PHYSICAL REVIEW A VOLUME 40, NUMBER 10 NOVEMBER 15, 1989

Avalanche dynamics in a deposition model with "sliding"

Z. Cheng and S. RednerCenter for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215

P. MeakinCentral Research and DeUelopment Department, E.I. DuPont de Nemours and Company, S'ilmington, Delaware 19880-0356

F. FamilyDepartment of Physics, Emory University, Atlanta, Georgia 30322

(Received 12 April 1989)

We introduce a simple model of "avalanches" in which there is continuous deposition of mass ona "tilted" substrate, with avalanches occurring whenever the mass at a given site reaches a preas-signed threshold value h, . The avalanche is defined to sweep away all mass which is downhill fromthe initiation event. Basic dynamical features are studied, including the distribution of avalanchesizes and the time intervals between avalanches. For a one-dimensional "slope" of length L, the

i —i //1

average avalanche size is found to scale as L ', and in the continuum limit the averaged steady-—j /pI

state mass distribution a distance x from the top of the slope scales as x '. Qualitatively similarresults are found for avalanches on a two-dimensional substrate. Contrasts between the presentavalanche model and models of self-organized criticality are discussed.

I. INTRODUCTION

In this paper, we investigate the statistical properties ofa simple "avalanche" model in which there is a continu-ous deposition of mass on a "tilted" substrate, withavalanches occurring whenever the mass at given sitereaches a preassigned threshold value. In an avalanche,mass is defined as falling along a preferred directionwhich we consider as being imposed by an external field,such as gravity, and in the process coalesces with and re-moves all other mass that it contacts. Our model maymimic processes such as the falling off of water dropletson a thread, e.g. , dew on a cobweb, or the flow of wateron an inclined plane, e.g. , rain on a window pane, orperhaps even real snow avalanches. The example of rainon a window pane appears to amenable to simple, yetquantitative experimental studies.

In addition to the potential connection with avalanchephenomena, we are also motivated by the "sandpile"models of self-organized criticality introduced by Bakand co-workers, ' which have generated considerable re-cent interest both theoretically ' and experimentally.Our avalanche model has several features in commonwith the sandpile models. There is a continuous input ofmass into the system, with transport being initiatedwhenever the threshold for flow is exceeded 1ocally. Inthe avalanche model, however, the mass transport afterthreshold has been attained has a catastrophic nature, asthe avalanche ends only when the boundary of the systemis reached. This feature strongly contrasts with the dissi-pation mechanism in the sandpile models. Due to thisdissipation, the sandpile model naturally evolves to aself-organized critical state in which many dynamicalquantities exhibit power-law correlations. ' However,

the connection between the sandpile model and potentialexperimental realizations has yet to be fully realized.Our model appears to provide a better physical picturefor certain types of transport in open systems, such aswater droplets running down a window pane. Our goal isto explore the dynamical behavior of the avalanche mod-el, and perhaps to gain general insights about genericmodels of open systems.

We have formulated both a lattice and an off-latticeversion of an avalanche model incorporating (i) dropletdeposition, (ii) droplet growth and coalescence, and (iii)ensuing avalanche. The first two features have alreadybeen studied extensively in a wide variety of contextssuch as the growth of thin films, ' the growth of breathfigures, ' ' " and heterogeneous nucleation. ' ' The possi-bility of allowing for avalanching allows the system toachieve a steady state in which there is a fluctuatingtransport of mass out of the end of the system. The be-havior of this mass flow is the main focus of this paper.

In the lattice version of the avalanche model, unitmasses are sequentially deposited at random on a d-dimensional substrate of linear dimension L. When themass at any lattice site reaches a threshold value h, anavalanche begins in which this critical mass nucleusslides "downhill" and collects with it any mass which itcontacts, either downstream or laterally, as illustrated inFig. 1. In the off-lattice model, there is sequential, ran-dom deposition of D-dimensional hyperspherical dropletsof diameter do on the substrate. When two droplets ofradii r,- and r2 touch or overlap, they coalesce into alarger droplet of radius r =(r, +r2 )', whose locationis at the center of mass of the two original droplets. Ifthis newly created droplet overlaps other droplet(s),coalescence continues until no overlaps remain. When

40 5922 1989 The American Physical Society

AVALANCHE DYNAMICS IN A DEPOSITION MODEL WITH ~ . . 5923

~ ~

~ ~~ ~J

J~ ~~ ~ J ~

t 5 ~ ~ ~

J r~ ~ 5 ~ ~

5 ~ir~ ~~ ~

J~ ~

32 LATTICE UNITS

(a)R

~ ~~ ~

~ ~ ~ ~~ ~ ~L g

~ ~ 5~ ~ ~ ~ ~

R~ ~~ ~~ ~

J

the droplet mass reaches a threshold value h„ it corn-mences sliding down the substrate, still continuing tocoalesce with other droplets in its path according to themechanism just outlined. In our simulations of theavalanche model in two dimensions, periodic boundaryconditions in the lateral direction are imposed.

We are interested in the statistical properties of theavalanches and in the nature of the steady-state mass dis-

tribution on the substrate. In this work, we will primari-ly focus on two characteristic properties of avalanches.These are, P(t) which is the distribution of times betweensuccessive avalanches, and R(m), the distribution ofavalanche sizes. The former quantity coincides with themass added between avalanches, if one mass is added tothe system per unit time and if the avalanche propagationtime is neglected compared to the time between massdeposition events. For both the lattice and off-lattice ver-sions of the model, we find that these dynamical quanti-ties obey simple scaling laws. However, the spatio tem-poral correlations of the ensuing steady state are not gen-erally of a long-range nature, i.e., this steady state doesnot appear to lie within the universality class of self-organized criticality.

