RSS feeds
РусскийEnglish
Issue 12, 2025
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

1994

 / 

August

←  →

Reviews of topical problems


Nonlinear Brownian motion


Lomonosov Moscow State University, Faculty of Physics, Leninskie Gory 1 build. 2, Moscow, 119991, 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.

Fulltext pdf (810 KB)
Fulltext is also available at DOI: 10.1070/PU1994v037n08ABEH000038
PACS: 05.40.+j
DOI: 10.1070/PU1994v037n08ABEH000038
URL: https://ufn.ru/en/articles/1994/8/b/
A1994PJ71400002
Citation: Klimontovich Yu L "Nonlinear Brownian motion" Phys. Usp. 37 737–766 (1994)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Климонтович Ю Л «Нелинейное броуновское движение» УФН 164 811–844 (1994); DOI: 10.3367/UFNr.0164.199408b.0811

References (57) Cited by (155) ↓ Similar articles (20)

  1. Solbakov V V, Zatsepa S N, Ivchenko A A Modeling, Analysis, Control and Removal of Oil and Hydrocarbon Spills Earth And Environmental Sciences Library Chapter 3 (2025) p. 61
  2. Oei N, Krenz N et al Phys. Rev. D 111 (7) (2025)
  3. Roberts C, Zhang Z et al Phys. Rev. Research 7 (3) (2025)
  4. Cai Y, Wang T, Zhao L Phys. Rev. E 111 (2) (2025)
  5. Yoshimori A, Das Sh P Phys. Rev. E 112 (4) (2025)
  6. Liang Y, Wei Q et al Physics of Fluids 37 (3) (2025)
  7. Quintanilha V S C, da Costa L B R et al Entropy 27 (2) 155 (2025)
  8. Milster S, Dzubiella J et al The Journal of Chemical Physics 162 (15) (2025)
  9. Pacheco-Pozo A, Sokolov I M et al New J. Phys. 27 (6) 064602 (2025)
  10. Patrón A, Sánchez-Rey B, Prados A Phys. Rev. E 109 (4) (2024)
  11. Al-Mendalawi M D Tdd 60 (2024)
  12. Shumsky S Studies In Computational Intelligence Vol. Biologically Inspired Cognitive Architectures 2023Hierarchical AGI from First Principles1130 Chapter 89 (2024) p. 823
  13. Pacheco-Pozo A, Balcerek M et al Phys. Rev. Lett. 133 (6) (2024)
  14. Lev B, Zagorodny A Ukr. J. Phys. 69 (8) 519 (2024)
  15. Akter Sh, Hossain A et al Journal Of Taibah University For Science 18 (1) (2024)
  16. García-Beltrán G, Mercado-Zúñiga C et al Optik 301 171693 (2024)
  17. Cai Y, Wang T, Zhao L Phys. Rev. Research 6 (2) (2024)
  18. Zhmakin A I Non-Fourier Heat Conduction Chapter 12 (2023) p. 289
  19. Silvano N O, Barci D G Physica A: Statistical Mechanics And Its Applications 630 129246 (2023)
  20. Strekalov M L Chemical Physics 573 112012 (2023)
  21. Cai Y, Wang T, Zhao L J Stat Phys 190 (12) (2023)
  22. Cai Y, Wang T, Zhao L J Stat Phys 190 (11) (2023)
  23. Solbakov V, Zatsepa S et al 2022 International Conference on Ocean Studies (ICOS), (2022) p. 110
  24. Goychuk I New J. Phys. 24 (4) 043018 (2022)
  25. de Pirey T A, Cugliandolo L F et al Advances In Physics 71 (1-2) 1 (2022)
  26. Megías A, Santos A, Prados A Phys. Rev. E 105 (5) (2022)
  27. Goychuk I Proc. Natl. Acad. Sci. U.S.A. 119 (48) (2022)
  28. Strekalov M L Computational And Theoretical Chemistry 1215 113832 (2022)
  29. Sandev T, Domazetoski V et al J. Phys. A: Math. Theor. 55 (7) 074003 (2022)
