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Renormalization group approach to dynamic fragmentation
Volume 147, number 8,9
PHYSICS LETI'ERS A
30 July 1990
Renormalization group approach to dynamic fragmentation J. K a m p h o r s t Leal d a Silva Departamento de Fisica, Instituto de Ci~ncias Exatas, Universidade Federal de Minas Gerais, C.P. 702, 30161 Belo Horizonte MG, Brazil Received 11 September 1989; revised manuscript received 11 April 1990; accepted for publication 18 May 1990 Communicatedby A.A. Maradudin
A renormalizationgroup approach to geometricalmodels of dynamic fragmentation in fractal and Euclidean systemsis developed. The fractal dimension 3 = 2B describing the irregularityand fragmentation of the systemis evaluated.
In the last years growth phenomena have received a lot of study, some of them concerned with geometrical models [ 1 ]. In this Letter we are interested in dynamic fragmentation, in some sense, the opposite phenomenon. It consists in the breakup of a connected system by some external attack (for a review see ref. [ 2 ] ). Several examples can be found in the literature: the fracture of solids after the formation of random cracks [ 3 ], the fragmentation of a reactive porous medium by chemical means [4], the fragmentation of a system under adiabatic expansion [ 5 ] and the geophysical formation of archipelagoes [ 6 ]. The several physical processes underlying the dynamic fragmentation - growth, propagation, bifurcation, stress reliving, energy consumption, interacting cracks and fractures - are very complicated and a complete theory is not available yet. So geometrical models have been introduced as an alternative theoretical approach [4,6,7]. Here we develop a renormalization group approach to dynamic fragmentation and evaluate the fractal dimension/I which measures the irregularity and fragmentation of the system. Consider the simple geometrical model for fragmentation dynamics introduced by G o m a s and Vasconcelos [6,7] in the kinetic regime (the external part of the System ~is totally exposed to the attack). In their model, the system is represented by occupied sites in a hypercubic lattice of coordination number Z. The attack o f the external fields is sire-
ulated by a rain of random drops. This rain is characterized by the number of drops falling by unit time on the lattice. I f a drop hits an occupied site with Z - 1 or less nearest neighbors the site is eliminated, otherwise it survives to the attack. Using computer simulation for two-dimensional Euclidean and fractal systems they have studied the growing of n(s, t), the average number per site at time t of clusters with s connected sites, as a function of time, due to the attack. It turns out that n(s, t) obeys the scaling relation n(s, t) ~ tWs - ~ f ( s / t z) where w, T and z are exponents defining universality classes. They have also evaluated the fractal dimension • of the archipelago of fragments at the time (to) when the system is disconnected with the largest number of different fragment sizes. At the beginning, the system consists of a large connected cluster but, as long as the time evolves, the fragmentation sets in and the system becomes a highly disconnected set (archipelago). This phenomenon can be characterized by the diversity D ( t ) [ 7 ], the number of different size of fragments existent at time t, and N ( t ) , the number of fragments at the same time. At t = 0 we have that D ( 0 ) = 1 and N ( 0 ) = 1. D ( t ) and N ( t ) grow as the time evolves and at certain time tc the diversity reaches its maximum. As the attack continues, the fragments break up in smaller ones implying that D ( t ) decreases and N ( t ) grows until its maximum, which occurs at a later time t~. After t', D ( t ) and N ( t ) decrease and in the long time regime both approach zero. Note that t~ is
0375-9601/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
507
Volume 147, number 8,9
PHYSICSLETTERSA
