Reviews of topical problems

Nonlinear Brownian motion

Lomonosov Moscow State University, Department of Physics, Vorobevy gory, Moscow, 119992, Russian Federation

The theory of Brownian motion as described by nonlinear Langevin equations and the corresponding Fokker-Planck equations is discussed. The general problems of the theory of nonlinear Brownian motion considered are: Brownian motion in a medium with nonlinear friction; the critical analysis of three forms of the relevant Langevin and Fokker-Planck equations (Ito’s form, Stratonovich’s form, and the kinetic form); the Smoluchowski equations and master equations for different cases; two methods of transition from master equation to Fokker-Planck equation; master equations for one-step processes; traditional and nontraditional definition of transition probabilities; evolution of free energy and entropy in Brownian motion; Lyapunov functionals. The following particular examples are considered: Brownian motion in self-oscillatory systems; H-theorem for the van der Pol oscillator; S-theorem; oscillator with inertial nonlinearity; bifurcation of energy of the limiting cycle; oscillator with multistable stationary states; oscillators in discrete time; bifurcations of energy of the limiting cycle and the period of oscillations; criterion of instability upon transition to discrete time, based on the H-theorem; Brownian motion of quantum atoms oscillators in the equilibrium electromagnetic field; Brownian motion in chemically reacting systems; partially ionised plasmas; the Malthus-Verhulst process.

Text can be downloaded in Russian. English translation is available on IOP Science.
PACS: 05.40.+j
DOI: 10.1070/PU1994v037n08ABEH000038
Citation: Klimontovich Yu L "Nonlinear Brownian motion" Phys. Usp. 37 737–766 (1994)
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Оригинал: Климонтович Ю Л «Нелинейное броуновское движение» УФН 164 811–844 (1994); DOI: 10.3367/UFNr.0164.199408b.0811

References (57) ↓ Cited by (102) Similar articles (20)

