Absence of the link between self-organized criticality and deterministic fixed energy sandpiles

0.12 MB·English

by  Su-Chan Park

Absence of the link between self-organized criticality and deterministic fixed energy
sandpiles
Su-Chan Park
Institut fu¨r Theoretische Physik, Universita¨t zu K¨oln, Zu¨lpicher Str. 77, 50937 K¨oln, Germany
(Dated: January 19, 2010)
Self-organized criticality (SOC) observed in an abelian sandpile model (ASM) has been believed
to be related to the absorbing phase transitions (APT) exhibited by a deterministic fixed energy
0
sandpile (DFES). We critically investigate the link between the SOC and the APT exhibited by
1
the DFES. In contrast to the widespread belief, a phase transition in the DFES is shown not to
0
bedefineduniquelybuttodependon initial conditions becauseof thenon-ergodicity of theDFES.
2
Furthermore,weshowthataphasetransition inthespreadingdynamicsoftheDFESisintimately

n related to thepercolation rather than the avalanche dynamics of the ASM. These results illustrate
a that the SOC exhibited by the ASM has nothing to do with phase transitions in the DFES. We
J discuss theimplication of ourresult to stochastic models.

9
1 PACSnumbers: 05.70.Fh,05.07Ln, 64.70.qj,64.60.ah

]
h
The concept of self-organized criticality (SOC) coined Asaneasycounterexample,itisclearthattheavalanche
c
by Bak, Tang, and Wiesenfeld (BTW) [1] has been dynamics of the one-dimensional BTW automata [19] is
e
m invoked to describe the ubiquity of criticality in na- completelydifferentfromdynamicsattransitionpointof
ture [2, 3]. Sandpile models with either determinis- its FES version [20]. Although the ASM and the DFES
-
t tic [1, 4] or stochastic [5] toppling rule have played in the one-dimensionalchain do not share criticality, the
a
t the role of Ising model in understanding SOC theoret- SOC critical density is identical to the transition point
s
. ically [6, 7]. Although sandpile models were devised of the DFES [19, 20]. Accordingly, it seems still valid
t
a as prototypes with the spontaneous emergence of crit- to anticipate that the mean energy in an ASM hovers
m icality without fine-tuning of any parameters, which is around the phase transition point ζ of the correspond-
c
- the characteristics of SOC [1], the similarity of SOC to ing DFES by dissipation (when activity is too strong)
d
standard (nonequilibrium) phase transitions had been and driving (when activity is absent) and that in the
n
discussed immediately after the emergence of the con- thermodynamic limit the meanenergy ζ of the ASM at
o S
c cept [8, 9]. The connection between SOC and standard criticality is identical to ζc. Meanwhile, the ASM on a
[ phase transitions, especially absorbing phase transitions one-dimensionalchainis rather pathologicalin the sense

