Model

Gene, allele, chromosome, cell, individual, chemical element, compound of elements, mineral, rock, community of organisms described on a sample plot selected, pixel of a cosmic image, car, plant, settlement, town, country, and so on may be elements of models. In all the cases, we have n elements, and each of them may be referred to one of the k classes according to its properties. In the process of interactions, the elements belonging to different classes may be assumed to form structures locally stable in time. It is unknown a priori what structures are stable or unstable, but their whole diversity is described by the formula I = »1!»2l»3l.. .nm\, where n is the number of elements in class i.

This is a large value. Using simple rearrangements, we obtain that this value is in general agreement with m i=m ln(I) = Nln(N) - £ n, ln(n,) = -N^NbN

where ni is the number of elements of class i, S = - Kp^p, is the Gibbs-Shannon's entropy (see Shannon-Wiener Index) (pi = ni/N is the probability of elements of class i in sample N, Kis the analog of Planck's constant).

Under equilibrium (derivatives are close to zero), in a linear case, the Gibbs's distribution has resulted. A. Levich in 1980 supposed the nonlinearity of relations to the property space and obtained the rank distributions:

• pi = y exp(—a,) - the Gibbs' rank distribution, that is, condition of linear dependence of a system on a resource;

• pi = y exp(A log(i)) = yi—A - the Zipf's rank distribution, that is, logarithmic dependence on a resource;

• pi = y exp(A log(a + i)) = y(a + i)—a - the Zipf-Mandelbrot's rank distribution, that is, logarithmic dependence on an resource, where a is the number of unoccupied (vacant) state with an unused resource;

• pi = y exp(A log(log i)) = y log i—A - the MacArthur's rank distribution (the broken stick), at twice logarithmic dependence on a resource.

Simple transformations on making the assumption that there is some class with only one element allow finding widespread relationships of the number of species with the volume of sampling N or with the area, where the sampling was made. Such relations obtained in island biogeography are true for any phenomenon.

If these relations are nonequilibrium, members with order >1 are included into rank distributions. These forms of distributions are typical in nature. If a system is nonstationary, Kulback's entropy is a measure of nonsta-tionarity. Under the same conditions, entropy of the nonstationary system is less than entropy nonequilibrium one, and the entropy of the nonequilibrium system is less than that of equilibrium one.

According to the model, diversity (entropy) of a system is the function of power or diversity of the environment and evolutionary parameters. The first parameter is identical to free energy of Gibbs (exergy in a nonstationary case), the second one to temperature. Thus, in the closed space, evolution of diversity corresponds to the thermodynamic model, and entropy increases in time.

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