next up previous
Next: Uwano Y. Up: Book of Abstracts Previous: Trubetskov D.I.

Uchaikin V.V., Gusarov G.G

Exactly resolvable model of random medium
based on the stable law theory

V.V. Uchaikin and G.G. Gusarov
Moscow State University Branch in Ulyanovsk, Ulyanovsk, Russia

One of the most interesting application of statistical mechanical concepts is the large-scale distribution of matter in the universe. Most of its properties have been obtained by a statistical correlation analysis for catalogs of galaxies and clusters [1,2]. The galaxies are considered as the smallest individual entities and their distribution is characterized by points tex2html_wrap_inline4426 in the 3-dimensional Euclidian space. The standard analysis consists in the calculation of two - point correlation function

displaymath2025

Observations show, that C(r) can be approximated by a power law

displaymath2028

This fact is the basis for the assumption about of a fractal structure of the universe [2].

At first the use of walk method for mathematical simulating such distribution of galaxies was offered by Mandelbrot [3]. The process starts from choosing the position where the first galaxy to be, the next galaxy is placed in a random isotropic direction and at a random distance l from the previous one while l is chosen from the distribution

displaymath2031

This procedure repeats many (maybe infinitely large number of) times and every time the value and direction of l are independently chosen. Based on such probability distribution the model cannot be solved exactly.

In present work it is shown that choice of 3-dimensional spherically symmetric stable distribution tex2html_wrap_inline4436 , having the same asymptotic behaviour as power law tex2html_wrap_inline4438 , instead of the tex2html_wrap_inline4440 leads to exact analytical expressions for irreducible correlation functions of different orders and the cut off probability 1 - q at each step:

displaymath2040

Here tex2html_wrap_inline4444 is the average number of all the particles per unit volume, tex2html_wrap_inline4446 and tex2html_wrap_inline4448 means the summation over all n! permutations of the arguments.

The characteristic exponent of stable law tex2html_wrap_inline4452 so the stable distribution densities form a whole family. The distribution tex2html_wrap_inline4454 is a normal (Gauss') law, tex2html_wrap_inline4456 is Cauchy's law. Other distributions tex2html_wrap_inline4436 cannot be expressed in elementary functions and are calculated and tabulated in this work.

  1. P.J.E.Peebles, The Large-Scale Structure of the Universe (Princeton, 1980).
  2. P.H.Coleman and L.Pietronero, Phys. Rep. 213 (1992) 311.
  3. B.B.Mandelbrot, Fractals: Form, Chance and Dimension (San Francisco, 1977).
  4. V.M.Zolotarev, One-dimensional Stable Distributions, in Russian (Moscow, 1983).


next up previous
Next: Uwano Y. Up: Book of Abstracts Previous: Trubetskov D.I.

Book of abstracts
ICND-96