By Dietrich Stauffer (auth.), H. Eugene Stanley, Nicole Ostrowsky (eds.)

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**Additional resources for Correlations and Connectivity: Geometric Aspects of Physics, Chemistry and Biology**

**Example text**

L l Consider an Ising configuration of spins: between nn parallel spins, introduce a bond being active with probability p. The bonds do not affect the energy of the pair of the spins, instead they are introduced to define the connectivity and therefore the clusters. How can we calculate the connectivity and thermal properties of the Ising model? For a given configuration of spins {Sd the weight is given by the Boltzmann factor e-,BH{Si}, where (J = l/kT. The weight associated with a given configuration of bonds C which is made of ICI bonds present and IBI bonds absent between parallel spins, is given by plCI(l - p)IBI.

Stanley and N. ), Correlations and Connectivity, 21-33. © 1990 Kluwer Academic Publishers. 2 These clusters are in fact too large, representing both correlations and pure geometrical effects. As an example at T = 00 in a typical configuration clusters of parallel spins are present although there is total absence of correlations. To eliminate the pure geometrical effect a different definition of cluster was proposed5 for the Ising model and generalized to the q-state Potts model 6 The idea was based on a site bond correlated percolation model which was introduced as a model for a sol-gel transition.

Curve shows evidence of multifractality with respect to t. \ \ .. \ o -1 \ -2 -3 -4 -5 \ -6~--~4--~-3~--~2--~-1~-±O--~--+-~3~ q JAN . POOLE • MAC ISAAC . HUNTER . 47 MANNA Consider a cluster or fractal of overall size R. Let g(r, R) be the probability density for finding a piece of the cluster a distance r from the origin from which the cluster was grown. For generalized scaling this "density profile" g(r, R) is invariant when all lengths are rescaled by I. That is, if we define r == r/I, and, R == R/l, then g(r, R) = I d- d, g(r, R), where df is the fractal dimensionality, and d is the dimension of space.