For a one-dimensional substrate, we have calculatedthe steady-state properties and the temporal correlationsof the avalanches. We have also studied the conditionsfor the initiation of the first avalanche in the system. Inaddition to analytical calculations, we have performed ex-tensive numerical simulations of this avalanche processfor both the lattice and continuum version of the modelfor one- and two-dimensional substrates. In the simula-tions of the lattice model, we injected between 1.5X10and 7.5X10 particles in one dimension, and between5 X 10 and 7. 5 X 10 particles in two dimensions in orderto study steady-state properties. For the continuummodel, the number of particles added was typically oneorder of magnitude less than in the lattice models. Thesesimulations required a total of approximately 600 h ofCPU time on an IBM 3090 computer.

In Sec. II of this paper, we begin by giving a simpleprobabilistic derivation for when the first avalanche is ex-pected to occur on a one-dimensional substrate. In Sec.III, we discuss the nature of the steady state in one di-mension. For the case of the threshold h, =2 the steadystate is found by a direct solution of the master equationsthat summarize the occupancy probability on the sub-strate. In the limit of h, ~ ~, we present a complemen-tary continuum approach from which we can derive thesteady-state avalanche properties, as well as the spatialdistribution of mass remaining on the substrate. In Sec.IV we present extensive simulation results for avalancheson two-dimensional substrates. Finally, in Sec. V we givea brief discussion of our results.

II. AVALANCHE INITIATION

~ ~

~ ~ ~I I32 LATTICE UNITS

FIG. 1. Illustration of the discrete version of the avalanchemodel on a 32 X 32 substrate for h, =3. The system is shown (a)just before and (b) just after the avalanche. For this system size,the wedge-shaped cleared region is apparent.

We first determine the condition necessary for the oc-currence of the initial avalanche for a one-dimensionallattice, as mass builds up on an initially empty system oflength L. For the first avalanche to occur, it is necessarythat h, particles land on the same lattice site. This eventcan be expected to occur on a system of length L whenthe probability of the event is of the order of 1/L. Usingthe Poisson distribution for the occupancies at each site,we therefore have

(h& '&~) 1

h, ! L

5924 &, S. REDNFR p MEAKIN, AND F. FAMILY

-IO—

2 ~SLOh

C

eL=1

where ( h & is thew e average mass atfi h'

avalanche

As

}1 th doccurrin assiso the order of

ass eposited on th e sub-

as L~~.A similar ar gument can b

h h fi 1

e into tw enmm e

py when refer-

njected. When this m'"

pproximatelarne site is a, —1 partic

ri ution,y equal to, from theom t e Poisson dis-

(h, —1)!

wher e n

(3

ere we now definsion

e neph&=m = . n ts

i y. Consequentlcoincid

pp o ated by

C

( ), i

-180

E 10—CL

12—

16—

180

-8

CL -12

L= 10

(b)

I

2 3 4

2 3

I

5 6 7ln (t)

he=2

5 6ln (m)

8 9 10 11

he=4

8 9 10 11

m

P(m)= g (1—c~nn=1

and expanding the produe product to lowest de produ or er leads to

P(m)-exp — „-exp I. (h &

C'

(4a)

(4b)

-20

-24 '

0ln(f)

10 12 14

Finall thd h }1

y, he probabilit tha

1 o —dP()etween successive avalanches

Chc= 2

he=4

p(t)= (h &

exph, I

L =10'

Since the 1e location of the in' '

bt t lo1 h 11

ris ic size w'

wi e unifwhich is of the

iform untild fM

poo, beyond

eatures are obia y cut off.

}lo F 2 ddto to th dig. 2. In aour simula

'

e times and size istribution f

sizes we hav 1

time unt'1un i the

first

'stributions. Specificallsidered the

y gs e nedby

e

~=/ tP(t),

and the averaverage size of the fie rst avalanch be y

-200 6 8

ln (m)10 12 14

FIG. 2. NuF . . umerical simune imension for v

t e discrete aval

th dit 'b to of

ava ancheues o h, . Shown in

e m), both foIn (c) and (d) th, t e respective

at large times for lar

= 10 . In (c), note the1 of A Th

anc es that ois peak c

ft thre ore coincides wit

e s ope is swe t cd

e rst avalanch e.u ion of timesu required to

5925

the threshoate whenhinges pn

1n

2. Thisnl on

olutiPnavalanche

'n g equals

depends oe initiatio ~

the systemy site

avalanc eolution 0

f lls on anthe fact t

dded to the syarticle land

at the evostem a s

s on an~~~l~~~h~ m

subseq

pr a Pd site an

'there is no

occupiet site

interest 0f

alreadyd on an emP y

h intljnsicthe Partic

In additionful check Pn

' lelan s

tot ethenu-

e event.as a use

avalanch'i also serves

exist

it as

bility that there exl simulatio

the proba'" tjme step~

mericacalculating

th L at theWe begjn by

rval of leng .relation

s pn an inthe recur 1

J particles otity obeysan

(9

(6b)

p(r), wwe findBy using Eq.a

the num

~-Lhand from

elthe pther ha

~ can be w

Onears that

with &

. 2, it also aPPl data of F gwer-law rela

ca 'dbythepor esente

(7b)

epra'

erical esmate~ fpr

L-dependentto the L

tiveolatewh jch

ate effecthen extrap

mptoticand a'

data sugges3) Theonents sre

(F'g '

th these exPova]ues pf bot

(J)Th&sqL

1 (').N

g —pNpN —

& J'=2+1

(' —1)+LpN( J) L

at anit th intervalaccoun

corn mod

fort ee an

h possib»itydditio"

while

first. term al s can acco

lanche, wcontaining J

the pccurrenin which t

thput tparticle wcounts for

l ontaining Jl ft

term acc'

