Growth with Memory. Institut de Physique Teorique, Universite de Fribourg, Perolles, Fribourg, CH-1700

Transcription

1 Growth with Memory Matteo Marsili 1 and Michele Vendruscolo 2 1 Institut de Physique Teorique, Universite de Fribourg, Perolles, Fribourg, CH International School for Advanced Studies (SISSA) Via Beirut 2-4, Trieste, Italy (November 21, 1997) Abstract A new model for cluster growth is introduced. The model is built with a long time memory which is similar to the one found in cluster growth in disordered media. The model, as a function of its parameter, interpolates between Eden growth, Invasion Percolation-like growth and has a behavior similar to self avoiding walks in another limit. PACS numbers: r, j, Hv SISSA ref. 122/96/CM/ 1 Typeset using REVTEX

2 Since the pioneering work of Eden [1], the study of growth processes has ourished into a very lively eld of statistical physics [2]. In spite of their simple denition, models like the one proposed by Eden, display very non-trivial properties which are not yet completely understood. The Eden model describes in the simplest possible way a stochastic growth process driven by an internal pressure [3]. It is curious to note that invasion percolation [4] (IP), a model for the more complex situation where this process occurs in a random medium, has probably been understood to a deeper extent [5,6] than the Eden model. In particular a link between the two models where used in ref. [7] in order to infer, from the known behavior of IP in high dimensions, that of Eden clusters and, more generally, of the KPZ universality class which it is supposed to belong to [2]. The purpose of this letter is to propose a new model which, to some extent, interpolates between these two models and which, moreover has strong similarities with self avoiding walks [8] in a limiting case. The basic ingredient of the model is a probability distribution for growth events which decays as a power law with respect to the \age" of the site where growth occurs. Before giving a more precise denition of the model, it is worth to spend some words on the motivation which led us to this model. The choice of the growth probability distribution introduces a long time memory. A similar memory was recently introduced and analyzed for models in the context of self organized criticality [9] (SOC). It was shown that the scenario of self organized criticality is intimately connected to the presence of memory. On the other hand it was found [9,10] that memory emerges naturally from dynamics in random media. Following similar lines, our intention here is to analyze the relation between memory, disorder and fractality, in the context of fractal growth. Let us remind the denition of the Eden model. At t = 0 the cluster C 0 i(0) consists of a seed particle at the origin i(0) = 0 of a d dimensional lattice [11]. At the rst time step a particle is created in one site i(1) neighbor to the cluster, so that the cluster becomes C 1 = f0; i(1)g. At a generic time t, a site i(t) is chosen randomly in the t?1 of perimeter sites of the cluster C t?1. It is occupied by a particle and added to the set of cluster sites S C t = C t?1 fi(t)g. In the Eden model the growth event occurs on a randomly chosen site of 2

3 the perimeter of C t, i.e. with a probability distribution which is uniform. Let us assign an age variable k i to each site of the lattice. This is set to k i = 0 initially. For a given site i, k i remains 0 until the site is reached by the cluster, or better when it enters the set of perimeter t. At this time we set k i = 1 and at each time step, unless site i grows, k i is increased by one. We call k i the age of site i. Indeed it mimics a local clock which starts to measure the time steps since the rst entry of site i in the set of perimeter sites. Age variables can be introduced in any model of cluster growth to quantify memory eects. A site i t \dies" when it grows, i.e. its age stops increasing. We shall later discuss the use of age variables in IP and Eden growth to quantify memory eects in these two models. Here we shall focus instead on a model whose dynamics is expressed directly in terms of age variables. The model is dened by assigning the probability i;t that site i t is selected and occupied i;t = 1;t k? i ; X i2@c t i;t = 1: (1) Therefore k i (t) play the role of dynamic variables in our model. The constant 1;t is xed by the normalization condition of i;t which implies that only one site at a time is allowed to grow. This condition, as discussed in ref. [9], can be seen as the result of a large separation of timescales which is typical in Self Organized Critical systems. Clearly when = 0 the model reduces to the Eden one, since any site has the same probability to grow. With > 0, the model describes a process where younger sites have more chances to grow than older ones. It might be that such a situation could be relevant for the growth of cell or bacteria colonies. Leaving aside such speculations, we note that recently it has been found that this dynamics bears many similarities with the process of Invasion Percolation [10] (see later). In order to understand the nature of the cluster grown by the above rule, let us focus on a particular site i which enters the perimeter set at time t 0. For this sate we can estimate the probability that it will not grow in the up to time t, which is 3