  30. Kulikov V, Kulikov A, Ignatyev A Studies In Systems, Decision And Control Vol. Society 5.0: Cyberspace for Advanced Human-Centered SocietyThe Stochastic and Singular Analysis of Fractal Signals in Cyber-Physical Systems of Biomedicine333 Chapter 19 (2021) p. 239
  31. Fujitani Y J. Fluid Mech. 907 (2021)
  32. Patrón A, Sánchez-Rey B, Prados A Phys. Rev. E 104 (6) (2021)
  33. Jumaev M, Sharipov M, Rizoqulov M J. Phys.: Conf. Ser. 1889 (2) 022076 (2021)
  34. Salimi M, Pasquier P Lecture Notes In Computer Science Vol. Advances in Swarm IntelligenceLiminal Tones: Swarm Aesthetics and Materiality in Sound Art12689 Chapter 5 (2021) p. 46
  35. Goychuk I, Pöschel T Phys. Rev. Lett. 127 (11) (2021)
  36. Rudyak V Ya J. Phys.: Conf. Ser. 1677 (1) 012152 (2020)
  37. Mancois V, Barré Ju et al Phys. Rev. A 102 (1) (2020)
  38. Serov A S, Laurent F et al Sci Rep 10 (1) (2020)
  39. Moreno M V, Barci D G, Arenas Z G Phys. Rev. E 101 (6) (2020)
  40. Müller B, Berner J et al New J. Phys. 22 (2) 023014 (2020)
  41. Santos A, Prados A Physics of Fluids 32 (7) (2020)
  42. Wang W, Cherstvy A G et al Phys. Rev. E 102 (1) (2020)
  43. Hirose Yu, Yasugahira Yu et al J. Phys. Soc. Jpn. 89 (7) 074004 (2020)
  44. Martin R J, Craster R V et al J. Phys. A: Math. Theor. 52 (8) 085002 (2019)
  45. Afsar O Physica D: Nonlinear Phenomena 390 62 (2019)
  46. Bo S, Lim S H, Eichhorn R J. Stat. Mech. 2019 (8) 084005 (2019)
  47. Khomenko A V Ukr. J. Phys. 64 (6) 487 (2019)
  48. Chétrite R, Gupta Sh et al EPL 124 (6) 60006 (2019)
  49. Khomenko A, Troshchenko D, Metlov L Phys. Rev. E 100 (2) (2019)
  50. Moreno M V, Barci D G, Arenas Z G Phys. Rev. E 99 (3) (2019)
  51. Ugrozov V V Colloid J 81 (2) 187 (2019)
  52. Kalinina E G, Pikalova E Yu Russ. Chem. Rev. 88 (12) 1179 (2019)
  53. Vulfson A N, Borodin O O Physics of Fluids 30 (9) (2018)
  54. Wirth A Journal of Physical Oceanography 48 (4) 831 (2018)
  55. Arenas Z G, Barci D G, Moreno M V Physica A: Statistical Mechanics And Its Applications 510 98 (2018)
  56. Berner J, Müller B et al Nat Commun 9 (1) (2018)
  57. Arkhincheev V E, Budazapov L V Nonlinear Systems - Modeling, Estimation, and Stability Chapter 7 (2018)
  58. Fier G, Hansmann D, Buceta R C Soft Matter 13 (18) 3385 (2017)
  59. Paul Sh, Laskar A et al Phys. Rev. E 96 (5) (2017)
  60. Singh R, Adhikari R European Journal Of Computational Mechanics 26 (1-2) 78 (2017)
  61. Apushkinskaya D, Apushkinskiy E J. Phys.: Conf. Ser. 936 012043 (2017)
  62. Seroussi I, Grebenkov D S et al Journal Of Magnetic Resonance 277 95 (2017)
  63. Hohmann M, Kindermann F et al Phys. Rev. Lett. 118 (26) (2017)
  64. Lasanta A, Vega R F et al Phys. Rev. Lett. 119 (14) (2017)
  65. Goychuk I Eur. Phys. J. B 90 (11) (2017)
  66. Bo S, Eichhorn R Phys. Rev. Lett. 119 (6) (2017)
  67. Mentrelli A, Pagnini G Ricerche Mat 65 (2) 523 (2016)
  68. Aron C, Barci D G et al J. Stat. Mech. 2016 (5) 053207 (2016)
  69. Kazakevičius R, Ruseckas Ju Phys. Rev. E 94 (3) (2016)
  70. Vulfson A N, Borodin O O Uspekhi Fizicheskikh Nauk 186 (2) 113 (2016)
  71. Bhattacharjee A K, Balakrishnan K et al The Journal of Chemical Physics 142 (22) (2015)
