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3S4
Appendix A
Algorithm for growing diffusion-limited aggregates
To generate large DLA clusters one has to use tricks because the orig
inal version of the model would require prohibitively large amounts of com
puter time. In the following, three simple procedures will briefly be described
which are helpful in reducing the computational time substantially and allow
one to grow - using a mainframe - aggregates consisting of several millions of
particles. Of course, a personal computer is much slower and the size of the
clusters one can generate in a PC is in the range of a few thousand particles.
i) The first trick is that we release the particles from a circle of radius R0 which is just a bit larger than the largest distance between the particles already belonging to the aggregate and the origin. This method is justified because the particles released very far from the cluster arrive (for the first time) at different points of a circle of radius larger than R0 and centred at the origin with the same probability.
ii) Whenever a randomly walking particle leaves the region which is inside the above circle, the distance AR of the particle from the circle is determined. Then the next step made by the particle is a jump of length AR from its actual position made in a randomly selected direction (the final position is the lattice site closest to the coordinates of the particle after the jump has taken place). Inside the circle of radius R0 the particle always undergoes a random walk jumping one lattice unit choosing one of the nearest neighbour sites randomly. If the particle making large jumps outside of the circle gets too fax (e.g. ten times R0), it is killed and a new particle is released from the circle R0.
iii) A further improvement can be achieved using the following algorithm. At the beginning one assigns a quite arbitrary value /' = lmax to all
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Appendix A 335
lattice sites. (A reasonable choice is in the range of 15-30.) Then, for each site from which the random walker could reach the seed (cluster) by making a jump of length V lattice units the value assigned to the site is changed to
/' if V < 'max- In o t h e r words> c l o s e t o t h e c l u s t e r l' i s t h e m a x i m u m s t e p
length a random walker may take without crossing the arms of the aggregate. Thus the particles jump a distance which is either V or AR (the latter choice is made if SR > U ) . After a new particle has been added to the cluster the I' values in the region where it landed are updated so that the next particle could not make a jump which would result in crossing any parts of the aggregate.
It is straightforward to extend the above method to the off-lattice case by making use of an underlying virtual lattice. In this case the J'-B are assigned to the lattice elements (for the square lattice these are the unit squares bounded by four bonds) and one keeps track of the distances from the cluster and the trajectory of a particle by detecting the lattice elements visited by the off-lattice walk. Below a sample program is given which was written in BASIC for the IBM PC with a colour graphics card. It generates DLA clusters on a square lattice. This short version does not contain trick iii) which is more effective when one grows very large clusters.
10 REM DIFFUSION-LIMITED AGGREGATION ON THE SQUARE LATTICE 20 REM SEED PARTICLE AT NO,NO 30 REM NEXT PARTICLE IS RELEASED FROM A CIRCLE AND UNDERGOES ,0 REM A RANDOM WALK. IT STICKS TO THE PARTICLE AT THE ORIGIN WHEN 50 REM ARRIVES AT AN ADJACENT SITE. A NEW PARTICLE IS RELEASED ... 60 REM N - LINEAR SIZE OF THE CELL 70 REM N9 - NUMBER OF PARTICLES TO BE DEPOSITED 80 REM A(I.J)-1 IF THERE IS AND A(I.J)-0 IF THERE IS NO PARTICLE 90 REM AT THE SITE I,J 100 OPTION BASE 1 110 SCREEN 2 120 DIM A(90,90),J$(100),B(10,10) 130 CLS 140 N=49 150 C=1 160 N9=50
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336 Appendix A
170 R1=2 180 R2=1 190 W=4 200 N0=(N+1)/2
210 CIRCLE(N0-8,N0-W)I3:CIRCLE(N0-8,N0^)I2:CIRCLE(N0-8,N0^)|1 220 R3=R1 230 R4=5*(R3+R2) 240 FOR 11=1 TO N 250 FOR 12=1 TO N 260 A(I1,I2)=0 270 NEXT 12
280 NEXT 11 290 10=1 300 REM PUT A PARTICLE AT THE ORIGIN 310 A(N0,N0)=1 3 2 0 REM START GROWING 330 RANDOMIZER 59) 340 FOR 19=2 TO N9 ( 350 F1=6.28319*RND(1) 360 I=INT((R3+R2)*SIN(F1)+N0+.5) 370 J=INT((R3+R2)*COS(F1)+N0+.5) 380 X1=I 390 Y1=J ,oo REM JUMP 0NT0 A CIRCLE 0F RADIUs R6 (wHjcH ̂ THE DISTANCE
