FINITE-SIZE SCALING FOR CELLULAR AUTOMATA WITH RANDOMIZED GRIDS AND FOR FRACTAL RANDOM FIELDS IN DISORDERED SYSTEMS

Abstract

The finite-size effect on the statistics of independent random point processes is analyzed in connection with the theory of cellular automata with randomized grids. The space distribution of a large but finite number N0 of independent particles confined in a large region of ds-dimensional Euclidean space of size VΣ is investigated by using the technique of characteristic functionals. Exact formal expressions are derived for all many-body correlation functions of the positions of the particles as well as for all cumulants of the concentration field. These functions are made up of the contributions of the different negative powers of the total number N0 of particles from the system [Formula: see text] where n =1, 2, … are the orders of the correlation functions or of the cumulants of the concentration field. In the thermodynamic limit N0, VΣ → ∞ with N0 / VΣ = constant only the terms An(0) survive; the other terms An(m) with m > 0 express the finite-size effects. It is shown that, even though the particles are independent, for a finite size of the system a correlation effect different from zero exists among their positions and this correlation vanishes in the limit of an infinite size. The correlation among the positions of the different particles is a finite-size effect due to the conservation of the total number of particles which is similar to the correlation among ideal bosons or fermions at low absolute temperatures. The stochastic properties of an additive scalar field generated by a random distribution of independent particles are investigated. The approach can be applied to the study of stochastic gravitational fluctuations generated by a random distribution of stars or galaxies, of the short-range mean field generated by the particles making up a disordered medium, or of the distribution of the offspring number generated by a plant population randomly distributed in space. Special attention is paid to the finite-size scaling corrections to the long-range self-similar fractal fields. The computations lead to the surprising result that for large values of the resulting field the finite size of the system has practically no influence on the tails of the probability densities of the resulting field, which obey a statistical fractal scaling law of the negative power law type. This apparent paradoxical effect is due to the fact that the very large values of the field corresponding to the tails of the probability densities are generated by the closest neighbor of the test particle considered and are not influenced by the more distant particles.