Active-Site Motion and Pattern Formation in Self-Organised Interface Depinning

4 Pages·0.1 MB·English

by  Supriya Krishnamurthy

Active-Site Motion and Pattern Formation in Self-Organised Interface Depinning
Supriya Krishnamurthy and Mustansir Barma
Theoretical Physics Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
e-mail: skrish,[email protected]
of interface readjustment and the long-ranged dynamics
We study a dynamically generated pattern in height gra-
of active-site motion. This provides an understanding of
dients, centered around the active growth site, in the steady
the form of the induced pattern as a function of spatial
state of a self-organised interface depinning model. The pat-
correlations in the activity.
ternhasapower-law tailand dependsoninterface slope. An
In our model the interface is taken to be a directed
approximateintegralequationrelatestheprofiletolocalinter-
path on a square lattice (Fig. 2), with tilted cylindri-
face readjustments and long-ranged jumps of the active site.
6 cal boundary conditions [4] which ensure that the mean
The pattern results in a two-point correlation function satu-
9 slope is preserved. To every bond k on the lattice is
9 ratingtoafinitevaluewhichdependsonsystemsize. Pattern
pre-assigned a fixed random number f drawn from the
1 formation is generic to systems in which the dynamics leads k
interval [0,1]. Motion of the interface is initiated at that
to correlated motion of the active site.
n bond, on the forward perimeter of the interface, which
a
PACS numbers: 47.54.+r, 68.10.Gw, 05.40.+j, 47.55.Mh carries the smallest random number. In order that the
J
length of the interface not change (an effect of surface
5
tension),itsdirectedwalkcharacterispreservedbylocal

A common feature of many self-organised systems is re-adjustments; if the chosen minimal bond has a posi-
1
v that their dynamics is extremal [1]: that is, activity is tive (negative)slope, the sequence of links with negative
triggered at that spot where a link is weakest, or a force
3 (positive) slope just below (on the left) also advance [5],
1 thestrongest,andnotstatisticallyuniformlyoverthefull as illustrated in Fig. 2. This model resembles that in-
0 system. As a result, the active site performs an erratic troduced by Sneppen [6] but differs from it in that the
1 motionwithnontrivialcorrelationsinspaceandtime[2].
length of the interface is a strict constant of the motion
0 In this Letter, we demonstrate an intriguing and generic
andthe f ’s areassociatedwith bonds ratherthan sites.
6 k
feature of such systems: the existence of a well-defined
Theinterfacedynamicsisequivalenttothedynamicsof
9
spatial pattern around this dynamical center of activity.
/ asystemofhard-coreparticlesonaring,withapositive-
t We study this phenomenon in a model of a self-
a slopelinkoftheinterfacerepresentedbyaparticle(nj =
m organizing interface in a random medium. In steady 1) and a negative-slope link by a hole (n = 0); see Fig.
j
state, we find that there is a nonzero time-averagedspa-
2. The difference in height of the interface between sites
-
d tial pattern of height differences, provided that it is re- j and j is given by h −h = j2 (2n −1). Each
n ferred always to the location of the active site at that 1 2 j2 j1 j=j1 j
o instant. Figure 1 is a schematic representation of the site j of the ring also carries the rPandom number fj as-
signed to the bond in front of the corresponding spot on
c
interface profile, time-averaged in a frame which keeps
: the interface. In each time step, activity is initiated at
v the active site at the origin. Interfaces in which r →−r
the sitewiththe minimumf . Ifthissite containsapar-
i symmetry is present (Fig. 1a), or absent (Fig. 1b), are j
X ticle (hole), it exchanges with the first hole (particle) on
shown, symmetry-breaking being induced by tilting the
r the left (right). All sites hopped over, including the two
a interface[3]. Inthe formercase,the profilehasacuspat which exchange the particle and hole, are refreshed by
the origin, while in the latter case the dominant feature
assigning a new set of f ’s. The overall particle density
j
is a larger slope near the origin — a local amplification
ρ is strictly conserved, and determines the mean slope
of the broken symmetry. The activity-centered pattern
m=2ρ−1 of the interface. There is a nonzero current,
(ACP) is defined in terms of height gradients,
which translates into a forward-advancing interface.
1 ¿Fromtimetotime,theinterfacealignsalongdirected
Ψ(r)≡ [h∇h(r+R(t))i−m], (1) spanningpaths ina percolationproblem,as inthe Snep-
2
pen model [7,8]. In our case, the corresponding perco-
where R(t) is the position of the active site at time t, lation problem is simpler, viz. directed bond percola-
h(r′)denotestheheightatthesiter′,h...iisatime aver- tion (DP) [9] on the dual lattice, or equivalently, diode-
age and m is the overall slope of the interface. ¿From resistor percolation (DRP) [10] on the original lattice.
numerical simulations, we find that Ψ(r) has a long- Foragivenconfiguration{f }andatrialthresholdvalue
k
ranged (power law) tail with exponents which differ in fo, occupy all bonds k′ with fk′ > fo by diodes (one-
tiltedanduntiltedinterfaces. Wealsofindthatthe ACP wayconnectionspointingagainstthedirectionofadvance
leads to an unusual finite size effect, namely the satura- of the interface, see Fig. 2), and place resistors (two-
tion of a two-point correlation function with increasing way connections) on the remaining bonds. The diode
separation. Finally, we present an approximate integral concentration is p = 1 − f . Infinite connected paths
o
equation which relates the ACP to the local dynamics
1
(stoppers) first form along the easy direction of DP at P+(l), decays asymptotically as P+(l) ∼ |l|−π+ with
p = pDP ≃ 0.6445 on the square lattice. An untilted π+ =2.00±0.02. TheoddpartP−(l)changessign(asim-
interface (m = 0) aligns along stoppers in the course of pliedby the crossingofthe curvesinFig. 4)andasymp-
its motion [7,8]. For m6=0, the corresponding threshold totically follows P−(l) ∼ |l|−π− with π− = 2.49±0.06.
pc(m)>pDP is that value of p for which the edge of the We verified that the values of π+ and π− are the same
DP cone [9] has slope m. As it evolves,a tilted interface for various ρ6=1/2.
of slope m aligns along stoppers of that slope [11]. Equation (2) can be solved by Fourier transform.
The pattern defined in (1) is associated with a spa- Defining Ψˆ(q)≡ +∞e2πiqrΨ(r)dr etc we find
−∞
tial profile of the density in the particle-hole model, as R
the heightgradientmapsontothe density. The defining Φˆ(q)Pˆ(q)
equation (1) now reads Ψ(r) = hn(r +R(t))i−ρ. We Ψˆ(q)= . (3)
1−Pˆ(q)
studied Ψ(r) by Monte Carlosimulation. In the untilted
case, Ψ(r) is an odd function (Fig. 3a) decaying asymp-
ThisequationindicatesthattheACParisesasanonlocal
totically as a power law |r|−θ with θ = 0.90±0.03. In
response to the local density-increment function Φ(r).
the tilted case (Fig. 3b), as there is no r → −r symme-
Notice that the denominator vanishes as q → 0. We
try, Ψ(r) does not have a definite parity. It is useful to
have solved (3) numerically, but it is more instructive to
separately analyse Ψ±(r)≡(Ψ(r)±Ψ(−r))/2. The odd examine the small-q (large-r) behaviour.
part decays as Ψ−(r) ∼ |r|−θ− with θ− = 1.04±0.05. In the untilted case, the r → −r symmetry implies
The even part Ψ+(r) ≈ −b(L)+a |r|−θ+ where θ+ = that P(l) is even, while Φ(r) and the profile Ψ(r) are
0.46±0.05 and b(L)→0 as the lattice size L→∞. both odd. Φ(r) is a short-ranged function with finite
The density profilesof Fig. 3 correspondto the height firstmoment φ . Hence, to first order,Φˆ(q)≈iφ q. The
1 1
patterns shown in Fig. 1. In the untilted case, on aver- power-law tails in P(l) imply that Pˆ(q) ≈ 1−A|q|π−1.
age,theactivesiteislocatedatthepeak(Fig. 1a)where Thus we find Ψ(r) ∼ sgn(r) |r|−(3−π) i.e. the profile is
f ’s which have not been sampled earlier are most likely
k an odd function, with a power law tail.
tooccur. Inthetiltedcasethereisalsoabootstrapeffect
In the tilted case, the absence of r → −r symmetry
at work. If ρ≥0.5, the active site is more likely to con-
implies that none of P(l), Φ(r) and Ψ(r) has a definite
tain a particle. Thus regions to the left are more often
parity. Since Φ(r) is short-ranged, Φˆ(q) ≈ iφ q +φ q2
refreshed, making the active site likely to move in this 1 2
as q → 0. There is no φ term, as the elementary step
direction and find itself amidst a particle cluster. This 0
of hopping a particle or hole conserves particle number,
leadstodensitieshigherthanρonbothsidesoftheactive
implying Φ(r)dr = 0. The q → 0 behaviour of Pˆ(q) is
site (Fig. 3b) for ρ sufficiently different from 0.5.
determineRd by the asymptotic power law decays of the
Morequantitatively,wemaywriteanintegralequation
for Ψ(r) for untilted and tilted interfaces. If the pattern even and odd parts P±(l) as |l| → ∞. Thus we have
is centered at R(t) at time t, the dynamics causes two Pˆ+(q)≈1−A|q|π+−1. WemighthaveexpectedPˆ−(q)≈
changes at the next instant: (i) A short-ranged read- Bq + C sgn(q) |q|π−−1, but in fact the mean velocity
justment of the interface changes the profile near R(t). lP(l)dl of the active site vanishes [13] implying B =0.
As a result, the local density to the right of the active RThustheintegralequationpredictsthattoleadingorder,
site becomes less than the density to its left. The aver- both Ψ+(r) and Ψ−(r) decay as powers ∼ |r|−θ±, with
age density change at site R(t)+r defines the density- θ++2π+−π− =3andθ−+π+ =3. Thepredictionπ++
increment function Φ(r). (ii) The active site jumps a θ− = 3 compares quite well with the numerical values
distancel ≡R(t+1)−R(t);thereisasubstantialproba- 3.15fortheuntiltedcaseand3.04forΨ−(r) inthetilted
bility P(l)for largejumps. Since the ACP is centeredat case. For Ψ+(r), however, the numerically determined
theactivesite,theresultof(i)followedby(ii)isthatthe value of θ++2π+−π−(≃ 1.97) deviates more from the
averageprofilereproducesitself,exceptthatitiscentered predicted value 3. The reason behind the discrepancy is
atthe shiftedsite R(t)+l. Botheffects areincorporated that (2) implicitly ignores correlations e.g. in successive
into the integral equation jump lengths of the active site.
To check this point, we numerically studied a model
+∞ with no correlations in successive jump lengths. At ev-
Ψ(r)= [Ψ(r−l)+Φ(r−l)]P(l)dl. (2)
Z ery instant the jump length of the active site was taken
−∞
from a power-lawdistribution P(l) chosensuitably so as
P(l) decays as a power for large l: P(l) ∼ |l|−π (Fig. to implement the zero velocity constraint. Short-ranged
4). In the untilted case P(l) = P(−l) holds because adjustments were taken to follow the same particle-hole
of r → −r symmetry. We find π = 2.25±0.05 which hoppingrulesastheextremalmodel. Thismodelissimi-
compareswellwithearlierdeterminedvaluesofπ forthe larinspiritto the L´evyflightinterface modelconsidered
Sneppen model [12,8,2]. In the tilted case (ρ 6= 1/2), in[12]. Theresultsinthiscasewerefullyconsistentwith
P(l) is not a symmetric function (Fig. 4) and it is con- the predictions θ++2π+−π− =θ−+π+ =3. Further,
venient to separately analyse the even and odd parts we also studied cases in which P(l) is not a power law.
P± ≡ (P(l) ± P(−l))/2. We find that the even part, For short ranged P(l), the ACP has the form of a kink
2
function, with a particle (hole) - rich region on the left organised interface depinning, namely the existence of a
(right) of the active site. In the limit of strictly infinite spatial pattern centered around the active site. We have
ranged P(l), tantamount to normal stochastic evolution shownthatthepatterninheightgradientshaspowerlaw
[5],theACPceasestoexist. Inallcaseswecomparedour tails which are sensitive to r →−r symmetry, brokenby
numerical results with those predicted by (2) and found tiltingtheinterface,andthatitexplainsthesaturationof
good agreement. thetwo-pointcorrelationfunctionwithincreasingsepara-
TheACPhasseveralinterestingconsequences. Forin- tion. The integral equation (2) provides an approximate
stance, we expect there to be a larger length of interface descriptionoftheformationofthepattern. Wehavealso
in a region of fixed size x around the active site, than shown that power-law patterns exist in other quantities,
in a regionopposite it. Accordingly, we monitored mean for instance the distribution of random numbers around
squared fluctuations of the height around the instanta- theminimum,inboththeinterfaceandevolutionmodels.
neous average,in regionsaroundandopposite the active Our studies suggest that systems which evolve through
site, and found a pronounced difference (factor ≃ 2, for extremaldynamics are likely to exhibit activity-centered
tilted and untilted cases with x = 256,L = 4096). This patterns.
effect is smaller at stoppers, indicating that the ACP it- We thankGautamMenonandDeepakDharforuseful
self is suppressed there. The excess length of interface discussions, and the referee for his comments.
associated with the ACP may provide a useful way to
identify the active region in experiment.
Another consequence of the ACP is that being
a one-point correlation function, albeit an unusual
one, it has a strong effect on the customary, space-
time averaged two-point correlation function C(∆r) ≡
{hn(r′)n(r′+∆r)i}−{hn(r′)ihn(r′+∆r)i}. Here r′ is [1] D. Wilkinson and J. Willemsen, J. Phys. A 16, 3365
a fixed site on the lattice, h...i stands for a time av- (1983); S. Roux and E. Guyon, J. Phys. A 22, 3693
erage and {...} stands for an average over all sites r′. (1989).
Numerical results for C(∆r) (Fig. 5) show that it sat- [2] S.Maslov,M.PaczuskiandP.Bak,Phys.Rev.Lett.73,
urates at a value Csat which decreases with increasing 2162 (1994); M. Paczuski, S. Maslov and P. Bak, Phys.
Rev. E, to appear.
size L [14]. This unusual behaviour can be understood
[3] L. -H. Tang, M. Kardar and D. Dhar, Phys. Rev. Lett.
in terms of the ACP. Consider the correlation function
74, 920 (1995).
Γ(r,∆r) = hδn(r+R(t))δn(r+∆r+R(t))i where R(t)
[4] M. Barma, J. Phys. A 25, L693 (1992).
is the location of the active site , r is the distance
[5] These rulesaresimilar tothoseoflow-noise Toom inter-
from the active site and δn(r +R(t)) ≡ n(r +R(t))−
face dynamics; see B. Derrida, J. Lebowitz, E. R. Speer
ρ − Ψ(r) is the fluctuation around the average ACP. and H.Spohn,J. Phys.A 24, 4805 (1991).
A reasonable expectation is that the δn’s are indepen- [6] K. Sneppen,Phys.Rev.Lett. 69, 3539 (1992).
dent for large separations ∆r, i.e. Γ(r,∆r) → 0 as [7] L.-H.TangandH.Leschhorn,Phys.Rev.Lett.70,3832
∆r → ∞. On averaging over r (which has the same
(1993); Z.Olami,I.Procaccia andR.Zeitak,Phys.Rev.
effect as the average over space-fixed sites r′), we see E 49, 1232 (1994).
that {hn(r)n(r+∆r)i} − {(ρ+Ψ(r))(ρ+Ψ(r+∆r))} [8] H. Leschhorn and L. -H. Tang, Phys. Rev. E 49, 1238
approaches zero as ∆r → ∞. This predicts the satura- (1994).
tionvalueCsat tobe{(ρ+Ψ(r))(ρ+Ψ(r+∆r))}−ρ2. To [9] W. Kinzel, in Percolation Structures and Processes, eds.
test this, we subtracted this from C(∆r) and found that G. Deutscher, R. Zallen and J. Adler (Hilger, Bristol,
the saturation effect is in fact suppressed strongly (Fig. 1983), p.425.
5), supporting our interpretation [15]. Customarily, the [10] D. Dhar, M. Barma and M. K. Phani, Phys. Rev. Lett.
large-separation saturation of C(∆r) is associated with 47, 1238 (1981).
nonzero values of hn(r′)i−ρ; the unusual aspect here is [11] Forothermodelsoftiltedinterfaces,seeL.A.N.Amaral,
that there is saturation even though hn(r′)i=ρ. A.-L.Bar´abasi, and H.E.Stanley,Phys.Rev.Lett. 73,
Though we have focussed on the ACP associatedwith 62 (1994) and Ref. [3].
a conservedquantity like the density, a conservationlaw [12] K. Sneppen and M. H. Jensen, Phys. Rev. Lett. 71, 101
(1993).
is not essential for the occurrence of a nontrivial ACP.
[13] Wenumerically verifiedthatthemeanvelocity vanishes.
Forinstance,onmonitoringthedistributionoff ’s,with
r
This is related to the probability of first return of the
r referred to the active site, we find profiles with power-
activity to a site decaying as a slow power of time.
lawtailsintheinterfacedepinningmodelwithandwith-
[14] Atmuchlargerr(∝L),C(r)changessign,aresultofthe
out tilt [16]. We also find similar patterns in the Bak-
particle conservation sum rule C(r)dr=0.
Sneppen model of evolution [17] and a modified asym-
[15] It is possible that another slighRtly different definition of
metric version of it [16]. This indicates that pattern for-
the pattern would eliminate the slight shoulder which
mation is a generic feature of systems evolving through
remains after subtraction (Fig. 5).
extremal dynamics.
[16] S. Krishnamurthyand M. Barma, unpublished.
In summary, we have explored a new aspect of self-
3
[17] P.BakandK.Sneppen,Phys.Rev.Lett.71,4083(1993).
Figure Captions
FIG.1. Schematicviewofaveragedinterfaceprofilearound
the active site A in (a) untilted (b) tilted interfaces. Tilt
breaks r→−r symmetry,an effect that is amplified near A.
FIG.2. Thebondmodel. ThetiltedinterfaceII′advances
alongtheextremalperimeterbondAandlocally readjuststo
align along the dashed line. At thenext instant, theactivity
moves from A to A′. The corresponding configuration and
local moves for the particle-hole model are also shown. SS′
is a stopper, whose perimeter is fully occupied by diodes.
FIG. 3. Density profiles in the (a) untilted (ρ = 0.5) and
(b) tilted (ρ=0.75) cases. Inset: Theodd part of theprofile
in tilted (open circles) and untilted (filled circles) cases. We
used L=16384 and averaged over108 configurations.
FIG. 4. Monte Carlo results for the probability distribu-
tionofthejumpoftheactivesiteforthreedifferentdensities,
ρ = 0.5 (plus sign), ρ = 0.75 (circles) and ρ = 0.84375 (tri-
angles). If ρ 6= 0.5, there is a left-right asymmetry and the
asymptotic slope differs from that for ρ = 0.5. We used L=
65536 and averaged over3.109 configurations.
FIG. 5. Density-density correlations for L = 4096 (open
circles) and for L = 16384 (squares). The saturation is re-
ducedstrongly (filledcircles) onsubtractingthecontribution
of the ACP.We averaged over106 configurations.
4