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 rando m distributions of independent particles in Euclidean space. The geome trical fractal model used in astrophysics and hydrodynamics assumes th at the number of particles enclosed in a hypersphere of radius r obeys a Poisson law v(N)(N!)(-1) exp (-v), where the average number of part icles is a power function of radius r: v similar to r(df) and d(f) is the dimension of a fractal structure embedded in the Euclidean space c onsidered. A statistical fractal distribution is introduced by assumin g that the probability density of the distance between two nearest par ticles has a long tail of the inverse power law type phi(0)(r)similar to r(-)((1+H)(r)) as r --> infinity, where H-r > 0 is a statistical fr actal exponent. The distribution of the number of particles enclosed i n a hypersphere of radius r is also Poissonian but the average number of points increases logarithmically rather than algebraically with the radius r: v similar to H(r)ln(r/r(0)) as r --> infinity. The spatial distribution of points corresponding to this statistical fractal model is much rarer than in the geometrical fractal case. An alternative ap proach is derived by assuming that the probability density of the dist ance between two particles has a very broad logarithmic tail phi(0)(r) dr similar to d[ln(m)(r/r(0)))]/[ln(m)(r/r(0))](1+H) > 0, r --> infini ty where ln(m)(r/r(0)) = ln...ln(r/r(0)) is the mth iterated logarithm of r/r(0). For such a logarithmic statistical fractal the number of p articles increases much more slowly with the radius r than in the 'pur e' statistical fractal case: v similar to H In-m+1(r/r(0)) as r --> in finity. By using a heuristic approach two general probabilistic models are derived which include both the geometrical and statistical fracta l models as particular cases; these models predict power law dependenc es of the average number of particles for small systems and logarithmi c dependences for large systems, respectively. The significance of the generalized models is elucidated by using the notion of a specific fr actal hypervolume <(omega)over tilde> which corresponds to a given par ticle. For the generalized model derived from the pure statistical fra ctal <(omega)over tilde> is made up of two additive contributions: a c onstant one which determines the behavior of the model for small syste ms and a linearly increasing contribution with the total fractal hyper volume which is predominant for large systems. A similar structure of <(omega)over tilde> exists for the generalized model leading to the lo garithmic statistical fractal regime.