greater than & only for finite systems. For infinite systems they could be equal [ 7 ]. At to, the fragmentation reaches its maximum in the sense that there is the largest number of different sizes of fragments. The archipelago is then sealing invariant and can be characterized by its fractal dimension A= 2B. It is well known that/t can be obtained from the plot of the number of fragments of sizes larger than k, X ( s > k), as a function o f k [8]. At &, X ( s > k ) ~ k -B. We adopt a position-space renormalization group approach (PSRG) [ 9 ], which has been used in the growth phenomena [ 10,11 ]. A fugacity K is associated to each occupied site, the lattice is divided in cells of linear dimension b and the cells are rescaled to a single site. The renormalized site will be occupied if and only if a connected path of nearest neighbors occupied sites spans the cell in such a way that the renormalization transformation K' =Rb(K) includes all possible paths spanning the cell. The archipelago fractal dimension A is found by evaluating d K ' / d K = b a at the critical fixed point K*. Note that fragmentation is a non-spanning problem, in the sense that we have not an infinite cluster of connected sites as in percolation or growth phenomena. However, a fragment is defined as a set of connected sites. So the connectivity plays here an important role. We need a transformation that changes the original archipelago into another one with a renormalized fngacity associated to each new site. It means that big connected sets of sites must be transformed into renormalized ones and that the smaller ones are eliminated. So, it is sensible to transform a cell with a spanning path of neighboring occupied sites into an occupied renormalized site. On the other hand, a cell without a spanning path is renormalized into a vacant site. We have taken into account the essential ways that a cell be connected to the neighboring cells and consequently be a part of a fragment. Thus the spanning rule is suitable for the renormalization of the archipelago. Note that in this simple transformation there are some known errors [9 ]. For example, cells that are connected in the original lattice can be disconnected in the rescaled lattice. Conversely, cells disconnected in the original lattice can be part of the same renormalized fragment. These problems are well discussed in ref. [9 ] and we expect them to be irrelevant. First let us consider the smallest cells (b = 2 ) of an 508
30 July 1990
Euclidean system in a square lattice. By definition, a cell is occupied if there is a connected path through the cell. The clusters entering the transformation are shown in fig. I a. We consider then, all different ways that a spanning duster can be obtained from the initial full occupied configuration by a fragmentation process. This is shown for a cluster of two occupied sites in fig. lb. The renormalization transformation is given by
K' = K 4 + 4 K 3 + 8 K 2 .
( 1)
This equation has two trivial fixed points (K* = 0 and K*=oo) and a critical one (K*=0.118) with exponent B = A/2 = 0.521. This value of B agrees with the numerical simulation value, Bs = 0.55 + 0.04, for twodimensional rings [ 7 ]. For cells with b = 3 we obtain K' = K 9 + 8K a + 60K 7+ 408K e + 1904K 5 + 4560K 4+ 3600K 3
(2)
and B=0.503. For cell-to-cell transformation ( b = 3) we find B=0.500. large cell renormalization group is necessary t o determine the asymptotic behavior of the B exponent, but all values obtained here agree with the numerical value. We have also evaluated the renormalization transformation for a cubic lattice with the smallest cells (b = 2 ), From the renormalized relation
•
•
•
0
•
•
•
•
•
•
0
0
•
01
•
02
•
0 2
•
01
a)
b) Fig. 1. (a) The set of spanning clusters in a 2 X 2 cell enteringthe RG transformation of an Euclidean system. Clusters of 3 and 2 sites obtained by permutations of the vacant sites shouldalso be considered. (b) The two waysthat a cluster of two occupiedsites can be obtained from the initial full cluster by a fragmentation process. Full circles representoccupied sites and the order of consumptionis indicated by the numbers.
Volume 147, number 8,9
PHYSICS LETTERS A
K' = K s + 7K 7+ 56K 6 + 336K 5+ 1632K 4 + 5760K 3+ 8640K 2 .
