Condensed Matter Physics, 1994, No. 4, p. 89-116, English
Title: From the hamiltonian mechanics to a continuous media. Dissipative struct ures. Criteria of self-organization
Authors: Yu.L. Klimontovich

The paper is aimed at presenting some main ideas and results of the modern statistical theory of macroscopic open systems.

We begin from the demonstration of the necessity and the possibility of the unified description of kinetic, hydrodynamic and diffusion processes in nonlinear macroscopic open systems on the base of a generalized kinetic equations.

A derivation of the generalized kinetic equations is based on the concrete physical definition of continuous media. A "point" of a continuous medium is determined by definition of physically infinitesimal scales. On the same base the definition of the Gibbs ensemble is given. The Boltzman gas and a fully ionized plasma as the test systems are used.

For the transition from the reversible Hamilton equations to the generalized kinetic equations the dynamic instability of the motion of particles plays the constructive role.

The kinetic equation for the Boltzman gas consists of the two dissipative terms: 1. The "collision integral" is defined by processes in a velocity space; 2.The additional dissipative term of the diffusion type in the coordinate space. Owing to the later the unified description of the kinetic, hydrodynamic and diffusion processes for all values of the Knudsen number becomes possible.

The $H$-theorem for he generalized kinetic equation is proved. An entropy production is defined by the sum of two independent positive terms corresponding to redistribution of the particles in velocity and coordinate space respectively.

An entropy flux also consists of two parts. The one is proportional to the entropy, and other one is proportional to the gradient of entropy. The existence of second term allows to give the general definition of the heat flux for any values of the Knudsen number which is proportional to the gradient of entropy. This general definition for small Knudsen number and a constant pressure leads to the Fourier law.

The equations of gas dynamic for special class of distribution functions follow from the generalized kinetic equation without the perturbation theory for the Knudsen number. These equations differ from the traditional ones by taking the self- diffusion processes into account.

The generalized kinetic equation for description the Brownian motion and of autowave processes in active media are considered. The connection with reaction diffusion equations - the Fisher-Kolmogorov-Petrovski-Piscunov and Ginzburg-Landau equations, are established. We discuss the connection between the diffusion of particles in a restricted system with the natural flicker ($1/f$) noise in passive an d active systems.

The Criteria of the relative degree of order of the states of open system - the criteria of self-organization, are presented.

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