(APTs) [10, 11], has been materialized by defining a that the probability of observing any finite avalanche in
1
v ‘fixed energy’ ensemble of the sandpile model, or a fixed the thermodynamic limit is zero [19]. Hence, one might
9 energy sandpile (FES) [12–15]. The idea of the FES has expect that except such pathological cases the program
5 elicited fruitful conclusions at least for stochastic sand- to associatethe SOCwith the DFES mayturnoutto be
3
pile models; the FES version of the stochastic sandpile successful.
3
model [5] is argued to form a universality class together
. Recently,however,eventhisexpectationhasbeenchal-
1 with other APT models [13], which, in turn, has moti-
lenged [18]. By measuring ζ in the thermodynamic
0 c
vated to pave “paths to self-organized criticality” from
0 limit which is preceded by the infinite time limit, Fey,
conventional phase transitions [16, 17].
1 Levine, and Wilson [18] argued that ζ is different from
c
: DespitethetriumphoftheideaoftheFES,skepticism ζ for many different models. Because phase transitions
v S
i againstthe SOC-FESlink has been unabated[6, 17,18]. innonequilibriumsystemsareusuallystudiedinthether-
X
In comparison to the achievement in the analytic study modynamiclimitfollowedbynotprecededbytheinfinite
r of deterministic sandpile models, also known as abelian time limit, the method employed in Ref. [18] might be
a
sandpile models (ASMs) [4], rigorousanalysis in favorof considered inconsistent with the philosophy of the FES.
the link between SOC and the FES still lacks. In this Then can we restore the link between the SOC and the
context, it is not surprising that the authors of Ref. [12] DFES if appropriate order of limits are taken?
also worried about the possibility that the FES may ex-
To answerthe above question, let us begin with defin-
plore transient rather than recurrent (SOC) configura-
ing the DFES. In the DFES on a graph with V ver-
tions. Thus,afirststeptoremovesuchskepticismwould
tices, each vertex starts with nonnegative integer energy
be to cement the link between the SOC in the ASM and
z (i = 1,...,V). A configuration change is governed
i
its FES version or deterministic FES (DFES). The pur-
by a V ×V integer matrix F which has properties such
poseofthis workistoinvestigateiftheabove-mentioned
that F ≤ 0 for i 6= j, F = 0 (energy conserva-
ij j ij
program can be accomplished.
tion),andforanypairi,jPofverticesthere isa sequence,
However, this program does not seem fully successful. say a connecting sequence, of vertices with finite length
2
l+2 (k = i,k ,...,k = j) such that F < 0 should be C after all vertices topple exactly once be-
0 1 l+1 kmkm+1 m
for all 0≤m≤l. Except the energy conservation, these cause adding energy of amount L to all vertices of a
i
properties are same as in the ASM [6]. For convenience, recurrent state of this ASM is an identity operation [4].
zc ≡ F −1 is introduced and called the critical energy That is, by the DFES toppling rule, C should come
i ii m
(at vertex i). We refer to a vertex i with energy higher back to C if the toppling order is taken exactly as the
m
than the critical energy as an unstable vertex and it can corresponding ASM, which implies A(C ) = 1. Note
m
topple, resulting in the energy redistribution such that that the minimum energy among recurrentstates of this
z → z −F for all j. It is clear that the total energy ASM is exactly Z [6]. Hence above consideration also
j j ij m
of a configuration C, E(C) = z , is conserved during provesthatA(C)ofarecurrentstateC ofacertainASM
i i
toppling events. P constructed by making some vertices dissipative should
Byasequenceoftopplingevents,asystemmayevolve be 1.
toaconfigurationwithoutanunstablevertex,whichwill Z is in many cases larger than Z , but the order
M m
be called an absorbing state. For convenience, we in- relation is reversed for the DFES on a tree. By a tree
troduce a function of configurations A(C) which takes 0 is meant a system with |F | = |F | ≤ 1 (i 6= j) and
ij ji
if the system starting from C will arrive at an absorbing a unique connecting sequence between any two different
stateafterafinitenumberoftopplingeventsand1other- vertices. In a tree, Z is V −2 and Z is V −1. Hence
M m
wise. Due to the abelian property of toppling events [4], foranyconfigurationonatreewithE(C)≥Z =V −1,
m
this function does not depend on the way how toppling A(C)=1.
events are ordered and, accordingly, is well-defined for Since A(C) = 0 for any C with E(C) < Z and
m
finite V. A(C)=1 for any C with E(C)>Z , one might expect
M
Forthe DFES,there aretwobounds Z andZ such that “the phase transition point” ζ should lie between
M m c
that ζ andζ ,whereζ (ζ )isthelimitingvalueofZ /V
m M m M m
(Z /V) in the thermodynamics limit V → ∞. Now we
M
ZM = min{E|∀C with E(C)>E,A(C)=1}, (1) will investigate if this claim is valid.
Z = max{E|∀ with E(C)<E,A(C)=0}. (2) In the literature, the similarity between this phase
