percolation

2 dynamical fluctuations of the nonequilibrium process in- threshold pc of the lattice for λ > λ∗ (transition “b” in
v fluences the properties of the phase transition. Fig. 1) has received much less attention.
5 In this Letter, we show that the interplay between
Our starting point is the contact process [8], a proto-
6 geometric criticality and dynamic fluctuations leads to
typical system exhibiting a nonequilibrium phase transi-
0
a novel universality class for this nonequilibrium phase
1 tion. It is defined on a d-dimensional hypercubic lattice
transition. Even though the transition is driven entirely
1 (d≥2). Eachsite can be active (occupied by a particle)
5 or inactive (empty). In the course of the time evolution, by the geometry of the lattice, the dynamical fluctua-
0 tions of the contact process enhance the singularities in
active sites infect their neighbors, or they spontaneously
/ allquantitiesinvolvingdynamiccorrelations. Ourresults
t become inactive. Specifically, the dynamics is given by
a
can be summarized as follows. The dynamical scaling is
m acontinuous-timeMarkovprocessduringwhichparticles
are created at empty sites at a rate λn/(2d) where n is
-
d thenumberofactivenearestneighborsites. Particlesare λ−1
n annihilatedatunitrate. Forsmallbirthrateλ,annihila- a
o λ −1 Inactive
tiondominates,andtheabsorbingstatewithoutanypar- c0
c
ticles is the only steady state (inactive phase). For large
:
v birth rate λ, there is a steady state with finite particle
i
X density (active phase). The two phases are separated by MCP
λ−1
r a nonequilibrium phase transition in the directed perco- ∗ Active
a lation [9, 10] universality class at some λ=λ0.
c b
We introduce quenched site dilution [11] by randomly
removing lattice sites with probability p. The result- 0 P P
c
ing phase diagram of the site-diluted contact process is
sketched in Fig. 1. For small impurity concentrations
FIG. 1: Schematic phase diagram of a site diluted contact
below the percolation threshold of the lattice, p < pc, processasfunctionofimpurityconcentrationpandbirthrate
the active phase survives, but the critical birth rate in- λ. There is a multicritical point at p = pc and λ = λ∗. The
creaseswithp(tocompensateforthemissingneighbors). phase transition (b) across the percolation threshold of the
Right at the percolation threshold the active phase sur- lattice is thetopic of this Letter.
2
not of conventional power-law form but activated, i.e., We nowstudy the time dependence ofthe density ρ(t)
the relation between correlation length ξ and correla- of active sites, starting from a completely active lattice.
⊥
tion time ξ is exponential, We first consider the contact process on a single per-
k
colation cluster of finite size (number of sites) s. For
lnξ ∼ξψ (1) λ > λ such a cluster is locally in the active phase. It
k ⊥ ∗
therefore has a metastable state with a nonzero density
with the critical exponent ψ being equal to the fractal
of active sites. This metastable state can decay into the
dimensionofthecriticalpercolationcluster,ψ =D . As
f inactivestateonlyviaararecollectivefluctuationinvolv-
a result, the long-time decay of the density ρ of active
ingallsitesofthecluster. Theprobabilityforsucharare
sites at p=p is ultra-slow,
c event decreases exponentially with the size s of the clus-
ρ(t)∼[ln(t/t )]−δ¯ . (2) ter. Therefore, the life time ts of the metastable active
0
state ona cluster increasesexponentially withits sizes,
The exponent δ¯ = β /(ν D ) is determined by D to-
c c f f t (s)∼t eA(λ)s , (7)
gether with the order parameter and correlation length s 0
exponents, β and ν , of the lattice percolation transi-
c c where t is some microscopic time scale. The prefactor
0
tion [17]. In contrast to the enhanced dynamical singu-
A(λ) vanishes at the multicritical point, A(λ )=0, and
∗
larities,the exponents ofstatic quantities likethe steady
increases with increasing λ. The number of sites s of a
statedensityρ andthespatialcorrelationlengthξ are
st ⊥ percolation cluster is connected to its linear size R via
s
identicalthecorrespondinglatticepercolationexponents, s∼RDf. Therefore,(7)establishestheexponentialrela-
s
ρst(p) ∼ |∆|βc (∆<0) , (3) tion betweenlength and time scales,lnts ∼RsDf leading
to activated dynamical scaling (1).