terva cn e

f a s&ngle palanche with J P

tion ot an ava an

form

cles can le

in the matbe in

can elumn vec

h d (Ftg'4)b rewritte '

tor whpse JEqU~

where NL + 1)X

tinn (9)p is the c+

L + 1 ) rnct

=M.PN —~'

mponent

(8)1/h,

b jljst1clementary Ph is initiate

jth our e eavalanc '

l pf

a reement'

e case, anan jnterva

agthe pff-lattic

deposited inwith

ment. Inh has been

lattice caser a mass

&

lent tp ihe awheneve»D

This is equivangth Ag

p 1/play1ng th role

ATE PROPE RTIESIII. STEADY ST

1

L0

=2MSSS g gshpld

1

L

jpn the distribu-ana y .erva s

l tic solutiol between

resent antime inte

W~ f avalanction o 0

L —10 0

—2L

hc

1

L

p.52

]/h050 =—

MODEL WITA DEPOSITIOE DYNAMICAVALANCH

steady

048—I-z046-zQp 44—Q

p42—

I

04o(~)

'

006 0.080 38

0'02 0041i2)

0.00 -

(~/L

I

o l0 O)2

h, =4

0I-

p.76t-l/hc

p74

p72LLIz p7oOQ~ p68

p )4

p66

p. tpp08p.p4 pp60.64

p02p.pp

~(L)f f he effect[(n[p( t

;„e exPoL )]

Dependenc;defined

« '(L )p(t L )/)~"" 'ln 'rl Shown ared f edslmilar y.de

of the numl evolution o. orre-

d

n oft e .ticle

h temporaddition co

4. Illustrationn Single-par

stem contain-

FIGthe system, "

th. For a yL- An

of pa ilcardstepo"robablllty

'cles inunit lenS

l —n &

s on s

a step octo an up

curs wit ph h can enste w 1C

any po'n

5926 Z. CHENG, S. REDNER, P. MEAKIN, AND F. FAMILY

From this, we find that p~( j) has the steady-state solution

Using Stirling s approximation, the steady-state mass onthe interval is peaked at a value that scales as &L, i.e., as

1 —1/AL ' with h, =-2.

From this distribution of steady-state occupancies, wecan now determine the distribution of avalanche sizesand the time between avalanches. For this purpose, notethat there is a unique location for the insertion of a parti-cle in an interval containing n & m —1 particles that leadsto an avalanche of mass m. Consequently, the steady-state distribution of avalanche sizes can be written as

R (m+1)=[p „(n)/L ]

m=1, 2, . . . , L .

g [p (n)/L]m=1 n=m

(12a)

By exploiting Eq. (9), the sum over p„(n) can be elim-inated to yield the closed-form expression

R(m+1)= p (m) .1 L+1

m+1 (12b)

L L

L —n —t+3 n+t —2X. XL L

(13a)

The factors in the product express the probability that inan interval which contains I masses, no subsequentavalanche occurs for the next t time units. Since onemass is added at each time step, t and m can again beused interchangeably in Eqs. (12) and (13). Now appeal-ing to Eq. (9), P(t) can be reduced to

1+—+ R (t)3 2L L2

L!(2L +t +1) '+' (L+1)"(L+1) L (A) !

~R(t)+g —,t=1,2, . . . , L+1,1(13b)

where the second relation continues to hold for the caset =m =1 if we continue to define R(1) via Eq. (12b). Thebasic qualitative features of these expressions for R ( m )

and P(t) is that they are similar in behavior to p„(m).

where (Af. ) is the average size of the avalanches in thesteady state.

By a very similar line of reasoning, the distribution oftime intervals between avalanches, or equivalently, themass added between avalanches, can be formally writtenas P(t) =P(t)/Q, P(t), with

They are relatively featureless for m & &L, and are ex-ponentially cut off for m )&L. Our simulation resultsfor R(m) and P(t) are shown in Figs. 2(c) and 2(d) forvarious values of h, . For h, =2, the simulation data arein excellent agreement with Eqs. (12) and (13).

B. Large threshold

For large values of h„numerical simulations indicatethat a steady state is reached relatively quickly, so thatmany properties of the steady state and the initialavalanche are fairly similar. Thus the exponent valuesa =a' = 1 —1/h, also appear to describe the scaling ofthe average avalanche size and the average time intervalbetween avalanches in the steady state. For large h„ thedistribution of avalanche sizes is still quite flat for sizesless than a characteristic size. However, for the distribu-tion of avalanche intervals the flat region is followed by apronounced minimum and then a sharp peak at the larg-est possible times [Figs. 2(c) and (d)].

This striking behavior can be understood from thepeculiar evolution of the system in the large h, limit.Starting with an empty substrate, there is a long quies-cent period during which mass builds up to a point that isclose to the thresho1d value at each site. When the firstavalanche finally does occur, the upstream portion of thesystem is still close to the avalanche threshold. Conse-quently this portion of the substrate is highly susceptibleto additional avalanching soon after the initial event.This suggests that there will follow a relatively short"burst" of avalanche activity in which the initiation pointof the avalanche moves upstream through the near-threshold portion of the system at each successive event.This burst terminates when an avalanche occurs at thetop of the slope, thereby sweeping the slope clean. Thelarge-time peak in the steady-state distribution P(t) origi-nates from the long time needed to generate an avalancheafter the slope has been swept clean. The location of thispeak quantitatively coincides with the distribution P(t)associated with the first avalanche.

We now develop this physical picture for the dynamicsin the large h, limit in order to provide quantitative pre-dictions for steady-state properties. Before the firstavalanche occurs, the mass at each site obeys a Gaussiandistribution p(h) whose average value (h ) increaseslinearly in time, and whose width is of the order of&(h ). The maximum mass at any site h, „ is deter-mined by the condition Jh p(h)dh =1/L. When h

max

reaches h, the first avalanche occurs. From the integralcondition for h „, we thereby find that the firstavalanche occurs at a time r, /L = TL —h, +2h, lnL, —and at a random location on the interval.