410 REM THE AGGREGATE) IF R6>2 420 R5=(l-N0)*(l-N0)+(J-N0)*(j-N0) 430 R5=SQR(R5) 440 R6=R5-(R3+R2) 450 IF R6 < 2 THEN GOTO 570 460 R7=6.28319*RND(1) 470 X1=X1+R6*SIN(R7) 480 Y1=Y1+R6*C0S(R7) 490 I=INT(X1+.5) 500 J=INT(Y1+.5) 510 R5=(I-N0)*(I-N0)+(J-N0)*(J-N0) 520 R5=SQR(R5)
530 IF R5 < 20.1 THEN CIRCLE(8*J,4*1),3 540 IF R5 < 20.1 THEN CIRCLE(8*J,4-1),3,0
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550 GOTO 760 560 REN WALK BY STEPS 570 K1=INT(RND(1)*4+1) 580 ON K1 GOTO 590,610,630,650 590 1=1+1 600 GOTO 660 610 1=1-1 620 GOTO 660 630 J=J-1 640 GOTO 660 650 J=J + 1 660 REM 670 M1=I-1 680 M2=J-1 690 P1=I+1 700 P2=J+1 710 CIRCLE(8*J,4*I),3: CIRCLE(8*J,4*1),3,0 720 A1=A(M1,J)+A(P1,J)+A(I,M2)+A(I,P2) 730 IF A1 >= .1 THEN GOTO 810 740 X1=I 750 Y1=J 760 R5=(I-N0)*(I-N0)+(J-N0)*(J-N0) 770 R5=SQR(R5) 780 IF R5 > R4 THEN GOTO 350 790 GOTO 400 800 REM FIX NEW PARTICLE 810 A(I,J)=1 820 10=10+1 830 IF R5 > R3 THEN R3=R5 840 R4=5*(R3+R2) 850 CIRCLE(J*8,I*W),3 860 NEXT 19 870 END
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338
Appendix B
Construction of a simple Hele-Shaw cell
In this Appendix a few details are given for those who are interested
in the construction of a Hele-Shaw cell. To build a versatile Hele-Shaw cell is
not particularly troublesome but the application of a few tricks makes the ex
periments easier to carry out. A schematic picture of a possible arrangement
is shown in Fig. B.l. The related information is given below.
Figure B . l . Schematic picture of a radial Hele-Shaw cell. Its cross-section is also indicated. The distance between the plates is denoted by b.
i) The sizes of the upper and lower plates are 27 x 27 cm and 34 x 34
cm, respectively. They are made of good quality glass of width 5mm.
ii) Air is injected into a viscous fluid through a hole of radius 3 m m drilled at the centre of the upper plate.
iii) In order to prevent the viscous fluid from flowing out, walls are at-
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Appendix B S39
tached to the plates in a manner shown in Fig. B . l .
iv) The distance between the plates is controlled by inserting between
them thin metallic strips.
v) At large pressures one needs to clamp the two glass plates together
either by using screws or a heavy frame.
The above basic arrangement can be modified in a number of ways and,
correspondingly, many types of viscous fingering patterns can be observed.
By inserting a third glass plate (in between the two original ones) with a mesh
etched on its surface, one is able to study the effects of anisotropy. If small
balls are spread randomly on the surface of the third plate the structure of
the interface becomes a random fractal, similar to the geometry of diffusion-
limited aggregates.
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340
Appendix C
Basic concepts underlying multifractal measures
In Chapter 3. we have discussed a formalism and algorithms which can be used to characterize the properties of fractal measures. Here we describe the underlying basic concepts presented (after this book was finished) in a recent paper by Mandelbrot (1988) to be published in Fluctuations and Pattern Formation edited by H. E.Stanley and N. Ostrowsky (Kluwer, Dordrecht and Boston).
In example 3.1 a fractal measure is constructed by recursion. This measure fails to have density, and it is not discrete. For example, if one goes from e = (1/3)* to e = ( l / 3 ) * + 1 , the sharing of the measure in an original interval of length e among its three parts is usually very uneven. Thus, the values of this measure in the k -> oo limit can not be described by any distribution function.
To define the appropriate quantity, we fix a given box size e, and for this e plot the corresponding distribution of the measure (number of boxes with the given value of the measure in them) using double logarithmic coordinates. Then a multifractal measure has the property that reducing both logarithmic coordinates by the same factor In e, the reduced plots of the distribution converge to a limit probability distribution p(a). The reduced horizontal logarithmic coordinate is nothing else than a as defined in Section 3.1. while the function f(a) is given by f(a) = p(a) + 1.
