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
Article

Abstract

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.

Keywords

Entropy Entropy Production Continuous Medium Knudsen Number Dissipative Term 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Klimontovich Yu.L. On Nonequilibrium Fluctuations in a Gas. //TMF. 1971. V. 8. P. 109.Google Scholar
  2. [2]
    Klimontovich Yu.L. Kinetic Theory of Non Ideal Gases and Non Ideal Plasmas. Moscow: Nauka, 1975; Oxford: Pergamon Press, 1982.Google Scholar
  3. [3]
    Klimontovich Yu.L. The Kinetic Theory of Electromagnetic Processes. Moscow: Nauka, 1980; Berlin-Heidelberg: Springer, 1983.Google Scholar
  4. [4]
    Klimontovich Yu.L. Statistical Physics. Moscow. Nauka, 1982; New York: Harwood Academic Publishers, 1986.Google Scholar
  5. [5]
    Klimontovich Yu.L. Turbulent Motion and the Structure of Chaos. Moscow: Nauka, 1990; Dordrecht: Kluwer Acad. Pub., 1991.Google Scholar
  6. [6]
    Klimontovich Yu.L. Statistical Theory of Open Systems. Kluwer Dordrecht: Academic Publishers, (in press).Google Scholar
  7. [7]
    Monin A.S., Yaglom A.M. Statistical Fluid Mechanics. Moscow: Nauka, 1965; MIT, 1971.Google Scholar
  8. [8]
    Landau L.D., Lifshits E.M. Fluid Mechanics. Moscow: Nauka, 1986; Oxford: Pergamon Press, 1959.Google Scholar
  9. [9]
    Klimontovich Yu.L. Statistical Theory for Non Equilibrium Processes in a Plasma. Moscow: Nauka, 1964; Oxford: Pergamon Press, 1967.Google Scholar
  10. [10]
    Krylov N.S. Works for the Foundation of Statistical Physics. Moscow: Nauka, 1950.Google Scholar
  11. [11]
    Prigogine I. From Being to Becoming. San Francisco: Freeman, 1980; Moscow: Nauka, 1985.Google Scholar
  12. [12]
    Prigogin I., Stengers I. Order out of Chaos. London: Heinemann, 1984; Moscow: Progress, 1986.Google Scholar
  13. [13]
    Romanovski Yu.M., Stepanova N.V., Chernavsky D.S. Mathematical Biology. Moscow: Nauka, 1984.Google Scholar
  14. [14]
    Klimontovich Yu.L. Entropy Evolution in Self-Organization Processes. H-Theorem and S-Theorem. // Physica. 1987. V. 142A. P. 390.Google Scholar
  15. [15]
    Lorenz E. Deterministic Nonperiodic Flow. // J.Atm. Sci. 1963. V. 20. P. 167.Google Scholar
  16. [16]
    Haken. Synergetics. Berlin-Heidelberg: Springer, 1978; Moscow: Mir, 1980.Google Scholar
  17. [17]
    Anishchenko V.S. Complicated Oscillations in Simple Systems. Moscow: Nauka, 1990.Google Scholar
  18. [18]
    Neimark Yu.I., Landa P.S. Stochastic and Chaotic Oscillations. Moscow: Nauka, 1987; Dordrecht: Kluwer Acad.Publ., 1992.Google Scholar
  19. [19]
    Lifshits E.M., Pitaevsky L.P. Statistical Physics. Moscow: Nauka, 1978.Google Scholar
  20. [20]
    Nicolis G., Prigogine I. Self Organization in Non Equilibrium Systems. New York: Wiley, 1977; Moscow: Mir, 1979.Google Scholar
  21. [21]
    Haken H. Advanced Synergetics. Berlin-Heidelberg: Springer, 1983; Moscow: Mir, 1985.Google Scholar
  22. [22]
    Michailov A.S. Foundations of Synergetics I. Berlin-Heidelberg: Springer, 1990.Google Scholar
  23. [23]
    Michailov A.S., Loskutov A.Yu. Foundations of Synergetics II. Berlin-Heidelberg: Springer, 1991.Google Scholar
  24. [24]
    Murray G. Lectures on Nonlinear Differential Equation Models in Biology. Oxford: Clarendon Press, 1977.Google Scholar
  25. [25]
    Vasiliev V.A., Romanovsky Yu.M., Yachno V.G. Autowaves. Moscow: Nauka, 1987.Google Scholar