  72. Search and Foraging 0 (2015) p. 171
  73. Martin R J, Craster R V, Kearney M J Proc. R. Soc. A. 471 (2179) 20150084 (2015)
  74. Khruschev S S, Abaturova A M et al BIOPHYSICS 60 (2) 212 (2015)
  75. Khruschev S S, Abaturova A M et al BIOPHYSICS 60 (4) 513 (2015)
  76. Mentrelli A, Pagnini G Journal Of Computational Physics 293 427 (2015)
  77. Godec A, Metzler R Phys. Rev. E 91 (5) (2015)
  78. Moreno M V, Arenas Z G, Barci D G Phys. Rev. E 91 (4) (2015)
  79. Khandekar Ch, Pick A et al Phys. Rev. B 91 (11) (2015)
  80. Lubashevsky I Stochastic Systems Theory And Its Applications (SSS) 2014 (0) 278 (2014)
  81. Pagnini G, Mentrelli A Nat. Hazards Earth Syst. Sci. 14 (8) 2249 (2014)
  82. Aron C, Barci D G et al J. Stat. Mech. 2014 (9) P09008 (2014)
  83. Pagnini G SEMA SIMAI Springer Series Vol. Advances in Differential Equations and ApplicationsFire Spotting Effects in Wildland Fire Propagation4 Chapter 20 (2014) p. 203
  84. Cherstvy A G, Chechkin A V, Metzler R Soft Matter 10 (10) 1591 (2014)
  85. Lisý V, Tóthová J, Glod L Int J Thermophys 35 (11) 2001 (2014)
  86. Chechkin A, Pavlyukevich I J. Phys. A: Math. Theor. 47 (34) 342001 (2014)
  87. Sarracino A Phys. Rev. E 88 (5) (2013)
  88. Polettini M J. Stat. Mech. 2013 (07) P07005 (2013)
  89. Arenas Z G, Barci D G Phys. Rev. E 85 (4) (2012)
  90. Romanczuk P, Bär M et al Eur. Phys. J. Spec. Top. 202 (1) 1 (2012)
  91. Arenas Z G, Barci D G J. Stat. Mech. 2012 (12) P12005 (2012)
  92. Dorofeyev I A, Vinogradov E A Physics Reports 504 (2-4) 75 (2011)
  93. Koide T, Kodama T Phys. Rev. E 83 (6) (2011)
  94. Wang Sh-H, Yang M, Wu D-J Chinese Phys. Lett. 28 (2) 020501 (2011)
  95. Almenar L, Rauscher M J. Phys.: Condens. Matter 23 (18) 184115 (2011)
  96. Rauscher M J. Phys.: Condens. Matter 22 (36) 364109 (2010)
  97. KHOMENKO ALEXEI V, LYASHENKO IAKOV A, BORISYUK VADIM N Fluct. Noise Lett. 09 (01) 19 (2010)
  98. Arenas Z G, Barci D G Phys. Rev. E 81 (5) (2010)
  99. Sokolov I M Chemical Physics 375 (2-3) 359 (2010)
  100. Dubkov A A Chemical Physics 375 (2-3) 364 (2010)
  101. Lev B I, Kiselev A D Phys. Rev. E 82 (3) (2010)
  102. Xing J J. Phys. A: Math. Theor. 43 (37) 375003 (2010)
  103. Ivanitskii G R Uspekhi Fizicheskikh Nauk 180 (4) 337 (2010)
  104. Chetrite R Phys. Rev. E 80 (5) (2009)
  105. Lubashevsky I, Friedrich R, Heuer A Phys. Rev. E 80 (3) (2009)
  106. Dunkel J, Hänggi P Physics Reports 471 (1) 1 (2009)
  107. Vinogradov E A, Dorofeyev I A Uspekhi Fizicheskikh Nauk 179 (5) 449 (2009)
  108. Dubkov A A, Hänggi P, Goychuk I J. Stat. Mech. 2009 (01) P01034 (2009)
  109. Arkhipov B, Koterov V et al Stud Appl Math 120 (1) 87 (2008)
  110. Ao P Commun. Theor. Phys. 49 (5) 1073 (2008)
  111. Plyukhin A V, Froese A M Phys. Rev. E 76 (3) (2007)
  112. Dunkel J, Hänggi P Physica A: Statistical Mechanics And Its Applications 374 (2) 559 (2007)
  113. Arkhipov B V, Koterov V N et al Comput. Math. And Math. Phys. 47 (2) 280 (2007)
  114. Lindner B New J. Phys. 9 (5) 136 (2007)
  115. Fröbrich P Nuclear Physics A 787 (1-4) 170 (2007)