Let us now generalize example 3.1 as done by Mandelbrot 1988, and use this new, multinomial version to demonstrate how one can relate the relevant quantities of the theory in a simple way. Denoting by P, the weights given to the ith box (t = 1,2, ...b) at the first step of the recursion we find
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Appendix C 341
that
a = - ^<p,- log 6 # and £ = - ^<£>ilog6 V?i, i «
where 6 is the box fractal dimension and a = log[//(e)]/loge with
Ai(6) = P * V l . . . P ^ .
Here <pi is the relative frequency of the digits i in the 6-base development of
x _ O.771772...r?jfc. The set of boxes with the same a is dominated by the term
corresponding to the highest dimension. This term maximizes the expression
- £ i ^ l o g 6 < / > i for given -<^ log 6 Pi = a and £V ifi = 1. To calculate ip{
one uses the classical method of Lagrange multipliers in which a multiplier
denoted by q ( -00 < q < 00) is introduced. The application of the method
gives
6? log* Pi p?
In terms of the cumulant generating function r(q) = - l o g a J ^ P / the La-
grange multipliers determine q and f(a) from a by
i q i
max 0 = f(a) = y p? '
This means that
In the above expressions q, r and / formally appear as the inverse tempera
ture, the Gibbs free energy, and the entropy, respectively. It is clear that for
such multinomial measures a > 0 and 6 > 0, therefore, / ( a ) > 0.
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342 Appendix C
The latter inequality is, however, not necessarily true for all multifrac-tal distributions. Obviously, if f(a) is viewed as a fractal dimension of the set characterized by the Holder exponent a, / (a) < 0 can not correspond to a fractal dimension in its usual sense. For the / (a) < 0 case Mandelbrot (J. Stat. Phys. 34 (1984) 895) introduced the term latent dimension, expressing that this property of the measure is present, but hidden, and can be observed using high-dimensional cuts from the measure (Mandelbrot 1988).
The above described multinomial measure can easily be genealized to the random case (Mandelbrot 1974). Clearly, if the weights P< are randomly redistributed before each stage of the recursion process, the / (a) of the resulting random multiplicative measure will be the same as that of the deterministic one. As a further step, one can also go from measures embedded into one dimension to multifractals defined on the set of points {x} in an ^-dimensional space, and suppose that b = BB with positive integers B and E. If b/B > 1, then the weights in the boxes along a one-dimensional cut from the measure can be regarded as statistically independent. Assuming independence means that the values P< of the random multiplier M can take on any of the values mj- = m ^ - i with the same probability.
Let us now express the amount of measure n(dx) in the B-adic interval of length B~k of the one-dimensional section assuming that this interval is at x = 0.771,-77*. Since we do not know the exact amount of the measure along the cut (it is normalized to be equal to 1 only in the ^-dimensional space), we need a prefactor fifa,...!,») which is due to stages of frequency larger than Bk. Thus
M(«K) = ft(»7i,...r7t)M(T71)....M(ri1,...rijb)...
Here the the multipliers M are identically distributed and independent. The same is valid for the random factors Q. Introducing aL = - ( l /*)pogB M(m)+ logs M(m,%) + ...] anda* = -(l/k)]ogBil gives a = aL + aH. For k - oo aH becomes negligible, while aL is the average of it independent random variables. According to the Cramer limit theorem, as k -» oo the quantity
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Appendix G 348
(l/fc)log6(probability density of <*L)
converges to a limit which is here denoted by p(a). However, this p(a) is
not universal in the sense that different multipliers M lead to different p(a)
distributions. It is easy to show that f(a) = p(a) + max / ( a ) = p(a) +
dimension of the measure's support.
Finally, a few comments should be made concerning the lognor-mal character of the distributions considered. If the sum logBM(?7i) + logB M(T7I , 772) + ••• b e n a v e d asymptotically as a Gaussian, a would follow a lognormal distribution. In this case f(a) would be a parabola. However, when the distribution is properly normalized, the above sum is not universal and the central limit theorem implies that p(a) and / ( a ) are parabolic only close to their maximum. Away from this maximum the behaviour of p(oc) and / ( a ) is determined by M and is parabolic only if the multipliers M are lognormal. For more details on the basic concepts of multifractals and the meaning of / ( a ) < 0 see Mandelbrot (1988).