(3)
we find that the critical fixed point is near zero ( K * = l . l × 1 0 -4) and B=0.498. This is the first evaluation of the archipelago fractal dimension for cubic systems. The next step is to study the fragmentation of a fractal system. Consider the attack on percolation clusters on a square lattice. A PSRG approach to this problem requires at least two parameters, a probability p that a site is occupied and a fugacity K associated with such occupied site. The renormalized transformation K' =Rb(K, p) contains all ways that a spanning cluster can be obtained by fragmentation from an initial configuration. But now there are several initial configurations, each of them occurring with a probability given by some polynomial function ofp. I n fact the initial clusters are the spanning ones entering the other renormalization transformation equation p ' =fb (P). We consider that all connected path traversing the cell in any direction contribute to J~(p). Therefore we have for the smallest cell ( b = 2 ) p' _~p4-1-4p3q-l-4p2q2 , K' = p 4 ( K 4 + 4 K 3 + S K 2 ) +4p3q(K3+ 2K 2) + 4p2q2K2 ,
(4)
where q = 1 - p . W h e n p = 1 we recover the Euclidean case studied above. The nontrivial fixed points are (p*=0.382, K*=0.488), (1, 0.11) and (0.382, 0). The B exponent evaluated at the (0.382, 0.488) fixed point is B = 0.553. It agrees with the numerical value Bs = 0.53 + 0.03 [ 7 ]. Our percolation threshold is far from the expected value (pc~0.593) [ 12]. So, we have also evaluated the B exponent using this value in (4) and found essentially the same exponent. It is interesting to mention that this exponent is close to the empirical exponent Be = 0.6 of the geophysical distribution of islands on the whole earth [ 8 ]. Let us discuss now the validity of our renormalization group approach. It is known that attempts to adapt real space renormalization methods to growth problems are still unclear [13,14]. In particular, Pietronem et al. [ 13,14 ] have discussed critically the PSRG method [ 10 ] and proposed a new method, the fixed scale transformation, to evaluate the fractal
30 July 1990
dimension for the dielectric breakdown model. It turns out that to compute the probability of an internal frozen configuration of a cell, we need consider infinity growth processes outside this cell. This point has been neglected in the PSRG method and it is the main reason for the failure of that approach. Further, if one were to lift the artificial restriction to configurations that span the cell in both directions, the fractal dimensions obtained by Gold et al. [ 10] are too low when compared with the expected values [ 13,14 ]. Some of these problems were already partially solved in ref. [11 ] through a phenomenological PSRG applied to growth phenomena. In that paper, the spanning rule which measures the spatial extent of relevant clusters has been changed by the use of the radius of gyration and the probability of a relevant configuration has been determined by Monte Carlo simulations. Furthermore, a phenomenological parameter has been introduced in order to improve the fractal dimension of the grown structure. In fragmentation the situation could be different. The consumption probability of a perimeter site of a cell depends on external processes. However to destroy an internal site of a cell we must have always a vacant site in the perimeter. The fragmentation process in a cell is from out sites to the inner ones. It means that in a cell, when we consider only the internal fragmentation process, the probability to destroy an inner site is considered. It is worth mentioning that we have not imposed any artificial restrictions in the spanning rule and the results obtained agree very well with the expected values. In summary, we have developed here the first renormalization group approach for geometrical models of dynamic fragmentation process. The fractal dimension 3 = 2B describing the fragmentation or the irregularity of the system has been evaluated. For one situation this is, to our knowledge, the first time that such evaluation is done. In other cases these exponents are compared with the numerical value of simulations with good agreement. I would like to thank S.L.A. de Queiroz and S. Oliffson Kamphorst for useful discussions. I am indebted to M.A. Gomes who has introduced me to fragmentation problems.
509
Volume 147, number 8,9
PHYSICS LETTERS A
References [ 1 ] H.I. Herrmann, Phys. Rep. 136 (1986) 153. [ 2 ] D.E. Crady and M.E. Kipp, J. Appl. Phys. 58 ( 1985 ) 1210. [ 3 ] R. Englman, Z. Jaeger and A. Levi, Philos. Mag. B 50 (1984) 307. [ 4 ] M. Sahimi and T.T. Tsotsis, Phys. Rev. Lett. 59 ( 1987 ) 888. [5] B.L. Holian and D.E. Grady, Phys. Rev. Lett. 60 (1988) 1355. [6]M.A.F. Gomes and G.L. Vasconcelos, in: Pro¢. 2nd Workshop on Scaling, fractais and nonlinear variability in geophysics (Paris, 1988). [ 7 ] M.A.F. Gomes and G.L. Vasconcelos, J. Phys. A 22 (1989) L757.
510
30 July 1990
[ 8 ] B.B. Mandelbrot, The fraetal geometry of nature (Freeman, San Francisco, 1982). [9] H.E. Stanley, P.J. Reynolds, S. Redner and F. Family, in: Real-space renormalization, eds. T.W. Burkhardt and J.M. van Leeuwen (Spinser, Berlin, 1982). [ 10] H. Gould, F. Family and H.E. Stanley, Phys. Rev. Lett. 50 (1983) 686. [ 11 ] J.L. Montag, F. Family, T. Vicsek and H. Nakanishi, Phys. Rev. A 32 (1985) 2557. [12] J.W. Essam, Rep. Prog. Phys. 43 (1980) 833. [ 13 ] L. Pietronero, A. Erzan and C. Evertsz, Phys. Rev. Lett. 61 (1988) 861. [ 14] L. Pietronero, A. Erzan and C. Evertsz, Physica A 151 (1988) 207.