  26. [26]
    Klimontovich Yu.L. Some Problems of the Statistical Description of Hydrodynamic Motion and Autowave Processes. // Physica. 1991. V. 179A. P. 471.Google Scholar
  27. [27]
    Klimontovich Yu.L. On the Necessity and the Possibility of the Unified Description of Kinetic and Hydrodynamic Processes. // TMF. 1992. V. 92. P. 312.Google Scholar
  28. [28]
    Klimontovich Yu.L. The Unified Description of Kinetic and Hydrodynamic Processes in Gases and Plasmas. // Physics Let.A. 1992. V. 170. P. 434.Google Scholar
  29. [29]
    Van Kampen N.G. Stocastic Processes in Physics and Chemistry. Amsterdam: North-Holland, 1983.Google Scholar
  30. [30]
    Risken H. The Fokker-Planck Equation. Berlin: Springer, 1984.Google Scholar
  31. [31]
    Gardiner C.W. Handbook of Stochastic Methods for Physics, Chemistry, and Natural Sciences. Berlin-Heidelberg: Springer, 1984.Google Scholar
  32. [32]
    Gantsevich S.V., Gurevich V.L., Katilus R. Theory of Fluctuations in Non Equilibrium Electron Gas. // Rivista del Nuovo Cimento. 1979. V. 2. P. 1.Google Scholar
  33. [33]
    Kogan Sh.M., Shul'man A.Ya. To the Theory of Fluctuations in a Nonequilibrium Gas. // ZhETF. 1969. V. 56. P. 862.Google Scholar
  34. [34]
    Lifshits E.M., Pitaevsky L.P. Statistical Physics. Moscow: Nauka, 1978.Google Scholar
  35. [35]
    Keizer J. Statistical Thermodynamics of Nonequilibrium Processes. Berlin-Heidelberg-New York: Springer, 1987.Google Scholar
  36. [36]
    Klimontovich Yu.L. Natural Flicker Noise. // Pis'ma v ZhTF. 1983. V. 9. P. 406.Google Scholar
  37. [37]
    Klimontovich Yu.L., Boon J.P. Natural Flicker Noise (1/f-noise) in Music. // Europhys. Lett. 1987. V. 3(4). P. 395.Google Scholar
  38. [38]
    Klimontovich Yu.L. Natural Flicker Noise (1/f-noise) and Superconductivity. // Pis'ma v ZhETF. 1990. V. 51(1). P. 43.Google Scholar
  39. [39]
    Kogan Sh.M. The Low Frequency Current Noise with Spectrum 1/f in Solid State. // Usp. Fiz. Nauk. 1985. V. 145. P. 285; Sov.Phys. Usp. 1985. V. 28. P. 171.Google Scholar
  40. [40]
    Voos R.F., Clarke J. “1/f Noise” in Music: Music from 1/f Noise. // J.Acoust. Soc.Am. 1978. V. 643(1). P. 258.Google Scholar
  41. [41]
    Klimontovich Yu.L. Entropy Decrease in the Processes of Self-Organization. S-Theorem. // Pis'ma v ZhTF. 1983. V. 9. P. 1089.Google Scholar
  42. [42]
    Klimontovich Yu.L. S-Theorem. // Z.Phys.B. 1987. V. 66. P. 125.Google Scholar
  43. [43]
    Klimontovich Yu.L. Problems in the Statistical Theory of Open Systems: Criteria for Relative Degree of Order of States in Self-organization Processes. // Usp. Fiz. Nauk. 1989. V. 158. P. 59; Sov. Phys. Usp. 1989. V. 32(5).Google Scholar
  44. [44]
    Ebeling W., Klimontovich Yu.L. Self-organization and Turbulance in Liquids. Leipzig: Teubner, 1984.Google Scholar
  45. [45]
    Haken H. Synergetic Computers and Cognition. A Top-Down Approach to Neural Nets. Berlin-Heidelberg: Springer-Verlag, 1991.Google Scholar
  46. [46]
    Fuchs A., Haken H. //Neural and Synergetic computers, / H.Haken. Berlin: Springer-Verlag, 1988. P. 16.Google Scholar
  47. [47]
    Akhromeyeva T.S., Kurdumov S.P., Malinetskii G.G., Samarski A.A. Chaos and Dissipative Structures in “Reaction-Diffusion” Systems. Moscow: Nauka, 1992.Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • Yu. L. Klimontovich

There are no affiliations available

Personalised recommendations