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Author Index
Aharony, A., 105, 130, 132, 304 Alexander, S., 131, 132, 317 Alexandrowiti, Z., 103, 107 Allain, C , 76, 77 Allen, N., 325, 327 Amit, D.J., 121 Amitrano, C , 156, 158 Amritkar, R.E., 193 de Arcangelis, L., 62 Armstrong, A.J., 253 Avnir D., 77
Balberg, I., 115 Ball, R.C., 138, 147-150, 156, 192,
195-197, 199, 200, 202, 216, 217, 221, 242, 243, 323
Ban, J., 251, 252 Bansil, R., 117 Barnes, P.D., 301 Barrow, J.D., 237 Bars, G.L., 317 Baudet, C , 304 Ben-Avraham, D., 130, 133 Ben-Jacob, E., 2, 292-294, 308
317, 321, 322, 323 Bensimon, D., 268, 297 Bentley, W.A., 2, 287, 288 Benzi, R., 47 Berne, B.J., 254 Besicovitch, A.S., 9, 16 Binder, K., 223 Boccarra, N., 4 Bories, S., 109 Borger, F., 304 Botet, R., 4, 42, 173, 179, 184, 186
187, 210, 213, 241 Brady, R.M., 147, 323 Brown, W.D., 216, 217
Bruinsma, R., 317 Bug, A.L., 231 Buka, A., 2, 312 Bunde, A., 108
Cannel, D.S., 76, 250 Cardoso, O., 308 Cardy, J.L., 108 Cates, M.E., 4 Chaikin, P.M., 304 Chandler, R., 109, 110 Chen, J.D., 302, 304 Chhabra, A., 115, 117 Clarke, R., 321-323 Ctement, E., 304 Cloitre, M., 76, 77 Coniglk), A., 62, 124, 156, 158
161, 162 Couder, Y., 308 Cserti, J., 277, 278 Cummings, P.T., 127
Daccord, G., 2, 281, 306, 307, 327-329 van Damme, H., 307 Daoud, M., 4 Debierre, J.M., 130, 220, 231 DeGregoria, A.J., 290, 300 Deutsch, J.M., 238, 239, 245 Deutscher, G., 2, 105, 313, 317 Dhar, D., 186 Dirks, A.G., 191, 193 Djordjevic, Z.V., 86 van Dongen, P.G.J., 237, 245 Dougherty, A., 2, 304, 321, 322 Duplantier, B., 126, 128 Dupuy, D., 308
Edwards, S.F., 203 Elam, W.T., 317
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346 Author Index
Engelman, R., 4 Ernst, M.H., 237, 241, 243, 245 Evertsi, C , 129
Family, F., 4, 88, 90-93,102, 106 107, 111, 142, 143,156, 181 187,192, 197, 199, 210, 225-235 235, 239, 242, 245, 280, 281, 288
Fanner, J.D., 43, 55, 57, 58 Feder, J., 4, 254, 302, 304 Fischer, M.E., 168 Flory, P., 151 Forrest, S.R., 209 Rreche, P., 185 Friedlander, S.K., 241 Arisen, U., 47
Garik, P., 2, 181, 317 Gaspari, G., 120 Gatinean, L., 307 Gefen, Y., 132 de Gennes, P.G., 79, 117, 121, 213 Gerard, N., 308 Gilmer, G.M., 191, 194 Godbey, Y., 2, 308 Goldenfeld, N.D., 2, 308, 317, 318 Gollub, J.P., 2, 304, 321, 322 Gould, H., 156 Gouyet, J.F., 129 Grassberger, P., 103, 104, 107, 108 Grier, D., 2, 315, 316, 321-323 Gubser, D.U., 317 Guinea, F., 330 Guyer, R.A., 131
Haanstra, H.B., 194 Halsey, T.C., 47, 51, 54, 56, 146,
156, 158, 160 Hausdorff, F., 9, 16 Hautman, J., 181 Havlin, S., 130, 133, 162, 164 Hayakawa, Y., 2, 158-160, 174,
179, 320, 322, 324 Hele-Shaw, J.S.S,, 298 Hentschel, H.G.E., 29, 47, 49, 51,
56, 156 Herrmann, H.J., 4, 102, 107, 108,
111, 114, 115, 117, 219 Hilfer, R., 317
Homsy, G.M., 270, 278, 300 Honda, K., 151, 174 Honjo, H., 2, 317, 318, 320, 322 Horkai, F., 251, 252 Horvath, V., 305, 309, 310, 312, 313 Huang, J.S., 250, 254-256 Huang, L.J., 317 Hulin, J.P. 304 Humpreys, W.J., 2, 287, 288 Hurd, A.J., 253
Itox, M., 234
Jaeger, Z., 4 Jan, N., 124 Jasnow, D., 300 Jen, E., 45 Jensen, M.H., 47, 51, 54, 56 Joag, P.S., 193 Jossang, T. 4, 11, 254, 302, 304 Jullien, R., 4, 42, 177, 179, 184,
186, 187, 199, 204, 210, 213, 216-220, 241
Kadanoff, L.P., 47, 51, 54, 56, 90, 193, 268, 277, 297