PHYSICS LETI'ERS A
30 July 1990
Renormalization group approach to dynamic fragmentation J. K a m p h o r s t Leal d a Silva Departamento de Fisica, Instituto de Ci~ncias Exatas, Universidade Federal de Minas Gerais, C.P. 702, 30161 Belo Horizonte MG, Brazil Received 11 September 1989; revised manuscript received 11 April 1990; accepted for publication 18 May 1990 Communicatedby A.A. Maradudin
A renormalizationgroup approach to geometricalmodels of dynamic fragmentation in fractal and Euclidean systemsis developed. The fractal dimension 3 = 2B describing the irregularityand fragmentation of the systemis evaluated.
In the last years growth phenomena have received a lot of study, some of them concerned with geometrical models [ 1 ]. In this Letter we are interested in dynamic fragmentation, in some sense, the opposite phenomenon. It consists in the breakup of a connected system by some external attack (for a review see ref. [ 2 ] ). Several examples can be found in the literature: the fracture of solids after the formation of random cracks [ 3 ], the fragmentation of a reactive porous medium by chemical means [4], the fragmentation of a system under adiabatic expansion [ 5 ] and the geophysical formation of archipelagoes [ 6 ]. The several physical processes underlying the dynamic fragmentation - growth, propagation, bifurcation, stress reliving, energy consumption, interacting cracks and fractures - are very complicated and a complete theory is not available yet. So geometrical models have been introduced as an alternative theoretical approach [4,6,7]. Here we develop a renormalization group approach to dynamic fragmentation and evaluate the fractal dimension/I which measures the irregularity and fragmentation of the system. Consider the simple geometrical model for fragmentation dynamics introduced by G o m a s and Vasconcelos [6,7] in the kinetic regime (the external part of the System ~is totally exposed to the attack). In their model, the system is represented by occupied sites in a hypercubic lattice of coordination number Z. The attack o f the external fields is sire-
ulated by a rain of random drops. This rain is characterized by the number of drops falling by unit time on the lattice. I f a drop hits an occupied site with Z - 1 or less nearest neighbors the site is eliminated, otherwise it survives to the attack. Using computer simulation for two-dimensional Euclidean and fractal systems they have studied the growing of n(s, t), the average number per site at time t of clusters with s connected sites, as a function of time, due to the attack. It turns out that n(s, t) obeys the scaling relation n(s, t) ~ tWs - ~ f ( s / t z) where w, T and z are exponents defining universality classes. They have also evaluated the fractal dimension • of the archipelago of fragments at the time (to) when the system is disconnected with the largest number of different fragment sizes. At the beginning, the system consists of a large connected cluster but, as long as the time evolves, the fragmentation sets in and the system becomes a highly disconnected set (archipelago). This phenomenon can be characterized by the diversity D ( t ) [ 7 ], the number of different size of fragments existent at time t, and N ( t ) , the number of fragments at the same time. At t = 0 we have that D ( 0 ) = 1 and N ( 0 ) = 1. D ( t ) and N ( t ) grow as the time evolves and at certain time tc the diversity reaches its maximum. As the attack continues, the fragments break up in smaller ones implying that D ( t ) decreases and N ( t ) grows until its maximum, which occurs at a later time t~. After t', D ( t ) and N ( t ) decrease and in the long time regime both approach zero. Note that t~ is
0375-9601/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
507
Volume 147, number 8,9
PHYSICSLETTERSA