m C
transition and the usual APT has been emphasized [13].
ItiseasytogetZ = zc. TheexistenceofZ isobvi- However,theDFESasanAPTmodelhasacertainprop-
M i i m
ous,buttofindZm forParbitraryF doesnotlooksimple. erty which is not possessed by usual APT models such
Fortunately, if F = 0, that is, if there is no greedy as the contact process (CP) [23]. In the CP, a finite
i ij
vertex [6], we Pcan show that Zm = − i6=jFij/2 = system should fall into an absorbing (particle vacuum)
iFii/2. To this end, we notice that AP(C) = 1 un- statewith probability1 unlessspontaneousdeathispro-
Pder the simultaneous parallel update (SPU) implies that hibited. Ontheotherhand,somefinitesystemsinDFES
the steady state is a limit cycle with a certain period cannotfallintoanabsorbingstate. Sincethephasetran-
T [21, 22]. In one period, every vertex should topple sition involves the thermodynamic limit as well as the
exactly same number of times, say W, because there is infinite time limit, the order of these two limits is cru-
no greedy vertex. Accordingly, the total energy flow be- cial. In the CP, if infinite time limit precedes the ther-
tween vertices is 2WZ . Hence the total energy should modynamic limit, there is no nontrivial phase transition
m
not be smaller than Z . (everywhere inactive). However, if these two limits are
m
To complete the proof, we will construct a configura- taken in the opposite order, there is a nontrivial phase
tion C with A(C ) = 1 and E(C ) = Z as follows. transitionwhichisthe mainconcerninthe literature. In
m m m m
Let P be a permutation of 1,...,V. At first, the vertex the DFES, however, nontrivial phase transition can be
P(1) = i is given energy F . Energy at other vertices defined even if we take the infinite time limit first [18].
ii
areassignedbythefollowingiteration;ifthevertexP(k) Our discussion about the APT exhibited by the DFES
has been assigned energy, the energy of vertex P(k+1) starts from this “unconventional” phase transition.
becomes − ′ F , where the sum is over all ver- To study an APT, it is also necessary to define the
j P(k+1),j
tices exceptPP(1),...,P(k+1). The finial configuration (mean) density of unstable vertices at stationarity, say
after all vertices are exhausted is C . It is obvious that ρ(C), where C is the initial configuration. If A(C) = 0,
m
E(C ) = Z . If an ASM is defined by the toppling ρ(C)isunambiguously0. However,ρ(C)dependsonhow
m m
matrix ∆ = F +δ L , where L = 1 if the energy the toppling events are ordered if A(C) = 1 [24]. For
ij ij ij i i
at vertex i of C is F and 0 otherwise, C should be example,the devil’s staircase[21]is observablewhenthe
m ii m
a recurrent state of this ASM. This is because by con- SPU is employed, but it should disappear if a random
struction it should successfully go through the burning sequential update rule is employed because the steady
test [4]. Note that the burning test to find a recurrent state is not a limit cycle any more. In any case, it is
state is applicable when there is no greedy vertex. If we trivially true that A(C) ≥ ρ(C) for all C. To simplify
addL toallverticestoC andperformthetopplingdy- the discussion, we will exclusively assume the SPU.
i m
namics according to ∆ (with dissipation), the final state Let P (C;ζ,V) be the probability that an initial con-
0
3
figuration of the DFES with V vertices is C. Here, where k is a non-negative integer and p (ζ) is a certain
o
ζ is the mean energy per vertex satisfying Vζ = functionofζ intherange0≤p (ζ)≤1. Due to thepar-
o
E(C)P (C;ζ,V). Let us introduce ity conservation at each vertex [18] and its resemblance
C 0
P to the DFES on a chain, one can easily find that if ζ
φ1(ζ;V)= A(C)P0(C;ζ,V), (3a) is smaller (larger) than 2+po(ζ), φ1(ζ;V) goes to zero
XC (nonzero) in the thermodynamic limit. That is, phase
φ2(ζ;V)= ρ(C)P0(C;ζ,V), (3b) transitions occur at points where 2+po(ζ)−ζ changes
XC sign. For example, let po(ζ) = [4ζ +1] (mod 2), where
[x] means the largest integer not greater than x and the
and define “transition points” (ℓ=1 or 2)
modulo 2 operation restricts the possible values to ei-
ther 0 or 1. Note that 2+ p (x) −x changes its sign
ζc(ℓ) =inf ζ lim φℓ(ζ;V)6=0 . (4) at 9,5,11. Hence, there is reoentrance behavior in the
ζ n (cid:12)V→∞ o 4 2 4
(cid:12) phase diagram(absorbing,active,absorbing,then active
(cid:12)
Since φ ≥ φ , ζ(2) cannot be smaller than ζ(1). By phases as ζ increases). For other initial condition, one
1 2 c c
definition, ζm ≤ζc(1) ≤ζM. Althoughtheequalityζc(1) = should use po(ζ)= ∞k=0p(2k+1;ζ) for the criterion of
ζ(2) seems plausible for any DFES, we could not find a phase transition. P
c
general proof. We can prove this equality only for the Asarguedinthebeginning,thedefinitionoftransition
DFES defined on a tree because the period of a limit pointsEq.(4)doesnotcomplywiththespiritoftheusual