Rang, K., 238, 242 Kapitulnik, A., 105, 324 Kardar, M., 200, 202, 204 Kaufman, J.H., 324 Kawasaki, K., 151 Kerstein, A.R., 230 Kert&z, J., 2, 102, 143, 166, 167,
188-190, 200, 273, 277, 278, 280-283, 304, 305, 309, 310, 312, 313
Kessler, A., 268, 290-292, 317, 321 Keyes, T., 108 Knoll, W., 317 Kolb, M., 144, 210-213, 212-219, 225, 242 KolmogOTov, A.N., 16 Kondo, H., 175 Koplik, J., 109, 110, 268, 290-292,
308, 321 Kremer, K., 124-126 Krug, J., 202
Landau, D.P., 4, 114, 115, 117 Langer, J.S., 268, 314
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Author Index 347
Laroche, C , 307 Leamy, H.J., 191, 194 Leath, P.L., 103 Lenormand, R., 109, 302-304, 327-329 Lereah, Y., 2, 317 Lerman, K., 109, 110 Levine, H., 268, 290-292, 308, 321 Levitz, P., 307 Leyvraz, F., 176, 242, 243 Libchaber, A., 300 Li, H.D., 317 Liang, S., 192, 193, 268, 277, 278, 297 di Liberto, F., 156, 158 Limaye, A.V., 193 Lin, M.Y., 250, 254-256 Lindsay, H.M., 304 Liu, B.X., 317 Louis, E., 330 Lugosi, M., 156 Lyklema, J.W., 123-126, 129
Madelaine, G.J., 234 Maher, J.V., 301 Majid, I., 80, 124 Makarov, N.G., 160 Maloy, K.J., 302, 304 Mandelbrot, B.B., 1, 9, 19, 29, 31-34,
40, 41, 43, 45, 47, 340-343 Manneville, P., 113 Margolina, A., 108 Martin, J.E., 76, 250 Matsushita, M., 2, 151, 158-160, 174,
179, 317, 318, 320, 322-324 Matthews-Morgan, D., 115 Mayer-Kress G., 45 McGrady, E.D., 237, 241 McKane, A.J., 204 McMurry, P.H., 234 Meakin, P., 4, 38, 48, 55, 83, 84,
106, 136-138, 140-142, 144-146, 148-150, 156-158, 161, 162, 164-168, 170, 176-178, 191, 195-197, 199, 200, 204, 210, 213, 214, 218, 227, 228-235, 238, 239, 241-243, 245, 280, 281, 283, 284, 317
Medalia, A., 212 Melroy, O.R., 324
Miller, A., 317 Miyazima, S., 231 Mocklet, R.C., 253 Montag, J.L., 142, 143 Moore, M.A., 204 Mohwald, H., 317 Mueller, T., 308 Mullins, W.W., 271, 301 Murat, M., 304 Muthukumar, M., 151
Nagatani, T., 156 Nakanishi, H., 88, 90-93, 142, 143 Nazzal, A.I., 324 Niemeyer, L., 172, 173, 325-327 Nieuwenheuzen, J.M., 194 Nittman, J., 2, 281, 285-287, 306, 307
Obukhov, S.P., 122 Ohtsuki, T., 108 Ohta, S., 317, 318, 320 Olbrecht, F., 307 Oliveria, M., 209, 247-249 Orbach, R., 132 O'Shaughnessy, B., 131 O'Sullivan, W.J., 253 Ostrowski, N., 4 Ott, E., 55 Oxaal, U., 304
Paladin, G., 47, 48 Palffy-Muhoray, P., 312 Pandey, R.B., 117 Parisi, G., 47, 121, 200, 202 Park, C.W., 270, 278, 300, 301 Paterson, L., 285, 301 Pecoria, R., 254 Peitgen, H.O., 27, 28 Peliti, L., 121, 122 Perrin, M.L., 234 Pfeifer, P., 77. 78 Pietronero, L., 4, 122, 172, 174,
175, 325-327 Platt, P., 102, 181, 288 Plischke, M., 142, 185 Procaccia, I., 29, 47, 49, 51, 54, 56,
131, 156, 158
Rabaud, N., 308
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348 Author Index
Racz, Z., 142, 162, 168-170, 185, 234, 237, 312
Radn6czy, G., 2, 315, 316 Ramanlal, P., 148, 149, 191, 195-197,
199, 200, 202, 204, 207, 292-294 Rammal, R., 132 Rauseo, S.N., 301 Redner, S., 62, 88, 91, 238, 242 Rlnyi, A., 51 Reynolds, P.J., 88, 91 Richter, P.H., 27, 28 Rikvold, P.A., 176 Robbins, M.O., 304 Rosenquist, E., 254 Rossi, G., 147 Rosso, M., 129 Rudnick, J., 200
Saffman, P.G., 300, 301 Saleur, H., 126, 128 Sano, M., 2, 320, 322 dos Santos, 86 Sander, L.M., 2, 4, 135, 136, 138, 140,