Since the upstream portion of the system is nearly fullat this time, the second avalanche almost surely occursupstream of the first avalanche. The length of thisupstream segment L', will typically be equal to L/2.Consequently, the typical time for the occurrence of thesecond avalanche ~2/L coincides with the time requiredfor the jtrst avalanche in a system of length L', i.e.,~2/L = TI '. Following this argument, the time delay un-

40 AVALANCHE DYNAMICS IN A DEPOSITION MODEL WITH. . . 5927

h,lnx

2

h,h(x)-2

(14)

til the i '" avalanche is r; /L —TI, where the typical dis-i —1

tance of the i'" avalanche from the top of the slope isI, —L /2', and the mass carried by this avalanche is

m, -l, h, . Eventually an avalanche occurs at the top ofthe slope after a time delay, 7 1/I h„after whichthe avalanche sequence starts over again.

From this picture of avalanche events, the densityprofile a distance x form the top of the slope, h (x), aver-aged over all realizations of the system will be of the or-der of one-half the time needed for an avalanche to occurat this position. Therefore,

1/2

tribution of time intervals between avalanches.Consider the difFerence sequence I r, I

=I T, ]

—.t t; j

which delineates the temporal occurrences of avalancheswhich begin at column L = 1. Since each T belongs toJsome interval [t, , t, +, ], and since at every t, the mass oncolumn L+ 1 is reset to zero, the statistics of '7 dependsJonly on the enclosing interval [t, , t;+, ]. For large L, theprobability of having more than one Y~ in a time interval[t, , t, +,] is small and therefore we make the "one-avalanche" approximation (Fig. 5) in which at mostone '7 occurs in any interval [t;, t; + &]. Within thisapproximation, the master equation that relates PL(t)=Prob( t = t; +, t, ) —to Pt +, ( t) =Prob( t = T; +, —T; )

reduces to

In addition, the typical size reduction of successiveavalanches by a factor of 2 suggests that the number ofavalanches in a renewal cycle is of the order of ln L.Consequently, the average avalanche mass is given by

PL+)(t)=Pt (t) g P((t')+ —,' g Pt (t+t')P)(t')

Lh,ML—

lnL(15)

+ —,'P, (t) g PL(t') .t'=t+1

(16)

These two formulas suggest that there is a crossover fromthe large-h, limit to the large-L limit when h, -lnL. Inthe large L limit, the density profile on the substrate canbe viewed as a continuous function. This forms the basisfor a continuum approximation which leads to detailedresults about the statistics of avalanches. This approachwill be treated in III C.

C. Finite deposition rate model for the thermodynamic limit

For arbitrary but large values of h, and for large L, theaverage density profile on the substrate becomes asmoothly varying function of position. To discussavalanche statistics, it proves very useful to introduce avariant of our original avalanche model in which eachlattice site may be occupied by an additional particle withprobability p at each time step. Since mass is deposited ata rate p per site, then pL particles will be added to thesubstrate in a unit time interval. In the limit where p ap-proaches 0 as 1/L, the behavior of this continuum"finite-p" model approaches that of the originalavalanche model in that one mass is added per unit timeinterval. However, this finite-p model lends itself natural-ly to a continuum description that facilitates the solutionin the large L limit.

For the finite-p model, let PL (t) be the probability dis-tribution for a time interval t between successive avalan-ches, where we now make explicit the fact that this distri-bution depends on the system length L. Furthermore,denote by t t, I the sequence of time intervals between suc-cessive avalanches in a system of length L. This coin-cides with the sequence of time intervals for avalanchesthat pass through column L of an infinite system. Wenow construct a master equation for PL (t) by relating thetime interval sequence I T; I associated with avalanchesthat pass through column L+1, to the time sequence

I t, I C: [ T; ], associated with avalanches that pass throughcolumn L of the same interval (Fig. 5). The solution tothis master equation provides information about the dis-

The first term on the right-hand side is the contribu-tion from a time interval [t;, t, + &] of duration t in whichno avalanche occurs on the (L + 1)th column, while thelast two terms correspond to the ways in which the timeinterval [t;, t, +&] of duration longer than t can be splitinto two subintervals, with one of duration equal to t, bythe occurrence of an avalanche on the (L + 1)th column.These equations are the discrete rate equations for the"fragmentation" of the time interval t, +1—t; due to theintroduction of a new avalanche event within the inter-val, in which L is the analog of the time and t the analogof the fragment sizes. '

We now take the continuum limit by letting p~0 andalso rescaling t by a factor of p, i.e., the deposition rateis now L particles in a unit time interval. With thisconvention, P, (t) becomes the Poisson distribution

h,.—1

P, (t)=t ' e 'llh, —1)!. Thus Eq. (16) can be recast as

FICz. 5. A schematic time sequence of avalanches for an in-terval of length I (solid peaks), and the additional avalanchesthat begin at column I+ l (dashed peaks). The "one-avalanche"approximation is based on assuming that no more than onesmall avalanche appears between large avalanches.