ξ ∼ |∆|νc . (4)
⊥
After having analyzed a single cluster we turn to the
where ∆ = p−p measures the distance from the per- full percolation problem. From classical percolation the-
c
colation threshold. Off criticality, i.e., away from the ory, we know that close to pc the number ns of occupied
percolationthreshold,wefindstrongGriffithseffects[18] clusters of size s per lattice site (excluding the infinite
characterized by a non-exponential density decay [12], cluster for p<pc) obeys the scaling form
ρ(t) ∼ (t/t )−d/z′ (p>p ) (5) n (∆)=s−τcf s∆1/σc . (8)
0 c s
(cid:16) (cid:17)
ρ(t)−ρ ∼ e−[(d/z′′)ln(t/t0)]1−1/d (p<p ) , (6)
st c The scaling function f(x) behaves as
where the nonuniversal exponents z′ and z′′ diverge as
f(x) ∼ exp(−B x) (p>p ) (9)
z′,z′′ ∼ ξDf for p → p . In the remainder of the Letter 1 c
⊥ c f(x) = const (p=p ) (10)
wesketchthederivationoftheseresultsandcalculateex- c
ponentvalues andadditionalobservables. We alsorelate f(x) ∼ exp[−(B2x)1−1/d] (p<pc). (11)
our results to the disordered 1D contact process [15, 16]
where B ,B are constants. The exponents τ and σ
and to the diluted quantum Ising model [19]. 1 2 c c
determine all critical exponents of the percolation tran-
Let us start by considering the steady state density
sition of the lattice including the correlation length ex-
ρ ofactivesites(i.e., the orderparameterofthe transi-
st ponent ν = (τ −1)/(dσ ), the order parameter ex-
tion). Anonzerosteadystatedensitycanonlydevelopon c c c
ponent β = (τ − 2)/σ , and the fractal dimension
theinfinitepercolationcluster;finiteclustersdonotcon- c c c
D =d/(τ −1) of the percolating cluster [7].
tributebecausetheyeventuallygointotheinactivestate f c
In order to obtain the total density of active sites for
via a rare fluctuation. For λ > λ , the infinite cluster
c the contact process on the diluted lattice, we sum the
is in the active phase. The total steady state density is
numberofactivesitesoverallpercolationclusters. Com-
proportionaltothenumberofsitesintheinfinitecluster,
bining the cluster size distribution (8) with the lifetime
ρst ∼ P∞(p) ∼ (pc −p)βc. The order parameter expo- of the metastable active state (7) leads to
nent of the nonequilibrium transition is therefore identi-
caltothatofthelatticepercolationtransition,β =β ,as
c
ρ(t,∆) ∼ ds s n (∆)exp[−t/t (s)] (12)
statedin(3). Todeterminethespatialcorrelationlength s s
Z
ξ we note that the correlations of the contact process
⊥
cannotextendbeyondtheconnectednesslengthξ ofthe Right at the percolation threshold, this reduces to
c
percolating lattice because sites on different percolation
clustersaredecoupled. Ontheotherhand,forλ>λ∗,all ρ(t,0)∼Z ds s1−τcexp[−(t/t0eAs)] . (13)
sites onthe same cluster are stronglycorrelatedin space
even though they collectively fluctuate in time. We thus The leading behavior of this integral can be found by
conclude ξ ∼ξ and ν =ν in agreement with (4). noticing that only islands with size s > s (t) =
c c min
3
A−1ln(t/t0) contribute at time t. The critical long-time Exponent d=2 d=3
dependence of the total density is thus given by β =βc 5/36 0.417
ρ(t,0)∼[ln(t/t )]2−τc (p=p ). (14) ν =νc 4/3 0.875
0 c ψ=Df =d−βc/νc 91/48 2.523
This completes the derivation of (2) with the critical ex- δ¯=βc/(νcDf) 5/91 0.188
ponent δ¯given by δ¯= τ −2 = β /(ν D ) in agreement
c c c f
with general scaling arguments [15, 16]. TABLE I: Critical exponents of the nonequilibrium phase
We now consider the behavior of the density off criti- transition at p=pc in two and three space dimensions.