142, 148, 149, 172, 191, 195-197, 199, 200, 202, 204, 207, 275, 292-294, 308, 315-317, 321-323
Sapoval, B., 129 Sarkar, S., 279, 300 Sato, S., 158-160, 179 Satpathy, S., 327 Sawada, Y., 2, 320-322, 324 Schaefer, D.W., 76, 250 Scher, H., 155 Schwarti, L.W., 290, 300 Sekerka, R.F., 271, 301 deSeze, L., 113 Shang, C.H., 317 Shraiman, B.I., 47,.51, 54, 56, 268, 297 Shlesinger, M., 4 Skjeltorp, A.T., 4, 253 von Smoluchowski, M., 234 Sole, K., 120 Souce*marianadin, A., 329 Sprague, J., 317 Squires, G.L., 76 Stanley, H.E., 2, 46, 86, 88, 91, 104, 107,
108, 111, 124, 156, 158, 161, 162, 176, 185, 281, 285-287, 306, 307
Stauffer, D., 104, 105, 114, 115, 117, 168, 185
Stell, G., 127 Stokes, J.P., 304 Sung, J., 250, 254-256 Sutherland, D.N., 191, 210, 214 Sze>, J., 277, 278
Tabeling, P., 300 Tang, L. 268, 281, 297 Tao, K., 317 Tavernier, P., 308 Taylor, G.I., 300, 301 Teixeira, J., 72 Tel, T., 13, 48, 57, 59, 60, 64,66,
68,69 Thorn, W., 308 Thompson, B.R., 147 Tihomirov, V.M., 16 Tokuyama, M., 151 Tosati, E., 4 Toulouse, G., 132 Tourboul, E., 329 Toyoki, H., 151, 174 Trugman, S.A., 126 Turban, L., 130, 220, 231 Turkevich, L.A., 155
Umberger, D.K., 45
van Vechten, D., 317 Vicsek, T., 2, 11, 13, 64, 66, 68,
69, 102, 106, 107, 111, 142-146, 150, 164, 166-171, 181, 187, 192, 197, 211, 225-235, 242, 273, 277-283, 288, 304, 305, 309, 310, 312, 313, 315, 316
Void, M.J., 191 Voss, R.F., 40 Vulpiani, A., 47, 48
Weierstrass, K., 41 Weinrib, A., 126 Weitz, D.A., 209, 242, 243, 247-250,
254-256, 304 White, W.h., 234 Wiesmann, H.J., 172-175, 325-327 Wilkinson, D., 110, 203, 302, 304 Willemsen, J., 109-111
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Author Index 349
Wiltzius, P., 76, 250 Witten, T.A., 4, 135, 136, 138,
140, 156, 158, 161, 162, 172, 192, 209, 221, 223, 242, 243, 275
Wolf, D.E., 186, 188-190, 200 Wolf., S.A., 317
Yorke, J.A., 55
Zabolitzky, J.G., 185 Zarcone, C , 302 Zhang, Y.C., 200, 202, 204 Ziff, R.M., 127, 128, 237, 241
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Subject Index
Aggregation, ballistic, 190, 194, 218 cluster-cluster, 209 deterministic, 171, 180, 286 diffusion-limited, 134, 213, 224 dynamics, 223 fast and slow, 249 noise-reduced, 281 reaction-limited, 216, 231, 242 steady-state, 234, 244 with disaggregation, 176
Algorithm for DLA, 334 Amplitude ratio, 114 Amorphous films, 315 Angle,
cone, 153 of incidence, 194
Anisotropy of, cluster-cluster aggregates, 212 DLA clusters, 147, 167, 282 Eden clusters, 185 growth, 147 surface tension, 292 viscous fingers, 309-312 walks, 120
Anomalous diffusion, 131 Asymptotic behaviour of,
DLA clusters, 147, 282 Eden clusters, 185
Attractor, 28
Backbone of percolation cluster, 116 Ballistic
aggregation, 190 deposition, 194
Box counting, 10, 65 Boundary integral method, 289 Breakdown dielectric, 172, 325
Brownian motion, 31
Cantor set, growing, 66 triadic, 24, 43-44
Capillary, force, 109 length, 270, 272
Carbon particle aggregates, 252 Causality bound, 139, 192 Chemical dissolution, 327 Cluster-cluster aggregation,
ballistic, 218 chemically or reaction-limited,
216, 231, 242 cluster-size distribution, 225, 235, 244 diffusion-limited, 213, 224 experiments, 246 fractal dimension, 212, 221, 247 hierarchical, 218 monodisperse, 217 poly disperse, 217 reversible, 237, 245 steady-state, 234