Z. CHENG, S. REDNER, P. MEAKIN, AND F. FAMILY

aP, (t)P—, (t)f P, (t')dt'

fjL 0

+ —,' PL t'+t P] t' dt'

0

+ —,'P, (t) f Pi(t')dt' . (17)

mh, +const Xx, x « 1C

(x ' /h, )exp( —x '/coh, ), x »1 (22)

Substituting Eqs. (21) and (22) into the definition (18)for PL(t), we find

Since the typical time between avalanches goes to zero asthe size of the system increases, we anticipate that theasymptotic behavior of PL(t) can be written in the scalingform'

1/hL ', small time

1 —1/h h —1 h . (23)L 't ' exp( Lt —'/h, !), large time

PL (t) =- TL 'p( t /TL ) . (18)

Here TL is the typical time between avalanches on an in-terval of length L, and the exponent value —1 is requiredby the normalization of the total probability. Substitut-ing this scaling form into the rate equations and making—xTLuse of the fact that as L ~ ~, e ~1 with x =—t/TL,the dependence on x and TL in Eq. (17) can be separatedinto two equations:

hc—

1

d TL TLTL CO

dL (h, —1)!h h —

1C

[p(x)+xp'(x )]co= — + f P(y)dyh, 2 x

(19)

+ ,' f y—' P(x+y)dy .

The first equation can be immediately solved to yield forthe typical time between avalanches:

TL =(L/h, !) '(coh, ) (20)

Since the average mass residing on the L'" column is pro-portional to TL, Eq. (20) also predicts that there is apower-law density profile as a function of distance fromthe beginning of the interval, with a characteristic ex-ponent —1/h, . A related quantity, which can be foundby this reasoning is the distribution of positions at whichan avalanche occurs, Q(x). Since Tt ' —j Q(x)dx, we

—1+ 1/hconclude that Q(x) -x ' for large x.

At this stage, we convert the integrodifferential equa-tion for the scaling function P(x) to a recursion relationbetween the moments mt, =—J 0 x "P(x)dx. Thus by multi-

plying the second half of Eq. (19) by x" and integratingover all x, we find

16)km' —my+ h

CC

1

2(k +h, )

P(k+ l, h, )

2

for k & 1 . (21)

where /3(n, m) is the f3 function. In both the limits x ~0and x ~ ~, these recursion relations can be solved toyield the asymptotic behavior of the moments. Then byinverting the Mellin transform, we find the followingasymptotic forms for P(x):

Here the crossover between small and large times isdetermined by comparing t with the quantity

1/h(h, !/coh, L ) '. For large times, Eq. (23) coincides withthe time interval distribution for the first avalanche givenin Eq. (5) when the time t is identified with ( h ) in (5).On the other hand, for small times, Eq. (22) approaches aconstant [Fig. 2(c)], while Eq. (5) has a power-law time

h —1

dependence t ' [Fig. 2(a)].

D. Autocorrelation functions for mass transport

We now study the autocorrelation functions associatedwith the mass transport to quantify the fluctuations inthe mass flow through the system. For this purpose, wedefine the autocorrelation function for the mass con-tained in the interval

c „,(t) = ( n (t')n (t'+ t ) ) —( n (t') )', (24)

where n(t) is the total mass in the system at time t, andthe angle brackets denote a time average over t'. Herewe define the time so that the deposition rate is one parti-cle per unit time interval. For the particular case h, =2,we can exploit the matrix formulation for the evolutionof the system upon single-particle addition, Eq. (10), toformally write the autocorrelation function as

OXp„(0)c „,(t)=(0, 1,2, . . . , L)M' 1Xp„(1) —(n(t))

L Xp (L)

(25)

Since the net mass leaving the system at time t is simplyn(t+1) —n(t) —1, the autocorrelation of the fiux cs„„(t)is-given by

c„„„(t)=2c„,(t) —c „,(t —1) c„,(t+1}. — (26)

By diagonalizing the matrix M for intervals up toL =400 and also by direct simulation of the autocorrela-tion function for an interval of length 25, 000, we obtainidentical results, which are shown in Fig. 6(a}. At shorttimes, there is an anticorrelation of the flux which merelyreflects the fact that immediately after one avalanche hasoccurred, there will be a finite time delay until the nextavalanche. At long times, the correlation function is flat,indicative of no long-term memory effects. This behaviorcontinues to hold for small values of h, & 2 [Fig. 6(b)].On the other hand, for large h, there is an oscillation in

40 AVALANCHE DYNAMICS IN A DEPOSITION MODEL WITH. . . 5929

the correlation function with a periodicity that coincideswith the "renewal" time, i.e., the time for the slope to beswept clean [Fig. 6(c)].

In two dimensions, we observe qualitatively similar be-havior, except for the oscillations associated with the

large h, limit. The absence of the oscillations stems fromthe lack of a well-defined renewal event. In both one andtwo dimensions, it appears that the correlation functiondecays rather quickly; there does not appear to be anyevidence for power-law behavior.

0.25-

0.25

-0.50

-0.75-

—1.00 I

1000 2000I

3000l

4000 5000

0.25,-

(b)

„—0.25

-0.50

-0.75-

—1.00 I

0.5I

1.0I

1.5I

2.0 2.5x10

1.0

(c)

0.5—

0

—0.5—

—1.0 I

0.5I

1.0I

1.5I

2.0l

2.5 3.0x10.

FIG. 6. Numerical results for the autocorrelation function ofthe mass flux in one dimension for the cases (a) h, =2 and (b)h, =5 for L =5000, and (c) h, =50 and L =500.

IV. AVALANCHES ON TWO-DIMENSIONALSUBSTRATES

In two dimensions, the time until the first avalancheand the size of this initial avalanche can also be obtainedby the statistical arguments given in Sec. II. For the ini-tial avalanche to occur, the density must again reach avalue where the most populated site contains h, particles.Using the Poisson distribution, this condition is reached

2(1 —1/A jwhen a total of L ' particles have been depositedon the substrate. Consequently, the exponenta =2( 1 —I /h, ) for a two-dimensional substrate.