  1. Davydov B I (Red.) Brounovskoe Dvizhenie (M.: ONTI, 1936)
  2. Leontovich M A Statisticheskaya Fizika (M.: Nauka, 1983)
  3. Chandrasekhar S Rev. Mod. Phys. 15 823 (1943)
  4. Lifshitz E M, Pitaevskii L P Physical Kinetics (Oxford: Pergamon Press, 1981)
  5. Stratonovich R L Voprosy Teorii Fluktuatsii V Radiotekhnike (Topics In The Theory Of Fluctuations In Radio Engineering) (Moscow: Sovet-skoe Radio, 1961)
  6. Rytov S M Vvedenie V Statisticheskuyu Radiofiziku (Introduction To Statistical Radiophysics) (Moscow: Nauka, 1976)
  7. Malakhov A N Fluktuatsii V Avtokolebatel’nykh Sistemakh (Fluctuations In Self-Oscillatory Systems) (Moscow: Nauka, 1968)
  8. Tikhonov V I Statisticheskaya Radiotekhnika (Statistical Radio Engineering) (Moscow: Sovet-skoe Radio, 1966)
  9. Klimontovich Yu L Statistical Physics (New York: Harwood Academic, 1986)
  10. Klyatskin V I, Tatarskii V I Usp. Fiz. Nauk 110 499 (1973); Klyatskin V I, Tatarskii V I Sov. Phys. Usp. 16 494 (1974)
  11. Van Kampen N G Stochastic Processes in Physics and Chemistry (Amsterdam: North Holland, 1981)
  12. Risken H The Fokker-Planck Equation (Berlin: Springer, 1984)
  13. Haken H Advanced Synergetics (Berlin: Springer, 1983)
  14. Gardiner C V Handbook Of Stochastic Methods (Berlin: Springer, 1983)
  15. Horstemke W, Lefever R Noise Induced Transitions (Berlin: Springer, 1984)
  16. Zel’dovich Ya B, Mikhailov A S Usp. Fiz. Nauk 153 469 (1987); Zel’dovich Ya B, Mikhailov A S Sov. Phys. Usp. 30 977 (1987)
  17. Klimontovich Yu L Turbulent Motion And The Structure Of Chaos (Dordrecht: Kluwer, 1991)
  18. Klimontovich Yu L Physica A (Ut Recht) 163 515 (1990)
  19. Klimontovich Yu L Physica A (Ut Recht) 182 121 (1992)
  20. Klimontovich Yu L, Kovalev A S, Landa P S Usp. Fiz. Nauk 106 279 (1972); Klimontovich Yu L, Kovalev A S, Landa P S Sov. Phys. Usp. 15 95 (1972)
  21. Lax M Fluctuation And Coherence Phenomena In Classical And Quantum Physics (New York: Gordon and Breach, 1968)
  22. Akhmanov S A, D’yakov Yu E, Chirkin A S Vvedenie V Statisticheskuyu Radiofiziku I Optiku (Introduction To Statistical Radiophysics And Optics) (Moscow: Nauka, 1981)
  23. Landa P S Avtokolebaniya V Sistemakh S Konechnym Chislom Stepenei Svobody (Self-Oscillations In Systems With Finite Number Of Degrees Of Freedom) (Moscow: Nauka, 1980)
  24. Stratonovich R L Nonlinear Nonequilibrium Thermodynamics (Berlin: Springer, 1992)
  25. Mikhailov A S Phys. Rep. 184 309 (1989)
  26. Moss F Stochastic Resonance From The Ice Ages To The Monkey’s Ear (St. Louis, M O: University of Missouri, 1992)
  27. Klimontovich Yu L Teor. Mat. Fiz. 92 312 (1992)
  28. Klimontovich Yu L Teor. Mat. Fiz. 96 3 (1993)
  29. Pontryagin L S, Andronov A A, Vitt A A Zh. Eksp. Teor. Fiz. 3 165 (1933)
  30. Hanggi P, Talkner P, Borkovec M Rev. Mod. Phys. 62 251 (1990)
  31. Klimontovich Yu L Usp. Fiz. Nauk 101 577 (1970); Klimontovich Yu L Sov. Phys. Usp. 13 480 (1971)
  32. Haken H Light Vol. Vol. 1 (Amsterdam: North Holland, 1981)
  33. Klimontovich Yu L The Kinetic Theory of Electromagnetic Processes (Berlin: Springer, 1983)
  34. Karlov N V, Kirichenko N A, Luk’yanchuk B Lazernaya Termokhimiya (Laser Thermochemistry) (Moscow: Nauka, 1992)
  35. Einstein A Verh. Dtsch. Phys. Ges. 18 318 (1916)
  36. Klimontovich Yu L Usp. Fiz. Nauk 151 309 (1987); Klimontovich Yu L Sov. Phys. Usp. 30 154 (1987)
  37. Andreev A V, Emel’yanov V I, Il’inskii Yu A Cooperative Effects In Optics (Bristol: Institute of Physics Publishing, 1993)
  38. Klimontovich Yu L Pis’ma Zh. Tekh. Fiz. 14 631 (1988); Klimontovich Yu L Sov. Tech. Phys. Lett. 14 281 (1988)
  39. Klimontovich Yu L Pis’ma Zh. Tekh. Fiz. 9 1089 (1983); Klimontovich Yu L Sov. Tech. Phys. Lett. 9 467 (1983)
  40. Klimontovich Yu L "Chaos, Solitons and Fractals", (in press)
  41. Landau L D, Lifshitz E M Statistical Physics Vol. 1, 3rd edition (Oxford: Pergammon Press, 1980)
  42. Patashinskii A Z, Pokrovskii V L Fluktuatsionnaya Teoriya Fazovykh Perekhodov (Fluctuation Theory Of Phase Transitions) (Moscow: Nauka, 1982)
  43. Anishchenko V S Slozhnye Kolebaniya V Prostykh Sistemakh (Complex Oscillations In Simple Systems) (Moscow: Nauka, 1988)
  44. Anishchenko V S, Astakhov V V Radiotekh. Elektron. 28 1109 (1983)
  45. Lorenz E J. Atmos. Sci. 20 1675 (1963)
  46. Ruelle D, Takens F Commun. Math. Phys. 20 167 (1971)
  47. Oraevskii A N Tr. Fiz. Inst. Akad. Nauk SSSR 171 3 (1986)
  48. Kravtsov Yu A (Ed.) Limits Of Predictability (Berlin: Springer, 1993)
  49. Feigenbaum M J Los Alamos Sci. 1 ((1)) 4 (1980)
  50. Anishchenko V S, Klimontovich Yu L Pis’ma Zh. Tekh. Fiz. 10 876 (1984); Anishchenko V S, Klimontovich Yu L Sov. Phys. Tech. Phys. 10 368 (1984)
  51. Lichtenberg A, Liebermann M Regular And Stochastic Dynamics (Berlin: Springer, 1982)
  52. Neimark Yu I, Landa P S Stochastic And Chaotic Oscillations (Dordrecht: Kluwer, 1992)
  53. Klimontovich Yu L Pis’ma Zh. Tekh. Fiz. 13 175 (1987); Klimontovich Yu L Sov. Phys. Tech. Phys. 13 72 (1987)
  54. Klimontovich Yu L, Chetverikov V I Pis’ma Zh. Tekh. Fiz. 13 977 (1987); Klimontovich Yu L, Chetverikov V I Sov. Phys. Tech. Phys. 13 407 (1987)
  55. Klimontovich Yu L, Kremp D Physica A (Utrecht) 109 512 (1981)
  56. Klimontovich Yu L, Kremp D, Kraeft W D Adv. Chem. Phys. 58 175 (1987)
  57. Belyi V V, Klimontovich Yu L Zh. Eksp. Teor. Fiz. 74 1660 (1978); Belyi V V, Klimontovich Yu L Zh. Sov. Phys. JETP 47 866 (1978)

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