Collision kernel (matrix), 240-241 Colloid,
gold, 247 silica, 250
Correction to scaling, 82, 146 Correlation length, 105 Correlation function
density-density, 22 lateral, 166 pair, 23 tangential, 144 three-point, 146
Critical, behaviour, 105
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352 Subject Index
dimension, 122, 204, 221, 237 exponents, 105 phenomena, 89 percolation cluster, 127 probability, 105
Crossover scale, 37, 183, 198, 215 Crystallisation, 314 Cumulative volume, 78 Curvature, 270, 300
Darcy's law, 299 Dendritic growth, 148, 273, 282,
293, 309, 323 Dense radial structure, 301, 317,
322, 325 Deposition,
ballistic, 194, 204 electrochemical, 320 diffusion-limited, 164
Deterministic aggregation, 171, 180, 286 growth, 289
Dielectric breakdown, 325 model, 172, 284
Diffusion constant, 213 length, 314, 316 equation, 314
Diffusion-limited aggregation off-lattice, 141 with sticking probability, 140, 147
Diffusion-limited deposition, 164 Diffusivity anomalous, 130 Dimension
box counting, 10, 65 critical, 122, 204, 221, 237, 239 embedding, 10 generalized, 52, 64, 159 information, 55 spectral, 132 sandbox, 65
Dissolution chemical, 327 Dispersion relation, 271 Distribution of
cluster size, 168, 223, 235 empty regions, 18, 78 mass, 64, 162
pore size, 78 tree size, 163, 323
Dynamic scaling, 223, 229
Eden model, model, 183 noise-reduced, 189 in strip geometry, 186
Electrochemical, deposition, 320 polymerization, 323
Embedding dimension, 10 Entropy,
configuration, 102 Renyi, 51
Equation, Darcy's, 299 diffusion, 314 Langevin, 200 Laplace's, 158, 172, 269 Navier-Stokes, 299 Smoluchowski, 239, 243
Enumeration, 85, 124 Examples, 24, 38, 44, 57, 65, 92
Fat fractals, 43, 193 Fingering viscous, 297 Fixed-point, 88 Flow in porous media, 302 Fluctuations, 271, 281, 317 Fluid,
anisotropic, 305, 310, 313 incompressible, 299 invading, 109 miscible, 306 non-Newtonian, 306 non-wetting, 109 trapped, 110, 304 wetting, 78
Fractal dimension, dimension, 14 global, 36 local, 36 spectrum of, 52
Fractal dimension of, projections, 17, 248 cross-sections, 17, 198 intersections, 17
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Subject Index 353
Fractal measure, 47, 157 Fractal Brownian,
curve, 38 motion, 38, 120 surface, 40
Function, correlation, 22, 144, 164 /(«), 50 scaling, 168, 193, 225, 235 self-affine, 33, 197 single-valued, 182 singular, 47 Weierstrass-Mandelbrot, 41
Gaps, 18, 32 Gelation,
kinetic, 112 time, 230
Generalized dimensions, 52 Gauss-Seidel overrelaxation, 174 Global fractal dimension, 36 Gold colloid aggregate, 247 Growth,
columnar, 92, 204 dense radial, 301, 317, 322, 325 dendritic, 148, 273, 282, 293
309, 323 probability scaling, 153, 157 sites, 103, 133 tip splitting, 279, 282, 310, 318
Harmonic measure, 158 Hele-Shaw cell,
anisotropic, 308 radial, 301 longitudinal, 298
Hull, Brown, 31 of percolation cluster, 126
Information dimension, 55 Intersection of sets, 17, 198 Invading fluid, 109, 302
Julia set, 27
Kinetic gelation, 112, 230 Kinetic term, 270, 300, 322
Laplace's equation, 158, 172, 269 Lattice animals, 179, 222 Length,
of shore, 10 correlation, 105
Light scattering, 73, 77, 254 Liquid crystal,
nematic, 310 smectic, 305, 313
Local fractal dimension, 36 Logarithmic correction, 203
