RSS feeds
РусскийEnglish
Issue 11, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

1998

 / 

March

←  →

Reviews of topical problems


Theory of stochastic systems with singular multiplicative noise


Sumy State University, ul. Rimskogo-Korsakova 2, Sumy, 244007, Ukraine

Noisy, interacting, stochastic systems are analyzed for the case in which their noise intensity varies with the hydrodynamic mode amplitude x according to the power law x2a, x \in [0, 1]. It is shown that the phase space domain of definition of the stochastic variable x forms a self-affine set of fractal dimensionality D = 2(1-a). Using the gauge procedure, a system of calculus is chosen which is not reducible either to the Ito case or the Stratonovich case. By generalizing the microscopic picture of phase transitions it is demonstrated that the system may reduce its symmetry (for 1 < D \leqslant 2) or lose ergodicity (for 0 < D \leqslant 1). Over the entire interval D \in [0, 2], a noise-induced transition is shown to be possible.

Fulltext pdf (523 KB)
Fulltext is also available at DOI: 10.1070/PU1998v041n03ABEH000377
PACS: 05.40.+j, 05.70.Fh, 64.60.−i, 82.20.Fd (all)
DOI: 10.1070/PU1998v041n03ABEH000377
URL: https://ufn.ru/en/articles/1998/3/c/
000073306100003
Citation: Olemskoi A I "Theory of stochastic systems with singular multiplicative noise" Phys. Usp. 41 269–301 (1998)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Олемской А И «Теория стохастических систем с сингулярным мультипликативным шумом» УФН 168 287–321 (1998); DOI: 10.3367/UFNr.0168.199803c.0287

References (60) ↓ Cited by (42) Similar articles (20)

