Entropy spectrum for distribution of history probabilities in growth models

Based on the multifractal formalism, we introduce a multifractal spectrum associated with entropy of clusters in growth models by calculating probability distributions of histories from an initial seed to a resultant cluster. The entropy spectrum is shown in a ballistic growth model producing fanlike clusters and Eden growth as examples. Numerical calculations of the models give results that suggest the existence of phase transitions in the spectrum. @S1063-651X~97!09506-8# PACS number~s!: 61.43.Hv, 64.60.Cn Considerable theoretical and experimental efforts have been undertaken to investigate scaling properties of fractal clusters and rough surfaces of compact clusters in the past few years @1#. Although our understanding of geometrical features of clusters is advanced by these efforts, most of them study only resultant patterns as snapshots at a moment rather than growth itself. For instance, the f (a) spectrum @2# of diffusion-limited aggregation ~DLA !@ 3 #gives us some information about the distribution of active or unscreened sites at the surface, but one could hardly predict how the pattern evolves. In such an analysis, one assumes that patterns should be self-similar in every time step and the fractal clusters are most probable. However, in what sense could we confirm that our observing patterns are so generic and most probable? To understand that, one has to take into account a huge number of possible ways of growth and those probabilities. Recently, Elezgaray et al. have introduced the ‘‘history probability,’’ that is, the probability of finding a history from a seed particle to a resultant cluster, and studied morphology selection mechanism of Laplacian growth @4#. If the distribution of history probabilities is given in some invariant form independent of the number of particles, one can know how often a cluster to be examined is observed in all possible patterns. To our knowledge, there is no formulation to characterize such statistics of clusters, except for a few studies on entropy @4‐6#. Our primary purpose in this Brief Report is to give a general formalism of the distribution of history probabilities for stochastic growth models in a similar way to the analysis of chaotic orbits in dynamical systems, i.e., the multifractal spectrum of entropy @7,8#. In addition, we would like to show a few examples for the formulation in wellknown models. Let us consider a growth model starting with a single seed particle on a lattice. Let s m be a cluster consisting of m particles and p($s m%) be the probability of finding a history set $s m%. Assuming that the conditional probability that s m becomes s m11 is proportional to its mass m at the some power g, the history probability takes the form