cality. Intheinactivephase,p>p ,thetimedependence
c
of the density is given by
Evaluating this integral analogously to the time-
dependent density (12) yields, for small fields h,
ρ(t,∆)∼ dss1−τcexp[−B s∆1/σc −(t/t eAs)]. (15)
1 0
Z
ρ (h) ∼ [ln(h /h)]−δ¯ (p=p ), (20)
For long times, the leading behavior of the integral can st 0 c
be calculated using the saddle-point method, giving ρst(h) ∼ (h/h0)d/z′ (p>pc), (21)
ρ(t,∆)∼t−(B1/A)∆1/σc (p>p ) (16) ρst(h) ∼ exp[−(d/z′′)ln(h0/h)]1−1/d (p<pc)(,22)
c
equivalent to (5). The nonuniversal exponent z′ is given whereh0 ∼1/t0. Atp=pc,therelationbetweendensity
by z′ =(Ad/B1)∆−1/σc ∼ξ⊥Df. and field is logarithmic,as expected from activated scal-
ing. Offcriticality,wefindstrongGriffithseffectssimilar
In the active phase, p < p , there is a nonzero steady
c
to those in the time-dependence of the density.
state density ρ coming from the infinite percolation
st
The above results can also be derived from a scaling
cluster. However, the approach of the density towards
theory. In the active phase, the density is proportional
this value is still determined by the slow decay of the
to the number of sites in the infinite percolation cluster.
metastable states of the finite percolation clusters
Thus, its scale dimension must be β /ν . Time must
c c
ρ(t,∆)−ρst(∆)∼ (17) enter via the scaling combination ln(t)bDf reflecting the
1−1/d exponentialdependenceofthelifetime(7)onthecluster
∼ dss1−τcexp − B s|∆|1/σc −(t/t eAs)
Z (cid:20) (cid:16) 2 (cid:17) 0 (cid:21) size. The field h, being a rate, scales like inverse time.
We therefore obtain the following scaling form:
Using the saddle point method to calculate the leading
long-time behavior gives (for p<pc) ρ[∆,ln(t),ln(1/h)]=bβc/νcρ[∆b−1/νc,ln(t)bψ,ln(1/h)bψ]
(23)
ρ(t,∆)−ρst(∆)∼e−[(B2/A)|∆|1/σcln(t/t0)]1−1/d . (18) wherebisanarbitrary(length)scalefactorandψ =Df.
This form is consistent with all our explicit results.
This completes the derivation of (6) with z′′ =
All critical exponents of the nonequilibrium phase
(Ad/B2)|∆|−1/σc ∼ ξ⊥Df. The non-exponential off- transitionaredeterminedbythe classicalpercolationex-
critical relaxationof the density (5,6) is characteristicof
ponents of the lattice. In two space dimensions, their
aGriffithsregioninthe contactprocess[12,18]. We also
values are known exactly and in three dimensions they
point out that time and spatial correlation length enter
are known numerically with high accuracy [7]. Table I
theseequationsintheformofthecombinationln(t)/ξDf
⊥ shows numerical exponent values for these cases.
again characteristic of activated scaling.
We also briefly comment on the early time behavior.