cycle on a tree is either 1 or 2 [25], which implies either APT.AproperdefinitionconsistentwiththeusualAPT
A(C)=2ρ(C) or A(C)=ρ(C) (that is, φ ≤φ ≤2φ ). is
2 1 2
Hence,ζ(2) =ζ(1) =1issatisfiedoneverytreeregardless
c c ζ(3) =inf{ζ| lim φ (ζ,t)6=0}, (6)
of the initial conditions. Note that the method used in c 3
ζ t→∞
Ref.[18]tofindthetransitionpointisequivalenttousing
φ (ζ,t)= lim n(C )P (C;ζ,V), (7)
φ1 with Poisson distributed initial condition. 3 V→∞XC t 0
Infact,the definition(4)makesitimpossible todefine
“the” transitionpoint. ζc(1) shoulddepend onP0. To ex- where Ct is the configuration at time t from the initial
plainwhy,letusdividethewholeconfigurationsintotwo configurationC,andn(C )isthedensityofunstablever-
t
classesSI andSA suchthatifC ∈SI (C ∈SA),A(C)= tices averaged over time up to t [ρ(C) = lim n(C )].
t→∞ t
0(1). Duetothenon-ergodicityoftheDFES[21],SI and But note that ζ(3) cannot be larger than ζ(2), so it is
c c
SA are disjoint and should exhaust all possible configu- (3)
clear that ζ cannot be defined uniquely, either.
c
rations. Let ζ (ζ;V) = P (C;ζ,V)E(C)/V ≥
A C∈SA 0 Since ζ ≤ ζ ≤ ζ for a certain ASM constructed
m S M
φ1Zm/V. If one can choosPe P0 such that in the thermo- fromtheDFESwithnegligibledissipativevertices,itmay
dynamic limit ζ →f(ζ)θ(ζ−ζ ), where ζ ≤ζ ≤ζ ,
A 0 m 0 M be possible to find an initial condition which yields the
θ is Heaviside step function, and f(ζ) which should be
criticality as in the ASM (fine-tuning of initial condi-
not larger than ζ is a certain function, this model shows
tions). However, as we have seen, recurrent states are
a (possibly discontinuous) transition at ζ(1) = ζ . Only
c 0 related to the active phase within our definition, so the
incaseζ =ζ asintheDFESonatree,the transition
m M criticality, if exists, has nothing to do with the recurrent
point (4) of the FES is unambiguously defined. Even
states. If the ASM forms a universality class, the de-
worse, certain P (C;ζ,V) can trigger reentrance behav-
0 tailed structure of dissipative vertices should not yield
ior in the sense that the set {ζ|φ (ζ;∞) = 0} is not
1 a considerable difference. Hence, no DFES can exhibit
connected. In this context, the disagreement between ζ
c the similar critical behavior as the ASM, if universality
(actually ζc(1)) andζS observedinRef. [18]is the generic hypothesis is valid.
feature of the DFES. ThedifferencebetweentheASMandtheDFESiseven
To illustrate, we will investigate the DFES on a more striking when we consider a spreading dynamics,
braceletgraphwithperiodicboundaryconditionsdefined or a microscopic APT (mAPT) [26]. The mAPT stud-
by Fij = 4δij −2δ|i−j|,1−2δ|i−j|,V−1. From the general ies how an initially localized unstable vertices spreads
consideration given above, ζm = 2 and ζM = 3. Hence, through a background inactive region with infinite vol-
ζc(1) shouldbelocatedbetween2and3. Nowassumethat ume (the density of unstable sites is zero for all time,
the initial energy distribution at each vertex is drawn hence the name “microscopic”). The order parameter in
from this case is the (survival) probability that the activity
persists indefinitely. For illustration, our discussion will
ζ k e−ζ/2
p(2k;ζ)=(1−p (ζ)) , be restricted to the two-dimensional BTW model [1].
o
(cid:18)2(cid:19) k!
First, we construct the background inactive region in
ζ−1 k e−(ζ−1)/2 such a way that each site is assigned energy 0, 1, 2, or 3
p(2k+1;ζ)=p (ζ) , (5)
o (cid:18) 2 (cid:19) k! withtheprobability(1−p3)p0,(1−p3)p1,(1−p3)p2,orp3,
4
respectively (0≤p ≤1 and p +p +p =1). By vary- comes meaningless, if the universality hypothesis is not
i 0 1 2
ing p ’s, one can study the mAPT with the background applicable to the FES versionor if the claim in Ref. [28]
i
density ranging from 0 to 3. After forming the back- is wrong. However, the numerical study in Ref. [28] is
ground density, we choose a site randomly (this is not rather convincing and the universality hypothesis seems
a mathematically rigorous statement, but the rigor can valid [13]. In any case, we believe that it is still neces-
be attained with ease) and perform the DFES dynam- sary to make the connection between the SFES and the
ics. Obviously,no unstable site appears with probability stochastic sandpile models more firmly.
1−p3. If p3 is larger than the site percolation thresh- TheauthorisgratefultoA.Feyforhelpfuldiscussions
old [27] and if the randomly chosen site is in the infinite andto J.Krugforcriticalreadingofthe manuscriptand
cluster of sites with energy 3, the spreading dynamics helpful discussions. Support by Deutsche Forschungsge-
neverdies. Hence, the criticaldensity ζmc is bounded by meinschaftwithinSFB680Molecular BasisofEvolution-
ary Innovations is gratefully acknowledged.
ζ ≤(1−p∗)(p +2p )+3p∗. (8)
mc 1 2
Hencebyvaryingp andp ,onecanhaveinfinitelymany
1 2
transition points. In striking contrast to the previous
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