  1. Gibbs J W Termodinamika. Statisticheskaya Mekhanika (Thermodynamics. Statistical Mechanics, Moscow: Nauka, 1982)
  2. Boltzmann L Vorlesungen über Gastheorie (Leipzig: J.A. Barth, 1896-1898) [Translated into Russian (Moscow: Gostekhteorizdat, 1956)]
  3. PoincarЉ H Izbrannye Trudy Vol. III (Selected Works, Moscow: Nauka, 1984)
  4. Feder J Fractals (New York: Plenum Press, 1988) [Translated into Russian (Moscow: Mir, 1991)]
  5. Haken H Synergetics (Springer Series in Synergetics, Vol. 1, Heidelberg: Springer, 1978) [Translated into Russian (Moscow: Mir, 1980)]
  6. Loskutov A Yu, Mikhaîlov A S Vvedenie v Sinergetiku (Introduction to Synergetics, Moscow: Mir, 1990)
  7. Schuster H G Deterministic Chaos: An Introduction (Weinheim, Federal Republic of Germany: Physik-Verlag; Deerfield Beach, FL: VCH Publishers, 1984) [Translated into Russian (Moscow: Mir, 1988)]
  8. Prigogine I Termodinamicheskaya Teoriya Struktury, Ustoîchivosti i Fluktuatsiî (Thermodynamic Theory of Structure, Stability and Fluctuations, Moscow: Mir, 1973)
  9. Nicolis G, Prigogine I Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order Through Fluctuations (New York: Wiley, 1977) [Translated into Russian (Moscow: Mir, 1979)]
  10. Haken H Synergetics. An Introduction. Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology (Springer Series in Synergetics, Vol. 1, Berlin, New York: Springer, 1983) [Translated into Russian (Moscow: Mir, 1985)]
  11. Haken H Information and Self-Organization (Springer Series in Synergetics, Vol. 40, Berlin, New York: Springer-Verlag, 1988) [Translated into Russian (Moscow: Mir, 1991)]
  12. Tyson J J The Belousov-Zhabotinskiî Reaction (Lecture Notes in Biomath.) 10, Berlin, New York: Springer-Verlag, 1976)
  13. Nicolis J Dynamics of Hierarchical Systems: An Evolutionary Approach (Springer Series in Synergetics, Vol. 25, Berlin, New York: Springer-Verlag, 1986) [Translated into Russian (Moscow: Mir, 1989)]
  14. Horsthemke W, Lefever R Noise-Induced Transitions (Springer Series in Synergetics, Vol. 15, Berlin, New York: Springer-Verlag, 1984) [Translated into Russian (Moscow: Mir, 1987)]
  15. Risken H The Fokker-Planck Equation (Springer Series in Synergetics, Vol. 18, Berlin, New York: Springer-Verlag, 1989)
  16. Gardiner C V Stokhasticheskie Metody v Estestvennykh Naukakh (Stochastic Methods in Natural Sciences, Moscow: Mir, 1986)
  17. Van Kampen N G Stochastic Processes in Physics and Chemistry (Amsterdam: North-Holland, 1981) [Translated into Russian (Moscow: Vysshaya Shkola, 1990)]
  18. Stratonovich R L Nelineînaya Neravnovesnaya Termodinamika (Nonlinear Nonequilibrium Thermodynamics, Moscow: Nauka, 1985) [Translated into English (Berlin, New York: Springer-Verlag, 1992)]
  19. Kuznetsov P I, Stratonovich R L, Tikhonov V I Zh. Eksp. Teor. Fiz. 28 509 (1955)
  20. Stratonovich R L, Landa P S Izv. Vyssh. Uchebn. Zaved. Radiofiz. 2 (1) 37 (1959)
  21. Stratonovich R L Izbrannye Voprosy Teorii Fluktuatsiî v Radiotekhnike (Selected Problems of the Theory of Fluctuations in Radio Engineering, Moscow: Sovetskoe Radio, 1961)
  22. Kubo R A, in Fluctuation, Relaxation and Resonance in Magnetic Systems (Ed. D Ter Haar, Edinburgh: Oliver and Boyd, 1962)
  23. Mayer-Kress G, Haken H J. Stat. Phys. 21 149 (1981)
  24. Crutchfield J P, Farmer J D, Huberman B A Phys. Rep. 92 45 (1982)
  25. Ito K Nagoya Math. J. 3 55 (1951)
  26. Stratonovich R L SIAM J. Control 4 362 (1966)
  27. Klimontovich Yu L Usp. Fiz. Nauk 164 811 (1994) [Phys. Usp. 37 (8) 737 (1994)]
  28. Landau L D, Lifshitz E M Statisticheskaya Fizika Chast’ 1 (Statistical Physics, Vol. 1, Moscow: Nauka, 1976) [Translated into English (Oxford: Pergamon Press, 1980)]
  29. Forster D Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Frontiers in Physics, 47, Reading, Mass.: W.A. Benjamin, Advanced Book Program, 1975) [Translated into Russian (Moscow: Atomizdat, 1980)]
  30. Landau L D, Lifshitz E M Mekhanika (Mechanics, Moscow: Nauka, 1988) [Translated into English (Oxford, New York: Pergamon Press, 1976)]
  31. Olemskoî A I, Koplyk I V Usp. Fiz. Nauk 165 1105 (1995) [Phys. Usp. 38 (10) 1061 (1995)]
  32. Katsnel’son A A, Olemskoî A I Mikroskopicheskaya Teoriya Neodnorodnykh Struktur (Microscopic Theory of Inhomogeneous Structures, Moscow: Izdatel’stvo MGU, 1987)
  33. Olemskoi A I, in Physics Reviews (Ed. I M Khalatnikov) 18 Part 1 1 (1996)
  34. Bogolyubov N N, Shirkov D V Vvedenie v Teoriyu Kvantovannykh Poleî (Introduction to the Theory of Quantized Fields, Moscow: Nauka, 1976) [Translated into English (New York: John Wiley, 1980)]
  35. Itzykson C, Zuber J-B Quantum Field Theory (New York: McGraw-Hill, 1980) [Translated into Russian (Moscow: Mir, 1984)]
  36. Parisi G, Sourlas N Phys. Rev. Lett. 43 744 (1979)
  37. Zinn-Justin J Quantum Field Theory and Critical Phenomena (Oxford: Clarendon Press; New York: Oxford Univ. Press, 1989)
  38. Lifshitz E M, Pitaevskiî L P Fizicheskaya Kinetika (Physical Kinetics, Moscow: Nauka, 1979) [Translated into English (Oxford: Pergamon Press, 1981)]
  39. Gnedenko B V Kurs Teorii Veroyatnosteî (A Course in Theory of Probabilities, Moscow: Nauka, 1967)
  40. Landau L D, Lifshitz E M Kvantovaya Mekhanika. Nerelyativistskaya Teoriya (Quantum Mechanics: Non-Relativistic Theory, Moscow: Nauka, 1984) [Translated into English (Oxford, New York: Pergamon Press, 1977)]
  41. Pawula R F IEEE Trans. Inform. Theory IT-13 33 (1967)
  42. Cooper F, Khare A, Sukhatme U Phys. Rep. 251 267 (1995)
  43. Lifshits I M, Slezov V V Zh. Eksp. Teor. Fiz. 35 479 (1958)
  44. Bray A J Adv. Phys. 43 357 (1994)
  45. Landau L D, Lifshitz E M Teoriya Polya (Field Theory, Moscow: Nauka, 1973) [Translated into English (Oxford: Pergamon Press, 1975)]
  46. Olemskoî A I, Kharchenko D O Izv. Vyssh. Uchebn. Zaved. Ser. Fiz. (2) 121 (1996)
  47. Feîgel’man M V, Tsvelik A M Zh. Eksp. Teor. Fiz. 83 1430 (1982) [Sov. Phys, JETP 56 823 (1982)]
  48. Akhiezer A I, Peletminskiî S V Polya i Fundamental’noye Vzaimodeîstvie (Fields and Fundamental Interaction, Kiev: Naukova Dumka, 1986)
  49. Berestetskiî V B, Lifshitz E M, Pitaevskiî L P Kvantovaya Élektrodinamika (Quantum Electrodynamics, Moscow, Nauka, 1989) [Translated into English (Oxford: Pergamon Press, 1982)]
  50. Palmer R G Adv. Phys. 31 669 (1982)
  51. Van den Broeck C, Parrondo J M R, Toral R Phys. Rev. Lett. 73 3395 (1994)
  52. Toropov E A, Kharchenko D O Izv. Vyssh. Uchebn. Zaved. Ser. Fiz. (4) 75 (1996)
  53. Kosevich A M Fizicheskaya Mekhanika Real’nykh Kristallov (Physical Mechanics of Real Crystals, Kiev: Naukova Dumka, 1981)
  54. Olemskoî A I, Kharchenko D O Metallofiz. (8) 3 (1996)
  55. Olemskoî A I, Flat A Ya Fiz. Tverd. Tela 35 542 (1993)
  56. Olemskoî A I, Khomenko A V Zh. Eksp. Teor. Fiz. 110 2144 (1996) [JETP 83 1180 (1996)]
  57. Ruelle D, Takens F, in Strannye Attraktory (Strange Attractors, Moscow: Mir, 1981)
  58. Lorenz E J Atmos. Sci. 20 1675 (1963)
  59. Olemskoî A I, Petrunin V A Izv. Vyssh. Uchebn. Zaved. Ser. Fiz. 30 (1) 82 (1987) [Sov. Phys. J. 30 (1) 61 (1987)]
  60. Olemskoî A I, Sklyar I A Usp. Fiz. Nauk 162 (6) 29 (1992) [Sov. Phys. Usp. 35 455 (1992)]
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions