Generalized scaling and multifractal phase transition ... - Science Direct

24 March 1997 PHYSICS LETTERS A

Physics Letters A 227 ( 1997) 287-290

ELSEVIER

Generalized scaling and multifractal phase transition in some physical and astrophysical processes A. Bershadskii P.0. Box 39953. Ramat-Aviv 61398, Tel-Aviv. Israel Received 22 November 1996; accepted for publication 8 January 1997 Communicated by V.M. Agranovich

Abstract In our recent Letter [Phys. Lett. A 222 ( 1996) 3751, it is shown (using data taken from the CfA2 catalogue) that there exists a phase transition from fractality to homogeneity in the large-scale universe. In the present Letter it is shown that this phenomenon is a consequence of a generalized scaling (some kind of self-similarity of random systems). It is also shown, by analysis of data obtained by different authors, that the generalized scaling and the multifractal phase transition to homogeneity occur in diffusive and growth processes as well as in percolation. Finally, recent data from the PerseusPisces survey are used to demonstrate universality of this phenomenon for the luminosity-space galaxy distributions of the large-scale universe.

1. Introduction In our recent paper [ 1] a multifractal phase transition to homogeneity in a large-scale luminosityspace galaxy distribution (for the data taken from the CfA2 catalogue) was studied using the HavlinBunde hypothesis [2]. In Ref. [2] this hypothesis has been rigorously proved for linear fractals and it is strongly supported for percolation systems by numerical simulations. Now we would like to show that the Havlin-Bunde hypothesis is a particular case of a generalized self-similarity of random systems anb (by comparison with data taken from different computer simulations and experiments) that a wide class of physical processes (percolative, diffusive and growth processes) satisfy this self-similarity. And finally, we show that the luminosity-space galaxy distribution from the Perseus-Pisces survey belongs to this class as well (cf. the analogous result for the data from

the CfA2 catalogue [ 11) . These facts could be an indication of the universality of this phenomenon.

2. Generalized

self-similarity

Let a random (positive) eralized scaling

field @(r, t) have the gen-

(ti”) N ($qf‘(?

(1)

where f(p) is some function of the moment order p (it is clear that random systems with ordinary scaling have a generalized scaling, see Section 3). Let us consider the map ICI+@

(h>l)

and define a generalized (WA)“)

N (WWP’.

0375-9601/97/$17.00 Copyright @ 1997 Published by Elsevier Science B.V. All rights reserved. PIISO375-9601(97)00064-9

(2) self-similarity

as (3)

288

A. Bershadskii/Physics

where f(p) is the same function as in ( 1). From ( 1) and (3) we can find f(p). Indeed, (@P) N ($)f(*P)

Letters A 227 (1997) 287-290

of turbulence [ 61 (for the velocity differences) and the result p365 2 1.65 was obtained, whereas formula ( 11) giVeS p365 N 1.63.

(4)

and

4. Ordinary scaling

($PP) = ((f/V)

w ((JIA))f’P’ N ($)f(A)f(P).

(5)

If there exists ordinary scaling

Then

(12)

(@P) N rG t f(G)

= f(A)f(P)*

(6)

then

This functional equation has the general solution f(P)

6.0-

(7)

=PY*

P”pq = l-q-

(P/n)&

(s/n)ln~

(13)

where y is some constant.

At the phase transition to homogeneity from ( 11) and ( 13) follows

3. Phase-transition

lp - (P/~>G =-P ln(pln)

to homogeneity

f& Numerical simulations performed in Ref. [ 21 show that the generalized self-similarity occurs in percolative systems [ 31. Moreover, it is shown in Ref. [ 21 that the case y -+ 1 corresponds to transition to homogeneity (see also Ref. [ 1 ] and Ref. [ 41, Section 1.1) . For the generalized self-similarity it is a real phase transition. To show this let us consider dimensionless moments

(V) fip = (+n)p/n

N (@)P’4PlW_

(8)

(q/n)&

4

ln(qln) ’

(14)

It is easy to show (see Ref. [5] ) that the general solution of this functional equation is (15)

lp = Ap ln(p/po).

where A and po are some constants. To demonstrate that the generalized self-similarity occurs not only in the percolative processes let us compare this result with the experimental data obtained for other physical systems. To do this it is suitable to rewrite (15) in the form

Then -5P --a+blnp.

F v N F;iw,

(9)

where the exponent Pn,,4=

PY- bln)nY qy- (q/n)nY.

(10)

Then P 10/n) lim pnpy = - ~

Y-+1

4 ln(q/n)

’

(11)

i.e. even after the transition to homogeneity the exponent pnpq has nontrivial value ( 11) (see also the next section and Ref. [ 1 ] ) . It is interesting that in our recent paper [ 51 the exponent p365 was calculated for data obtained in different laboratories and direct numerical simulations

(16)

P

If we choose the axes of coordinates (y, x) so that y = lp/p and x = lnp, then Eq. (16) (and (15)) is represented by a straight line. We start from the kinetic surface roughening [ 7,8]. The appropriately normalized qth-order correlation function of the height differences can be represented in the following scaling form, cp(r) = (Jh(x+

r) - h(n)IP) - &.

Fig. 1 (adapted from Ref. [ 7 ] ) shows results of recent large-scale simulations of kinetic surface roughening with power-law-distributed amplitudes of uncorrelated noise. The authors of Ref. [7] already pointed out that the sharp change at p - 3 (In p N 1) can be

289

A. Bershadskii/Physics Letters A 227 (1997) 287-290

0.7

.

$

0’s . 0.3 -

0

,

1

2.0

1.5

In

1

P

Fig. 1. The scaling exponent of the pth-order correlation function of the height differences obtained in large-scale computer simulations of kinetic surface roughening with power-law-distributed amplitudes of uncorrelated noise (adapted from Ref. [7] ). The straight line corresponds to F!.q.(16).

an indication of a “phase transition” (in our terms this is the “phase transition” from random fractality to homogeneity, see the previous section). The straight line in Fig. 1 is drawn for comparison with ( 16). It is known that for large p the main contribution in the (I,V) is given by relatively large values of $( r, t) (see, for instance, Ref. [9] ). Then, it is natural to expect that just for the relatively large p the moments (t,V’)should exhibit some universal properties of selfsimilarity (due to the elimination of a low-amplitude noise effects) (see Fig. 1) . The next example of this approach applicability will be diffusion of a passive scalar. We use recent experimental data on multifractality of passive scalar concentration &Ee~rzces in the atmosphere cP(r) = (10(x + r) - O(x)IP) N rsp.

P

‘n

I

2.5

2

Fig. 2. The scaling exponent of the concentration differences moments obtained in the atmosphere [ lo]. The straight line corresponds to Eq. (16). The same data (for integer values of p) were also obtained in the experiment [ 1 I 1.

5. Generalized scaling and generalized dimension

Suppose that the total volume of a sample consists of a d-dimensional cube of size L. We divide this volume into N boxes of linear size r (N - ( L/r)d). We label each box by the index i and construct for each box the measure function of a field p( x, r) , (17) where ui is the volume of the ith box. Then the generalized dimension, D,, can be introduced by the following scaling relationship (see, for instance, Ref. [ 121 and references therein) N

z, =

c

[pj(r)]‘N

r(p-*)D~,

(18)

i=l

Let us define a local r-averaging

k(r)

L,(r) =

rd

(19)

’

Then Fig. 2 shows the experimental data obtained in the atmosphere (25 m above ground) [ lo]. The passive scalar (0) in this experiment was a temperature. The straight line is drawn in Fig. 2 for comparison with Eq. ( 16). The same values of lP were also obtained in another recent experiment [ 111.

([-iqr)]P)

_

Cil’TL,(r)lP N N

r(D,-d)(p-I)

.

(20)

If the local r-averaged field, L,, satisfies the generalized self-similarity, then we obtain from the previous section

290

A. BershadskiUPhysics

Cd- D,>)(P- 1) =&W/PO),

(21)

where A and pe are some constants. If the measure chosen is adequate to the multifractal process (under consideration), then (in a formal way) the constant po = 1 in relation (21) (cf. 1.h.s. and r.h.s. terms in (2 1) ) . An interesting example of a percolative system with generalized scaling and multifractal phase transition one can find in Ref. [ 131. To obtain the generalized dimension of the percolation cluster surface (in twodimensional space) the authors of Ref. [ 131 used the so-called surface mass exponent yn, such that D,=Dy,

(n=q-

l),

(22)

where D = 1.89 is the fractal dimension of the set on which the adequate (so-called harmonic) measure resides. If one seeks the generalized scaling here, then one should use d = 1 in (21) (let us recall that the space is two-dimensional in this simulation). If we denote a = Y

(d-D,)(qqln(q)

1)

(23)

’

then the condition ay = const

(24)

is an indication that we are dealing with the generalized scaling (21). We obtain from the data of the computer simulation of Ref. [ 131

Letters A 227 (1997) 287-290

A good agreement with (24) occurs for 9 > 4 in this case as well. Another example of the generalized scaling one can find in the data represented in a recent paper [ 141. In this paper a set of D, for the luminosity-space galaxy distribution of the Perseus-Pisces survey was obtained. We use these data to calculate aq (d = 3 in this case), al.5 N 0.83, u3 N 0.80,

ad.5 E 0.82,

a3 N 0.08,

a4 E 0.09,

as 2 0.09,

a6 c” 0.09,

a7 = 0.09,

as ? 0.09,

ag cu 0.09.

ISI [91 [ 101

(26) I121 I131

(cf. (22) ) and we obtain a2 N 0.08,

a3 2 0.10,

a4 N 0.11,

as N 0.11,

@j !? 0.11,

a7 Y

us ‘v 0.11,

a9 2

0.11.

1141

0.11,

[ISI

(27)

a4 N

0.82,

and A. Aharony, Introduction to percolation theory (Taylor and Francis, London, 1992). J. Beck and W.W.L Chen, Irregularities of (Cambridge Univ. Press, Cambridge, 1987). A. Bershadskii, J. Phys. A 29 ( 1996) L453. R. Benzi et al., Europhys. Lett. 32 (1995) 709. A-L. Barabasi, R. Bourbonnais, M. Jensen, J. Kertesz, T. Vicsek and Y-C. Zhang, Phys. Rev. A 45 ( 1992) R6951. A-L. Barabasi and T. Vicsek, Phys. Rev. A 44 ( 1991) 2730. A. Bershadskii and A. Tsinober, Phys. L&t. A 165 ( 1992) 37. F. Schmitt, D. Schertzer, S. Lovejoy and Y. Brunet, Europhys. Lett. 34 (1996) 195. C.G. Ruiz, C. Baudet and S. Ciliberto, Europhys. Len. 32 (1995) 413. H.E. Stanley and P. Meakin, Nature 335 (1988) 405. P Meakin, A. Coniglio, H.E. Stanley and T.A. Witten, Phys. Rev. A 34 (1986) 3325. ES. Labini, A. Gabrielli, M. Montuori and L. Pietronero, Physica A 226 (1996) 195. P Garrido, S. Lovejoy and D. Schertzer, Physica A 225 (1996) 294.

I31 D. Stauffer

(25)

(n=q-1)

0.80,

Ill A. Bershadskii, Phys. Lett. A 222 (1996) 375. 121 S. Havhn and A. Bunde, Physica D 38 ( 1989) 184.

[III

D,=1.7Oy,

>

a.5 E 0.81.

References

151 [61 [71

An analogous result could be obtained from the computer simulation of semilattice DLA aggregates [ 131. In this case

q5

~2.5 N 0.80,

One can see good agreement with (24) beginning from q = 2.5. It should be noted that the capacity dimension of the total sample, DO, turns out to be approximately equal to 2 in this case [ 141. An analogous result was obtained in Ref. [ 1] for the data taken from the CfA2 catalogue [ 151, which could be an indication of universality of the generalized scaling in the large-scale universe. Thus, we could claim that the generalized scaling and the multifractal phase transition to homogeneity occur for a wide class of physical and astrophysical processes.

[41

a2 N 0.06,

a2 E 0.76,