An avalanche in two dimensions can accrete mass inthe lateral as well as in the longitudinal direction. In thelattice model, stationary mass which is laterally nearestneighbor to the avalanche is defined to become part ofthe avalanche (Fig. 1). In the continuum case, any sta-tionary droplet which is touched by the avalanche is con-sidered to join the avalanche. Owing to this lateral massaccretion, the mass of the initial avalanche has a differentdependence on system size than in one dimension. Thissize dependence can be found in terms of the shape of theregion which is swept out by the avalanche. Since thefirst avalanche occurs when the density reaches a value of—2/A,the order of L ', the cleared region will have awedge shape whose opening angle a is given by—2/h, ,

tan (a/2) ~L '. Multiplying the area of this wedgeby the initial mass density leads to the mass of the initialavalanche scaling as L with a'=(2 —4/h, ). Analogousarguments yield the same values for the exponents o. anda' in the two-dimensional continuum model.

We used numerical simulations to test many of thesepredictions. Figure 7 shows the distributions of massadded until an avalanche, P(t), and the mass removed byan avalanche R(m), for both the first avalanche, and inthe steady state. These results were obtained fromsingle-particle deposition on a square of linear dimensionL =512 for various thresholds h, . Estimates for the ex-ponents a and a' defined in Eqs. (7) were obtained by ex-trapolating simulation results for L in the range 32—512to L ~ ~. Figure 8 shows the L dependence of the mea-sured deviations of a(L) and a'(L) from their asymptoticvalues of 2(1 —I /h, ) and 2(1 —2/h, ), respectively. Thebehavior as L ~ ~ is in good agreement with ourtheoretical expectation.

A noteworthy feature of the distribution of initialavalanche sizes in two dimensions is the power-law be-havior at small sizes, R(m) —m ', as shown in Fig.7(b). This behavior stems from the fact that the locationof the initial avalanche is uniformly distributed and thatthe size of the initial avalanche varies as the square of thedistance from the bottom of the slope. Consequently, byfirst writing the size distribution as a function of the loca-tion of the avalanche and then changing variables from

5930 Z. CHENG, S. REDNER, P. MEAKIN, AND F. FAMILY

(a)

10— he =5

CL14-

16-

IS—

20 i

6/

8ln(f)

10

hc =1i

12 14

10—

12—

14-51

160 6

ln (m)10 12

avalanche location to avalanche size will lead to thepower-law behavior observed in the figure.

Simulations were also performed for the off-latticeavalanche model. Figures 9 and 10 are "snapshots" ofthe system at various stages of evolution; these impart a

helpful visualization of the avalanche process. Imme i-ately after the first avalanche, there is a characteristicwedge-shaped area that is cleared (Fig. 9). The steady-state configurations (Fig. 10) are relatively inhomogene-ous and exhibit regions of large droplets which are justabout to avalanche, as well as empty regions whereavalanches have recently occurred. For these off-latticesimulations, the diameter of the deposited droplets was1.5, and the substrate sizes ranged from 64X64 to 1024X 1024.

The shape of the cleared area can be found by a simplegeometric argument (Fig. 11). When a moving droplet ofmass m and radius m ~m' falls a distance dy, it ac-cretes additional mass dm which is proportional to thedifferential area swept out by the avalanche, pwdy, wherepis esuth bstrate mass density. This leads to a mass

"orthewhich increases with fall distance y as (py),or t eI /( —1)width of the wedge growing as (py) . For droplets

of dimensionality D =2, the width of the wedge increaseslinearly with y, while for D )2, the width grows moreslowly.

In the off-lattice model, we also measured the distribu-tion of mass added to initiate the first avalanche and themass removed by the first avalanche. We consideredboth two- and three- dimensional droplets on a two-dimensional substrate. For droplets of spatial dimensionD =2 the distributions shown in Fig. 12 are very similar,both qualitatively and quantitatively, to those of the lat-tice model on a two-dimensional substrate, as shown inFig. 7. From the data for the average mass added to gen-erate the first avalanche, M, for h, = 3, 5, and 100, we es-timate the exponent a to be very close to our theoretical

004— hc = 2

0.02-

—16—

0 I 2 3 4 5 6 7 8 9 10 I I 12 13ln(f)

oo 0.00-

—0.02-

-004-(.)

Q.OOO 0.025 0.050 0.075ly

0.100 0.125 0.150

h-8— 0.02—

hc= 5

—12— rg 0.00-

-16 — 5

0 I 2 3 4 5 6 7 8 9 10 11 12 13tn (m)

FIG. 7. Simulations results for the discrete avalanche modelon a 512 X 512 substrate. Shown are (a) the distribution otimes (or mass added) before the first avalanche and (b) the sizedistribution of the initial avalanche. In (c) and (d) the time andsize distributions are shown in the steady state.

—004 ~(b)

0.000 0.025i

0.050 0.0752lg

0.100 0.125 0.150

FIG. 8. Dependence of 6+ =a(L) —o. and 6a'=6a'=a'(L) —a' onlattice size L for (a) h, =2 and (b) h, =5. Here +=2(1—lib, )

AVALANCHE DYNAMICS IN A DEPOSITION MODEL WITH. . . 5931

0 ~ ~

00 0 'Ila

0DOODD

00 0 0 0

4 0 0 oD 0

4p Qd"..'.D8o0 O~Q 0

S 0

goa.o& g .QoDO a O

0o o

o 0 Q,4Dap ~ a 0 '4

~ 0 0

004040 04

oo40

oa P

0 0 0

o oooooa 0OPO O

0o0 p ap D

0 P ~d'0

popoQ

0Q

(Q) O

Smax = 10"D=2~d=2

170.7 do

'4 a Q uo oO o & 0 od OD& "o p D

~ 4

D

4p 0 c oo O 4~ 0 4 aoo D

0 0 0~ &oaf,Da Q0 4 a 0 4 S~g o

a 0 0 lbo

oo 0

4 04 0 4 0 0 0 0 4 0

oo 4

QOD 40 ao 0 0 o

o 4 uL 4 0 '~ a Q 0 0 o ooa o

00 o 0 ay 4 o 000 4 00 0 O4 p

40 44 400 0 0 4 0

~0 4 000 0 Qo 0 0 ~ 40 0 ~044 4 a ~ 0 4 Q O~~

0 04 ~ 0 4 ~ 0 ~ po

o 0 0 D 0 0 ~QQ Doa

0 Q O Q g p Q S 0 o4 0 0 0 DOQ o 0 o o ~ 0 0Cl 0 0 — 0 a Do- 'D 4 0

8

0

~.oQ P~

~ d. .Qo':