Mandelbrot set, 29 Mean cluster size, 226, 238, 242 Mean-field theory of,
DLA, 150 ballistic aggregation, 204 CCA, 239
Measure, deterministic, 55 fractal, 47 harmonic, 158 Hausdorff, 16 Lebesque, 65 multifractal, 48 growth probability, 157
Morphological phases, 282, 308-311 Mullins-Sekerka instability, 271 Multifractal,
geometry, 63 measure, 48
spectrum, 52, 59, 67 Multifractality in,
DLA, 156, 294 viscous fingering, 307 solidification, 319 screened growth, 177 DBM, 175
Navier-Stokes equation, 299 Nematic liquid crystal, 310 Neutron scattering, 73 Noise reduction, 189, 281, 285 Non-universality, 193, 227
Off-lattice simulation, 141, 218
Pair correlation function, 23 Peano curve, 26, 44, 301
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354 Subject Index
Percolation, backbone 116 continuum 115 cluster, 105 fractal dimension, 105 hull, 126 invasion, 109, 302 spreading, 103 threshold probability, 105
Perimeter, 128, 184, 285 Plaster, 327 Porod's law, 75 Polymerisation, 257, 323 Probability,
critical, 105 growth, 153, 157 measure, 157
Radial viscous fingering, 301 Radius of,
curvature, 270, 300 of gyration, 51
Random,. addition successive, 40 fractal, 22 media, 302-306 walk, 31, 117
Ratio, amplitude, 114 length to width, 149, 167, 284
Recursion, 55 Relaxation time, 235 Renormaliiation group,
dynamic, 202 Monte Carlo, 91, 142 position space, 86
Rlnyi, entropy, 51 information, 102
Resistor network, 61 Restructuring, 199-200, 203 Rough surface 184, 318 Rules, 17
Scale in variance, 15, 23 Scaling,
correction to, 82, 146 of the growth probability, 153
function 168, 193, 225, 235 of size distribution, 168,170, 225, 235 of the surface width, 187, 199 relation, 169, 202, 226, 242, 245
Scattering, curve, 75 light, 73, 77, 250, 254 neutron, 73 X-ray, 73, 250
Screened growth model, 175 Screening length, 151 Self-affine,
fractal, 42 function, 33 statistically, 33 surface, 40, 182, 198 transformation, 37
Set, cros section of, 17 intersection of, 17 Julia, 27 Mandelbrot, 29 product of, 17 project of, 17, 248 triadic Cantor, 24, 43-44 union of, 17
Sierpinski gasket, 27, 130 Silica colloid, 250 Smoluchowski equation, 239, 243-244 Snowflakes, 286-289 Spanning cluster, 90 Spectral dimension, 132 Statistics, of clusters, 168, 225, 235 Structure factor, 74, 254 Sutherland's ghost model, 221 Surface,
fractional Brownian, 40 tension, 252, 267, 270, 276, 292
Surface width of clusters, ballistic, 199 Eden, 187 intrinsic, 188
Tangent rule, 194, 204 Tangential correlations, 144 Thin films,
amorphous, 314-317 ion irradiated, 317
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sputter deposited, 317 Tip splitting growth, 280, 282, 310, 318 Transformation,
affine, 37 renormalization, 86
Transition of morphological phases, 282, 308, 311
Transmission electron microscopy, 247, 316
Trees in deposits 163, 323
Union of fractals, 17 Universality, 140, 204
Subject Index 355
Viscous fingering, and anisotropy, 308 and randomness, 303, 305 in porous media, 302, 304 radial, 301 three-dimensional, 304
Voltage distribution, 61
Walk, Laplacian, 129 self-avoiding, 85, 123 self-intersecting, 118 true self-avoiding, 92, 121
Weierstrass-Mandelbrot function, 41 Woods metal, 328
Zinc sulfate, 322
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