Theoretical and Mathematical Physics

, Volume 96, Issue 3, pp 1035–1056 | Cite as

From the Hamiltonian mechanics to a continuous media. Dissipative structures. Criteria of self-organization

  • 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 based on 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 basis, the definition of the Gibbs ensemble for nonequlibrium process is given. The Boltzmann 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 generalized kinetic equation for the Boltzmann gas consists of the two dissipative terms: 1) the “collision integral,” defined by the processes in a velocity space; 2) an additional dissipative term of the diffusion type in the coordinate space. Owing to the latter the unified description of the kinetic, hydrodynamic, and diffusion processes for all values of the Knudsen number becomes possible.

The H-theorem for the generalized kinetic equation is proved. The 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. One is proportional to the entropy, and the other is proportional to the gradient of entropy. The existence of the second term allows one to give a 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 constant pressure leads to the Fourier law.

The equations of gas dynamic for a 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 describing the Brownian motion and of autowave processes in active media is considered. The connection with reaction diffusion equations, the Fisher-Kolmogorov-Petrovski-Piskunov and Ginzburg-Landau equations, is established. We discuss the connection between the diffusion of particles in a restricted system with the natural flicker (1/f) noise in passive and active systems.

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


Entropy Entropy Production Continuous Medium Knudsen Number Dissipative Term 
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© Plenum Publishing Corporation 1993

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  • Yu. L. Klimontovich

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