&0 =2~smax = ~0&g "~~~~ )t o„

170.7 d0

&~4 g,op +

09' ~

o ~o . ~X)O %ADO O Q

QQ ogo

o 0~ octo

d O Oooo. &

&o'A~% 0ko

170.7 d &70.7 d,

FIG. 9. "Snapshots" of two-dimensional off'-lattice avalanche models just after the first avalanche has occurred for various values

of D and h, . The initial droplet diameter is 1.5 and the linear dimension of the system is 256.

prediction of a = 1 —1/h, for both two- and three-dimensional droplets. However, for the exponent a' rela-tively large corrections to the asymptotic scaling behav-ior of a(L) on L exist. This appears to stem from thefinite initial width mo-h, ' of the wedge swept out bythe first avalanche.

This finite initial width of the wedge gives rise to a sub-stantial contribution to the mass of an avalanche which islinear in y, in addition to the nonlinear contribution. Toaccount for this linear component, we consider the quan-tity AL':

W' =W —ah,"DpI. , (27)

which is simply the mass of the avalanche with the linearcontribution, whose amplitude is governed by an adjust-able parameter a, subtracted out. By this analysis, theeffective values of the exponent a'(L) were found to con-verge to a value close to 2 —4/h, as 1.~~ for two-dimensional droplets.

For three-dimensional droplets, the fact that the regioncleared by the avalanche has a width which grows moreslowly than linear in the fall distance leads to qualitative-ly different predictions for the exponent a'. In this case,the area swept out by the initial avalanche scales asp' ' "yo ' ", where yo~I. is the distance from the

5932 Z. CHENG, S. REDNER, P. MEAKIN, AND F. FAMILY

- va0 o

Cb oo agQ&:~.p~

O~ ~'Q 0 g ~ yP+Q0 0 p ~ ~So

0 0 ~~, o, goo oS'a~ o ~ 0 ~o

p

o a O

a~p-.o ~io 0 eq~ o. ~

o +Oo

0 ooao O O o ~ 4 ~

o o o 0 ~ s+ 4~+oooo ooo 0 y yo eP ~ q ~ op~o~oooooon+aooo ~ + ~ + ~ ~ joO ~ o a~+ ~ ~ ~gogo e 0 ooo o ~ ~ ~ 9 ~' ~ '

~ pab o o~eOI ~ ~ o~ r 0 ~ o+ ~o4 & oo o o 4 & po a ~ yo 0

O

0 D=2II" a=2

4

gaoo o oo ohio Q 0 o 0 ~ 0 ) ~ o

o ~o n ~ a o sa

170.7 do

~oa oaO+p'

~ go

oo ~

sg '&oQo+

o

oo oO+

o ~(

~4c.~8,

(gGg

cA

..s @CP~g~.54+'4:Po~.o &

~

g d = 2 'g. .~gPPa:: '

O~ 0~ oo ~ ~ 0

~ ~ Oot $~Q,

~5m" -'4''"-"-'.170.7 do

i

sp~ mL

&.&. o

pa++gwA

)0

Qg 4s

o o&&oo~ooo

Vj;$

j(c), Sm

~ D

odQ

170.7 do

..ig(

.' „)0O'Q

8(0

~ ~

FIG. 10. Snapshots of the continuum system in the steady state for various values of D and h, .

start of the avalanche to the bottom of the slope. Conse-quently, the mass in this cleared area scales as(pl. )

' ", and this leads to

believe that the discrepancy stems from our simulationsbeing too small to view the mass in the incoming path ofthe avalanche as a continuum.

D 2D —1 h,

(28) V. DISCUSSION

Our numerical results for large h, are in good agreementwith this prediction (Fig. 13), but not so good for smallervalues of h, . Numerically, we find a'=1.0 and 0.4S, re-spectively, for h, = 5 and 3, while the correspondingvalues from Eq. (28) are 0.9 and 0.5. In these cases, we

We have introduced a simple avalanche model inwhich there is a continuous input of mass on a tilted sub-strate, with avalanches occurring whenever the mass at agiven location reaches a preassigned threshold value. In-spired in part by the sandpile models of Bak and co-workers, ' one of the goals of our study is to ascertain

AVALANCHE DYNAMICS IN A DEPOSITION MODEL WITH. . . 5933

IIo

~l

09

o -10-C:—-11—

-12—

-13—

0=3, d

OFF- L

he=5c= 1O

(a)

hc= 25

-142

S

3 6 7ln (t)

8 9 1O

100I

ll 12

0

0 o0 (b)

w( )

FIG. 11. Growth in the width of the cleared region as adroplet whose radius is proportional to m ' falls a distance dy.The mass within the region swept out by the falling dropletpmdy contributes to the increase in the droplet mass.