greater than & only for finite systems. For infinite systems they could be equal [ 7 ]. At to, the fragmentation reaches its maximum in the sense that there is the largest number of different sizes of fragments. The archipelago is then sealing invariant and can be characterized by its fractal dimension A= 2B. It is well known that/t can be obtained from the plot of the number of fragments of sizes larger than k, X ( s > k), as a function o f k [8]. At &, X ( s > k ) ~ k -B. We adopt a position-space renormalization group approach (PSRG) [ 9 ], which has been used in the growth phenomena [ 10,11 ]. A fugacity K is associated to each occupied site, the lattice is divided in cells of linear dimension b and the cells are rescaled to a single site. The renormalized site will be occupied if and only if a connected path of nearest neighbors occupied sites spans the cell in such a way that the renormalization transformation K' =Rb(K) includes all possible paths spanning the cell. The archipelago fractal dimension A is found by evaluating d K ' / d K = b a at the critical fixed point K*. Note that fragmentation is a non-spanning problem, in the sense that we have not an infinite cluster of connected sites as in percolation or growth phenomena. However, a fragment is defined as a set of connected sites. So the connectivity plays here an important role. We need a transformation that changes the original archipelago into another one with a renormalized fngacity associated to each new site. It means that big connected sets of sites must be transformed into renormalized ones and that the smaller ones are eliminated. So, it is sensible to transform a cell with a spanning path of neighboring occupied sites into an occupied renormalized site. On the other hand, a cell without a spanning path is renormalized into a vacant site. We have taken into account the essential ways that a cell be connected to the neighboring cells and consequently be a part of a fragment. Thus the spanning rule is suitable for the renormalization of the archipelago. Note that in this simple transformation there are some known errors [9 ]. For example, cells that are connected in the original lattice can be disconnected in the rescaled lattice. Conversely, cells disconnected in the original lattice can be part of the same renormalized fragment. These problems are well discussed in ref. [9 ] and we expect them to be irrelevant. First let us consider the smallest cells (b = 2 ) of an 508
30 July 1990
Euclidean system in a square lattice. By definition, a cell is occupied if there is a connected path through the cell. The clusters entering the transformation are shown in fig. I a. We consider then, all different ways that a spanning duster can be obtained from the initial full occupied configuration by a fragmentation process. This is shown for a cluster of two occupied sites in fig. lb. The renormalization transformation is given by
K' = K 4 + 4 K 3 + 8 K 2 .
( 1)
This equation has two trivial fixed points (K* = 0 and K*=oo) and a critical one (K*=0.118) with exponent B = A/2 = 0.521. This value of B agrees with the numerical simulation value, Bs = 0.55 + 0.04, for twodimensional rings [ 7 ]. For cells with b = 3 we obtain K' = K 9 + 8K a + 60K 7+ 408K e + 1904K 5 + 4560K 4+ 3600K 3
(2)
and B=0.503. For cell-to-cell transformation ( b = 3) we find B=0.500. large cell renormalization group is necessary t o determine the asymptotic behavior of the B exponent, but all values obtained here agree with the numerical value. We have also evaluated the renormalization transformation for a cubic lattice with the smallest cells (b = 2 ), From the renormalized relation
•
•
•
0
•
•
•
•
•
•
0
0
•
01
•
02
•
0 2
•
01
a)
b) Fig. 1. (a) The set of spanning clusters in a 2 X 2 cell enteringthe RG transformation of an Euclidean system. Clusters of 3 and 2 sites obtained by permutations of the vacant sites shouldalso be considered. (b) The two waysthat a cluster of two occupiedsites can be obtained from the initial full cluster by a fragmentation process. Full circles representoccupied sites and the order of consumptionis indicated by the numbers.
Volume 147, number 8,9
PHYSICS LETTERS A
K' = K s + 7K 7+ 56K 6 + 336K 5+ 1632K 4 + 5760K 3+ 8640K 2 .