4 P fi t g = ty k=t 0 [1? 1;k (k? t 0 + 1)? ]: (2) Ignoring the dependence on k of 1;k for the moment, it is easy to see that this probability vanishes as k! 1 if 1 and stays nite for > 1. In order to see this it is enough to exponentiate eq. 2 and to study the convergence of the sum at the exponent. This suggests that for 1 any site which enters in the perimeter will, sooner or later, grow. On the other hand there will be a nite probability, for > 1, that a given site will not grow. This is the rst analogy with IP. Indeed in IP the probability that a site does not grow, once it is reached by the cluster, is exactly 1? p c [4], where p c is the critical percolation threshold. This feature, like in IP [5], implies that the surface to volume ratio j@c t j=jc t j of the clusters grown for > 1 is nite. Observing that, for any compact cluster in d dimensions this ratio vanishes as t?1, we argue that the clusters for > 1 will have a fractal dimension D f () < d. Introducing the number n k;t of sites t with age k, we can restate the result saying that j@c t j lim = lim t!1 jc t j n k;t > 0 for > 1: (3) k;t!1 A little more rened calculation, taking into account the dependence of 1;t on t (which derives from the asymptotic behavior of n k;t through the normalization condition) conrms this conclusions. Simple scaling suggests 1;t t?1 for < 1 which leads to compact Eden-like clusters. On the other hand for > 1 we nd eq. (3), 1;t ' 1;1 > 0 and therefore fractal clusters. The similarity with IP goes far beyond these observations, as shown by the data in gure 1. This reports the frequency with which sites with age k did grow in a simulation of IP as a function of k. This is in striking agreement with the behavior of eq. 1. The exponent inferred from the t is IP = 1:35. The statistical properties of the process are qualitatively similar to those of IP in the interval 1 < < 2. It is easy to argue that at = 2 a further qualitative change in the process occurs. Let us consider an asymptotic cluster t 1 and focus on the subset O k;t t of the sites with age larger than k. Clearly, at later times, O k;t = O k+s;t+s, and the probability that growth will occur in this \older" region at time t + s is clearly 4

5 P stop (k + s; t + s) = X q>k+s n q;t+s 1;t+s q? : (4) We can apply a standard 0?1 law of probability [12] to nd out how many events of growth will occur in the set O k;t after time t. More precisely we can conclude that, if 1X s=0 P stop (k + s; t + s) < 1; only a nite number of sites in O k;t will grow. If the above series diverges, instead, one can conclude that an innite number of events will occur. Clearly, by eq. (4) the series converges for > 2 and diverges otherwise. Actually the number of sites in O k;t is always nite. However we can consider t 1 as large as we want, so that the above conclusion can be translated by saying that for > 2 there will be a nite number of events occurring on the sites that are older than k at time t 1, whereas for < 2 the number of events will diverge as t! 1. The dierence is, in other words, that for > 2 one can ideally consider that a fraction of the interface is frozen in the sense that almost no growth will occur on it. This is similar to what happens [13] in other growth models like Diusion Limited Aggregation [14]. On the other hand for < 2 one can never exclude that growth will occur on a set of very old sites (which is exactly what happens in IP). In gure 2 we report numerical results for the fractal dimension D f () obtained from the scaling of the radius of gyration R(t) t 1=D f in d = 2. The inset shows a typical plot ln R vs. ln t from which the exponent was tted. For < 1 the clusters were found compact with D f = d = 2. We also included the point relative to the fractal dimension of IP clusters D IP f = 91=48 for = 1:35, which falls nicely on the curve D f (). The shape of D f () is highly non-trivial, we were unable to nd any sensible Flory type approximation. Note, in particular, that close to = 2 the fractal dimension displays a minimum. Yet another interesting information on the process is obtained by considering the statistics of avalanches. The dynamics of avalanches, which is typical of self organized processes, is naturally described in terms of age variables. Let us recall briey the arguments of ref. [9]. The time series fk i(t) g of the ages of the sites grown contains the information about the causal relations among dierent events. It is clear, for example, that the growth of 5

6 a site of age s + 1 is not causally connected with the previous s events, whereas it was related with the event s + 1 time steps before in which that site rst entered the t. In the SOC language, we can say that this event \stops" the ongoing avalanches of size s, i.e. it interrupts the causal connections between events. Therefore the probability that an avalanche is stopped at time t is given by P stop (s; t) above. Clearly once an avalanche is stopped it becomes inactive. Therefore the probability that an avalanche is active is P act (s; t) = Q k=1;s[1? P stop (t? k; s? k)] and the probability of an avalanche of size s is therefore P (s) = P act (s; t)p stop (s; t). It is easy to see that, in our case, lim s;t!1 P act (s; t) > 0 is nite for > 2 whereas it vanishes for 1 < < 2. In other words, for > 2, there is a nite probability P act (1) that an avalanche will never be stopped which means that all future events will be causally connected to the initial event. Furthermore one can argue that the exponent of the avalanche distribution, for nite avalanches, behaves as P (s) s? with =? 1. For < 2 instead all avalanches are nite. The above arguments suggest a stretched exponential form for P (s) which was conrmed numerically. This contrasts with the avalanche size distribution in IP which has a power law character and suggests that our model with = 1:35 is not equivalent to IP. Note that a probability k;t can also be introduced in IP [10]. However its dependence on the past history is much more complex than the one our model assumes and most probably contains time correlations which are not properly accounted by eq. (1). It is also interesting to consider the limit! 1. In this case only the site with k = 1 can grow. This suggests that the process reproduces a kinetic self avoiding walk in this limit. As! 1, in d = 2, we nd D f ()! 1:5 (we found D f ( = 40) ' 1:50). Note that our model is dierent from the True Self Avoiding Walk [15] because it cannot cross itself. The walk can form loops and when this happens either it enters the loop or it continues outside. In the former case it is easy to see that it lls up the loop and then it restarts from the last external site. If it continues outside, it will never ll the loop. Since the two events occur each with a nite probability, the walk will leave empty loops of any size. With respect to higher dimensions, it would be interesting to investigate the issue of the 6

7 upper critical dimensionality d c () of our model. At this level, we conjecture that, since self intersections are irrelevant for d > 4, d c = 4 for = 1, and presumably for > 2. The similarity with IP for 1 < < 2 suggest instead that d c = 6 in this interval. Finally, for < 1, the d c should coincide with that of the Eden model, which is an unsettled issue yet. We believe these discussion, in spite of its very rough arguments, shows that our model could also be an interesting tool to investigate the relations between the geometry of high dimensional spaces and stochastic growth processes. As the Eden model has shown, this is an issue which has is not yet well understood and which has shown a surprising richness [16]. 7

8 REFERENCES [1] M. Eden Fourth Berkeley Symposium on Mathematics, Statistics and Probability Vol. 4, ed. F. Neyman (Berkeley, CA: Univ. of California, 1961). [2] T. Halpin-Healy and Y.-C. Zhang, 1995, Phys. Rep [3] A. Maritan, F. Toigo, J. Koplik and J. R. Banavar, Phys. Rev. Lett. 69, 3193 (1992); M. Marsili, A. Maritan, F. Toigo and J. R. Banavar, Rev. Mod. Phys. XX, XXXX (Oct. 1996). [4] R. Lenormand and S. Bories: C. R. Acad. Sci. 291, 279 (1980); D. Wilkinson and J. F. Willemsen: J. Phys. A 16, 3365 (1983); M. Cieplak and M. O. Robbins, Phys. Rev. Lett. 60, 2042{2045 (1988). [5] J. T. Chayes, L. Chayes and C. M. Newman: Commun. Math. Phys. 101, 383 (1985). [6] R. Caero, A. Gabrielli, M. Marsili and L. Pietronero, Phys. Rev. E, XX, XXX (1996). [7] M. Cieplak, A. Maritan and J. R. Banavar, Phys. Rev. Lett. 76, 3754 (1996). [8] P. G. de Gennes: Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979). [9] M. Marsili, G. Caldarelli and M. Vendruscolo, Phys. Rev. E 53, 13, (1995). [10] M. Marsili, J. Stat. Phys. 77, 733 (1994). [11] We consider here a radial geometry for growth. The discussion generalizes trivially to a strip geometry. [12] W. Feller, An Introduction to Probability Theory Vol. 1, 200, John Wiley and Sons (1968). [13] A. Erzan, L. Pietronero and A. Vespignani, Rev. Mod. Phys. 63, 545 (1995). [14] T. A. Witten and L. M. Sander, Phys. Rev. Lett

9 [15] L. Peliti and L. Pietronero Riv. Nuovo Cimento, 10, 1 (1987). [16] G. Parisi and Y. C. Zhang, Phys. Rev. Lett. 53, 1791 (1984); D. Dhar in On Growth and Form, eds N. Ostrowsky and H. E. Stanley, p. 288 (Martinus, Boston 1986). 9

10 FIGURES FIG. 1. Frequency k of growth events on sites with age k for Invasion Percolation. The t has slope IP = 1:35. FIG. 2. The fractal dimension D f () as a function of (). The solid line is an interpolation intended only to guide eyes. The 2 point is the measured fractal dimension of invasion percolation D IP f ' 1:89. The inset shows a log-log plot of R(t) t 1=D f for = 2:25. 10