E-6—

c-8

—10

—121 5 6

ln (m)

oo

10

=25-

-12-13-14-15

0 1 2 3 4 5 6ln(t)

7 8 9 IO 1 I

-4Ece -6

SL

-10

-12

D=

OF

4 5ln (m)

FIG. 12. Distribution of (a) the times until the first avalancheP (t), and (b) the mass removed by the first avalanche, R (m), ob-tained from o6'-lattice simulations of two-dimensional (D =2)droplets for several values of h, .

the type of dynamical behavior that can occur in opensystems for which transport is inhibited by an intrinsicthreshold. We anticipate that experimental studies ofthis system will provide the impetus for additionaltheoretical work.

A basic feature of our model is that once an avalanche

FIG. 13. A plot similar to Fig. 12, except for three-dimensional droplets (D =3).

starts, it necessarily propagates instantaneously to theend of the system. Thus our model is dominated by iner-tial effects, whereas the self-organized criticality modelsof Bak and co-workers are strongly dissipative in nature.This seems to be an essential reason for why power-lawdecays in the autocorrelation functions associated withmass transport do not arise in our avalanche model. Ex-perimentally, it is not obvious how to design a systemwhich will be dominated by dissipation. In the two re-cent experiments on real sandpiles, ' the systems have tobe driven a finite "distance" from the critical state in or-der to generate mass transport. In this aspect, the experi-ment of Jaeger, Liu, and Nagel exhibits several featureswhich are qualitatively similar to our model. Because thesystem must be driven by a finite amount to generateavalanches, there is a characteristic relaxation timewhich is related in some fashion to this driving distance.However, in the sandpile experiment it seems that theavalanche involves the entire surface, as the slope is de-scribed always by a single angle. This would correspondto the maximal avalanches that occur in our model. Itmight be interesting to perform sandpile experiments inmuch larger systems, so that size-dependent effects, simi-lar to what we observe, might be studied. Furthermore,large sizes might allow one to observe the dissipation thatcould give rise to power-law dynamical behaviors.

While inertial effects appear to play a primary role ingoverning the avalanche dynamics in our model, it is evi-dent that in an experimental realization of water dropssliding down an inclined plane, an avalanche is not neces-sarily catastrophic nor is it instantaneous. Owing to wet-

5934 Z. CHENG, S. REDNER, P. MEAKIN, AND F. FAMILY 40

ting eAects, a sliding drop may leave behind a trail ofliquid, and this mass loss can eventually cause the dropletto stop. This stopping mechanism appears to be analo-gous to the dissipation in the sandpile models. Conse-quently, by generalizing our model to allow for the possi-bility that an avalanche can lose mass and also can stopwhen a lower mass threshold is reached, one might recov-er some of the scaling properties of the sandpile model.It should therefore be interesting to study experimentallythe dynamics of liquid droplets sliding down both wettingand nonwetting substrates.

For our avalanche model, the dynamical behavior de-pends crucially on the ratio of the threshold h, to the log-arithm of the system size L (in one dimension). For smallthresholds, the cleared region downhill from theavalanche can quickly fill up to the avalanche thresholdagain. Consequently, the location and size of successiveavalanches is essentially uncorrelated. For large thresh-olds, a relatively predictable sequence is followed. Thereis a long quiescent period where mass builds up to thethreshold height on the substrate. Then there follows ashort period of intense activity in which each avalanchebreaks off uphill from the previous event. This phaseends when a avalanche begins at the top of the slope, thussweeping the system clean. In the case where L ~ ~, itis possible to formulate a continuum approach that pre-

diets the system size and threshold dependence of manydynamical quantities.

There are several additional aspects of the avalanchemodel which could be modified in order to explore theinfluence of such variations on the dynamics of the sys-tem. These include allowing for a finite propagation timefor an avalanche, and also allowing for an incompletesweeping away of the already-deposited mass as theavalanche passes by. By varying the parameters associat-ed with these generalizations, one might gain a better un-derstanding of underlying mechanisms of the power-lawdecay of correlations in the sandpile models. These gen-eralizations may also prove to be useful in interpretingexperimental systems of fluid drops sliding down inclinedsubstr ates.

ACKNOW LEDGMENTS

We thank D. ben-Avraham, P. Bak, and C. L. Henleyfor helpful discussions. This work was supported in partby the U.S. Army Research Office (Boston University),and by the Office of Naval Research and the PetroleumResearch Fund Administered by the American ChemicalSociety (Emory University). This financial assistance isgratefully acknowledged.

'P. Bak, C. Tang, and K. %'iesenfeld, Phys. Rev. Lett. 59, 381(1987); Phys. Rev. A 38, 364 (1988).

~C. Tang and P. Bak, Phys. Rev. Lett. 60, 2347 (1988); J. Stat.Phys. 51, 797 (1988).

3T. Hwa and M. Kardar, Phys. Rev. Lett. 62, 1818 (1989).4L. P. Kadanoft; S. R. Nagel, L. Wu, and S.-m. Zhou, Phys.

Rev. A 39, 6524 (1989).5J. Souletie, J. Phys. (Paris) 44, 1095 (1983).H. M. Jaeger, C. Liu, and S. R. Nagel, Phys. Rev. Lett. 62, 40

(1989).7P. Evesque and J. Rajchenbach, Phys. Rev. Lett. 62, 44 (1989).sB. Lewis and J. C. Anderson, Nucleation and Growth of Thin

Films (Academic, New York, 1978).F. Family and P. Meakin, Phys. Rev. Lett. 61, 428 (1988).

~oD. Beysens and C. M. Knobler, Phys. Rev. Lett. 57, 1433(1986).J. L. Viovy, D. Beysens, and C. M. Knobler, Phys. Rev. A 37,4965 (1988).

'2B. J. Mason, The Physics of Clouds (Oxford University Press,London, 1957).P. Meakin and F. Family (unpublished).See, e.g. , Z. Cheng and S. Redner, Phys. Rev. Lett. 60, 2450(1988).

'5T. Vicsek and F. Family, Phys. Rev. Lett. 52, 1669 (1984).