We now turn to the influence of an external source
Forλ>λ eachpercolationclusterislocallyintheactive
∗
field h that describes spontaneous particle creation at a
phase. Startingfromasingleactiveseed,thecloudofac-
rate h ateachlattice site. To determine the steady state
tive sites thus initially growsballistically, i.e., the radius
density as a function of h we again start by considering
of the cloud grows linearly with time, until a metastable
a single percolation cluster of size s. For λ > λ , the
∗ state is reached in which it covers the entire percolation
cluster is active if at least one particle has been spon-
cluster. Thetimerequiredforthisinitialspreadingonan
ttaimneeouts(lsy)c=reatteedAso.n Foonre somfatlhlehs, tshiteesavweirtahgine nthuembliefer islandofsizesisti(s)∼Rs ∼s1/Df. Asdiscussedabove,
s 0 themetastablestatedecaysonlyatthemuchlargertime
of particles created on a cluster of size s within time ts scale t (s) ∼ eAs. We thus arrive at the somewhat sur-
is M (h) = hst = hst eAs. If M > 1, the cluster is s
s s 0 s prising conclusion that the early time behavior of the
(almost) always active. If M < 1, it is active with a
s contactprocessonourdilutedlatticeismuchfasterthan
probability proportional to M . The total steady state
s the logarithmically slow long-time decay of the density.
density is obtained by summing over all clusters
Intheremainingparagraphs,wediscussthegenerality
of our results, compare them to the transition in the di-
ρ (h,∆)∼ ds s n (∆)min[1,M (h)] . (19)
st Z s s lutedquantumIsingmodel[19]andtotherecentlyfound
4
infinite-randomnesscriticalpointinarandom1Dcontact that peculiar disorder effects may be responsible for this
process [15, 16]. We also compare to a generalclassifica- in at least some of the experiments.
tion of phase transitions with quenched disorder [20].
This work has been supported in part by the NSF
The logarithmic time and field dependencies (2) and
under grant nos. DMR-0339147 and PHY99-07949, by
(20) at the nonequilibrium phase transition at p = p
c Research Corporation and by the University of Missouri
as well as the strong Griffiths effects in its vicinity are
ResearchBoard. Partsofthis workhavebeenperformed
the direct result of combining the spectrum of percola-
at the Aspen Center for Physics and the Kavli Institute
tion cluster sizes (8) with the exponential dependence
for Theoretical Physics, Santa Barbara.
(7) of the life time on the cluster size. We therefore
expect similar behavior in other diluted equilibrium or
nonequilibrium systems that share this exponential rela-
tion betweenlength and time scales. One example is the
diluted transverse field Ising model [19]. In this system, [1] B. Schmittmann and R.K.P. Zia, in Phase transitions
the energy gap of a cluster decreases exponentially with and critical phenomena, edited by C. Domb and J.L.
itssize. Asaresult,the scalingbehavioratthequantum Lebowitz, Vol. 17, (Academic, New York 1995), p. 1.
[2] J. Marro and R. Dickman, Nonequilibrium Phase Tran-
phase transition across the percolation threshold of the
sitions in Lattice Models (Cambridge University Press,
lattice is very similar to the one found in this paper.
Cambridge, England, 1996).
Recently, the critical point of the 1D contact pro- [3] R. Dickman, in Nonequilibrium Statistical Mechanics in
cess with spatial disorder was found to be of infinite- One Dimension,edited byV.Privman (Cambridge Uni-
randomness type [15, 16]. Analogous behavior is ex- versity Press, Cambridge, England, 1997), p. 51.
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p inthesite-dilutedcontactprocess(transitionainFig.
c Phys. A 38, R79(2005).
1). Ourcriticalpointatthe percolationthresholdshares
[7] D. Stauffer and A. Aharony, Introduction to Percolation
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Z. Phys. B 47, 365 (1982).
Moreover, the early time behavior is different (logarith-
[11] Our percolating lattice is externally given. This differs
mically slow at the generic infinite randomness critical
from the general epidemic process with immunization
point but of power-law type at our transition).
wherethenonequilibriumprocessitselfdynamicallygen-
Lastly, we point out that our results are in agreement erates a percolation cluster of inaccesible sites [26].
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(1998).
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activatedscalingaccompaniedbystrong,power-lawGrif- (2005).
fiths effects. This case is realized in random transverse [17] In the following, lattice percolation exponents carry a
subscript c while the exponents of the nonequilibrium
field Ising magnets [19, 21] as well as in our diluted con-
transitionofthecontactprocessdonotcarryasubscript.
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on a diluted lattice has unusual properties. The inter-
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