(3)
we find that the critical fixed point is near zero ( K * = l . l × 1 0 -4) and B=0.498. This is the first evaluation of the archipelago fractal dimension for cubic systems. The next step is to study the fragmentation of a fractal system. Consider the attack on percolation clusters on a square lattice. A PSRG approach to this problem requires at least two parameters, a probability p that a site is occupied and a fugacity K associated with such occupied site. The renormalized transformation K' =Rb(K, p) contains all ways that a spanning cluster can be obtained by fragmentation from an initial configuration. But now there are several initial configurations, each of them occurring with a probability given by some polynomial function ofp. I n fact the initial clusters are the spanning ones entering the other renormalization transformation equation p ' =fb (P). We consider that all connected path traversing the cell in any direction contribute to J~(p). Therefore we have for the smallest cell ( b = 2 ) p' _~p4-1-4p3q-l-4p2q2 , K' = p 4 ( K 4 + 4 K 3 + S K 2 ) +4p3q(K3+ 2K 2) + 4p2q2K2 ,
(4)
where q = 1 - p . W h e n p = 1 we recover the Euclidean case studied above. The nontrivial fixed points are (p*=0.382, K*=0.488), (1, 0.11) and (0.382, 0). The B exponent evaluated at the (0.382, 0.488) fixed point is B = 0.553. It agrees with the numerical value Bs = 0.53 + 0.03 [ 7 ]. Our percolation threshold is far from the expected value (pc~0.593) [ 12]. So, we have also evaluated the B exponent using this value in (4) and found essentially the same exponent. It is interesting to mention that this exponent is close to the empirical exponent Be = 0.6 of the geophysical distribution of islands on the whole earth [ 8 ]. Let us discuss now the validity of our renormalization group approach. It is known that attempts to adapt real space renormalization methods to growth problems are still unclear [13,14]. In particular, Pietronem et al. [ 13,14 ] have discussed critically the PSRG method [ 10 ] and proposed a new method, the fixed scale transformation, to evaluate the fractal
30 July 1990
dimension for the dielectric breakdown model. It turns out that to compute the probability of an internal frozen configuration of a cell, we need consider infinity growth processes outside this cell. This point has been neglected in the PSRG method and it is the main reason for the failure of that approach. Further, if one were to lift the artificial restriction to configurations that span the cell in both directions, the fractal dimensions obtained by Gold et al. [ 10] are too low when compared with the expected values [ 13,14 ]. Some of these problems were already partially solved in ref. [11 ] through a phenomenological PSRG applied to growth phenomena. In that paper, the spanning rule which measures the spatial extent of relevant clusters has been changed by the use of the radius of gyration and the probability of a relevant configuration has been determined by Monte Carlo simulations. Furthermore, a phenomenological parameter has been introduced in order to improve the fractal dimension of the grown structure. In fragmentation the situation could be different. The consumption probability of a perimeter site of a cell depends on external processes. However to destroy an internal site of a cell we must have always a vacant site in the perimeter. The fragmentation process in a cell is from out sites to the inner ones. It means that in a cell, when we consider only the internal fragmentation process, the probability to destroy an inner site is considered. It is worth mentioning that we have not imposed any artificial restrictions in the spanning rule and the results obtained agree very well with the expected values. In summary, we have developed here the first renormalization group approach for geometrical models of dynamic fragmentation process. The fractal dimension 3 = 2B describing the fragmentation or the irregularity of the system has been evaluated. For one situation this is, to our knowledge, the first time that such evaluation is done. In other cases these exponents are compared with the numerical value of simulations with good agreement. I would like to thank S.L.A. de Queiroz and S. Oliffson Kamphorst for useful discussions. I am indebted to M.A. Gomes who has introduced me to fragmentation problems.
509
Volume 147, number 8,9
PHYSICS LETTERS A
References [ 1 ] H.I. Herrmann, Phys. Rep. 136 (1986) 153. [ 2 ] D.E. Crady and M.E. Kipp, J. Appl. Phys. 58 ( 1985 ) 1210. [ 3 ] R. Englman, Z. Jaeger and A. Levi, Philos. Mag. B 50 (1984) 307. [ 4 ] M. Sahimi and T.T. Tsotsis, Phys. Rev. Lett. 59 ( 1987 ) 888. [5] B.L. Holian and D.E. Grady, Phys. Rev. Lett. 60 (1988) 1355. [6]M.A.F. Gomes and G.L. Vasconcelos, in: Pro¢. 2nd Workshop on Scaling, fractais and nonlinear variability in geophysics (Paris, 1988). [ 7 ] M.A.F. Gomes and G.L. Vasconcelos, J. Phys. A 22 (1989) L757.
510
30 July 1990
[ 8 ] B.B. Mandelbrot, The fraetal geometry of nature (Freeman, San Francisco, 1982). [9] H.E. Stanley, P.J. Reynolds, S. Redner and F. Family, in: Real-space renormalization, eds. T.W. Burkhardt and J.M. van Leeuwen (Spinser, Berlin, 1982). [ 10] H. Gould, F. Family and H.E. Stanley, Phys. Rev. Lett. 50 (1983) 686. [ 11 ] J.L. Montag, F. Family, T. Vicsek and H. Nakanishi, Phys. Rev. A 32 (1985) 2557. [12] J.W. Essam, Rep. Prog. Phys. 43 (1980) 833. [ 13 ] L. Pietronero, A. Erzan and C. Evertsz, Phys. Rev. Lett. 61 (1988) 861. [ 14] L. Pietronero, A. Erzan and C. Evertsz, Physica A 151 (1988) 207.