  116. Muzychuk O V Radiophys Quantum Electron 49 (8) 645 (2006)
  117. Dunkel J, Hänggi P Phys. Rev. E 74 (6) (2006)
  118. CHETVERIKOV A P, EBELING W, VELARDE M G Int. J. Bifurcation Chaos 16 (06) 1613 (2006)
  119. Antonov V A, Baranov A S, Gnedin Yu N Sol Syst Res 40 (3) 251 (2006)
  120. Dunkel J, Hänggi P Phys. Rev. E 74 (5) (2006)
  121. ERDMANN UDO, EBELING WERNER Int. J. Bifurcation Chaos 15 (11) 3623 (2005)
  122. Chetverikov A P, Ebeling W, Velarde M G Eur. Phys. J. B 44 (4) 509 (2005)
  123. Dunkel J, Hänggi P Phys. Rev. E 72 (3) (2005)
  124. Dunkel J, Hänggi P Phys. Rev. E 71 (1) (2005)
  125. Ebeling W, Dunkel J et al J. Phys.: Conf. Ser. 11 89 (2005)
  126. Kun W, Yu-Gang M et al Chinese Phys. Lett. 22 (1) 53 (2005)
  127. Ma Y G, Wang K et al Phys. Rev. C 72 (6) (2005)
  128. Grafov B M Russian Journal Of Electrochemistry 40 (6) 662 (2004)
  129. ERDMANN UDO, EBELING WERNER Fluct. Noise Lett. 03 (02) L145 (2003)
  130. Muzychuk O V Radiophysics And Quantum Electronics 46 (2) 150 (2003)
  131. Bonitz M, Ebeling W, Romanovsky Yu M Contrib. Plasma Phys. 43 (5-6) 247 (2003)
  132. Grafov B M Russian Journal Of Electrochemistry 39 (9) 999 (2003)
  133. Erdmann U, Ebeling W, Anishchenko V S Phys. Rev. E 65 (6) (2002)
  134. Muzychuk O V Radiophysics And Quantum Electronics 45 (9) 733 (2002)
  135. El-Wakil S A, Elhanbaly A, Elgarayhi A Chaos, Solitons & Fractals 12 (8) 1385 (2001)
  136. Ebeling W, Schweitzer F Theory Biosci. 120 (3-4) 207 (2001)
  137. El-Zant A A Astrophysics And Space Science 276 (2-4) 1023 (2001)
  138. Ebeling W, Schweitzer F Integrative Systems Approaches to Natural and Social Dynamics Chapter 9 (2001) p. 119
  139. KOLEVA MARIA K, COVACHEV VALÉRY C Fluct. Noise Lett. 01 (02) R131 (2001)
  140. Dubkov A A, Malakhov A N, Saichev A I Radiophys Quantum Electron 43 (4) 335 (2000)
  141. Muzychuk O V Radiophys Quantum Electron 43 (5) 422 (2000)
  142. Koleva M K, Eiyias A E, Petrov L A Metal-Ligand Interactions in Chemistry, Physics and Biology Chapter 15 (2000) p. 353
  143. Garbaczewski P Physica A: Statistical Mechanics And Its Applications 285 (1-2) 187 (2000)
  144. Gettinger A E, Gontchar I I J. Phys. G: Nucl. Part. Phys. 26 (4) 347 (2000)
  145. Klimontovich Yu L Uspekhi Fizicheskikh Nauk 169 (1) 39 (1999)
  146. Prokop’ev E P Tech. Phys. Lett. 25 (8) 650 (1999)
  147. Vanag V K Uspekhi Fizicheskikh Nauk 169 (5) 481 (1999)
  148. Klimontovich Yu L Contrib. Plasma Phys. 38 (3) 387 (1998)
  149. Allahverdyan A E, Saakian D B Phys. Rev. E 58 (1) 1148 (1998)
  150. Niemiec M, Gadomski A et al Physica A: Statistical Mechanics And Its Applications 248 (3-4) 365 (1998)
  151. Olemskoi A I Uspekhi Fizicheskikh Nauk 168 (3) 287 (1998) [Olemskoi A I Phys.-Usp. 41 (3) 269 (1998)]
  152. Bellotti U, Bornatici M, Engelmann F Riv. Nuovo Cim. 20 (5) 1 (1997)
  153. Gontchar I, Litnevsky L A, Fröbrich P Computer Physics Communications 107 (1-3) 223 (1997)
  154. Klimontovich Yu L Phys.-Usp. 40 (1) 21 (1997)
  155. Vanag V K J. Phys. Chem. 100 (27) 11336 (1996)
© 1918–2025 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions