Crossover from geometrical to stochastic fractal statistics for translationally invariant random distributions of independent particles in n-dimensional Euclidean space

The paper deals with the statistics of translationally invariant random distributions of independent particles in Euclidean space. The geometrical fractal model used in astrophysics and hydrodynamics assumes that the number of particles enclosed in a hypersphere of radius r obeys a Poisson law vN(N!)−1exp(−v), where the average number of particles is a power function of radius r: v rdf and df is the dimension of a fractal structure embedded in the Euclidean space considered. A statistical fractal distribution is introduced by assuming that the probability density of the distance between two nearest particles has a long tail of the inverse power law type φ0(r) r−(1 + Hr) as r → ∞, where Hr > 0 is a statistical fractal exponent. The distribution of the number of particles enclosed in a hypersphere of radius r is also Poissonian but the average number of points increases logarithmically rather than algebraically with the radius as r → ∞. The spatial distribution of points corresponding to this statistical fractal model is much rarer than in the geometrical fractal case. An alternative approach is derived by assuming that the probability density of the distance between two particles has a very broad logarithmic tail , is the mth iterated logarithm of . For such a logarithmic statistical fractal the number of particles increases much more slowly with the radius r than in the ‘pure’ statistical fractal case: as r → ∞. By using a heuristic approach two general probabilistic models are derived which include both the geometrical and statistical fractal models as particular cases; these models predict power law dependences of the average number of particles for small systems and logarithmic dependences for large systems, respectively. The significance of the generalized models is elucidated by using the notion of a specific fractal hypervolume gw which corresponds to a given particle. For the generalized model derived from the pure statistical fractal gw is made up of two additive contributions: a constant one which determines the behavior of the model for small systems and a linearly increasing contribution with the total fractal hypervolume which is predominant for large systems. A similar structure of gw exists for the generalized model leading to the logarithmic statistical fractal regime.