Issues

 / 

1994

 / 

August

  

Reviews of topical problems


Nonlinear Brownian motion


Lomonosov Moscow State University, Department 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 is available at IOP
PACS: 05.40.+j
DOI: 10.1070/PU1994v037n08ABEH000038
URL: https://ufn.ru/en/articles/1994/8/b/
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 (113) ↓ Similar articles (20)

  1. Santos A, Prados A Physics Of Fluids 32 072010 (2020)
  2. Serov A S, Laurent F et al Sci Rep 10 (1) (2020)
  3. Wang W, Cherstvy A G et al Phys. Rev. E 102 (1) (2020)
  4. Mancois V, Barré Ju et al Phys. Rev. A 102 (1) (2020)
  5. Hirose Yu, Yasugahira Yu et al J. Phys. Soc. Jpn. 89 074004 (2020)
  6. Müller B, Berner J et al New J. Phys. 22 023014 (2020)
  7. Moreno M V, Barci D G, Arenas Z G Phys. Rev. E 101 (6) (2020)
  8. Ugrozov V V Colloid J 81 187 (2019)
  9. Martin R J, Craster R V et al J. Phys. A: Math. Theor. 52 085002 (2019)
  10. Moutal N, Grebenkov D J Sci Comput 81 1630 (2019)
  11. Bo S, Lim S H, Eichhorn R J. Stat. Mech. 2019 084005 (2019)
  12. Afsar O Physica D: Nonlinear Phenomena 390 62 (2019)
  13. Khomenko A V Ukr. J. Phys. 64 487 (2019)
  14. Chétrite R, Gupta Sh et al EPL 124 60006 (2019)
  15. Moreno M V, Barci D G, Arenas Z G Phys. Rev. E 99 (3) (2019)
  16. Khomenko A, Troshchenko D, Metlov L Phys. Rev. E 100 (2) (2019)
  17. Arenas Z G, Barci D G, Moreno M V Physica A: Statistical Mechanics And Its Applications 510 98 (2018)
  18. Vulfson A N, Borodin O O Physics Of Fluids 30 095103 (2018)
  19. Wirth A 48 831 (2018)
  20. Berner J, Müller B et al Nat Commun 9 (1) (2018)
  21. Arkhincheev V E, Budazapov L V Nonlinear Systems - Modeling, Estimation, and Stability Chapter 7 (2018)
  22. Lasanta A, Vega R F et al Phys. Rev. Lett. 119 (14) (2017)
  23. Goychuk I Eur. Phys. J. B 90 (11) (2017)
  24. Paul Sh, Laskar A et al Phys. Rev. E 96 (5) (2017)
  25. Fier G, Hansmann D, Buceta R C Soft Matter 13 3385 (2017)
  26. Singh R, Adhikari R European Journal Of Computational Mechanics 26 78 (2017)
  27. Seroussi I, Grebenkov D S et al Journal Of Magnetic Resonance 277 95 (2017)
  28. Bo S, Eichhorn R Phys. Rev. Lett. 119 (6) (2017)
  29. Hohmann M, Kindermann F et al Phys. Rev. Lett. 118 (26) (2017)
  30. Apushkinskaya D, Apushkinskiy E J. Phys.: Conf. Ser. 936 012043 (2017)
  31. Vulfson A N, Borodin O O Uspekhi Fizicheskikh Nauk 186 113 (2016)
  32. Aron C, Barci D G et al J. Stat. Mech. 2016 053207 (2016)
  33. Kazakevičius R, Ruseckas Ju Phys. Rev. E 94 (3) (2016)
  34. Mentrelli A, Pagnini G Ricerche Mat 65 523 (2016)
  35. Mentrelli A, Pagnini G Journal Of Computational Physics 293 427 (2015)
  36. Khruschev S S, Abaturova A M et al BIOPHYSICS 60 212 (2015)
  37. Khandekar Ch, Pick A et al Phys. Rev. B 91 (11) (2015)
  38. Moreno M V, Arenas Z G, Barci D G Phys. Rev. E 91 (4) (2015)
  39. Khruschev S S, Abaturova A M et al BIOPHYSICS 60 513 (2015)
  40. Search and Foraging (2015) p. 225
  41. Bhattacharjee A K, Balakrishnan K et al J. Chem. Phys. 142 224107 (2015)
  42. Martin R J, Craster R V, Kearney M J Proc. R. Soc. A 471 20150084 (2015)
  43. Godec A, Metzler R Phys. Rev. E 91 (5) (2015)
  44. Pagnini G SEMA SIMAI Springer Series Vol. Advances in Differential Equations and Applications4 Chapter 20 (2014) p. 203
  45. Lubashevsky I Stochastic Systems Theory And Its Applications (SSS) 2014 278 (2014)
  46. Aron C, Barci D G et al J. Stat. Mech. 2014 P09008 (2014)
  47. Chechkin A, Pavlyukevich I J. Phys. A: Math. Theor. 47 342001 (2014)
  48. Pagnini G, Mentrelli A Nat. Hazards Earth Syst. Sci. 14 2249 (2014)
  49. Cherstvy A G, Chechkin A V, Metzler R Soft Matter 10 1591 (2014)
  50. Lisý V, Tóthová J, Glod L Int J Thermophys 35 2001 (2014)
  51. Pagnini G, Mentrelli A Nat. Hazards Earth Syst. Sci. Discuss. 1 6521 (2013)
  52. Polettini M J. Stat. Mech. 2013 P07005 (2013)
  53. Sarracino A Phys. Rev. E 88 (5) (2013)
  54. Arenas Z G, Barci D G J. Stat. Mech. 2012 P12005 (2012)
  55. Romanczuk P, Bär M et al Eur. Phys. J. Spec. Top. 202 1 (2012)
  56. Arenas Z G, Barci D G Phys. Rev. E 85 (4) (2012)
  57. Almenar L, Rauscher M J. Phys.: Condens. Matter 23 184115 (2011)
  58. Wang Sh-H, Yang M, Wu D-J Chinese Phys. Lett. 28 020501 (2011)
  59. Dorofeyev I A, Vinogradov E A Physics Reports 504 75 (2011)
  60. Koide T, Kodama T Phys. Rev. E 83 (6) (2011)
  61. Lev B I, Kiselev A D Phys. Rev. E 82 (3) (2010)
  62. Ivanitskii G R Uspekhi Fizicheskikh Nauk 180 337 (2010)
  63. Xing J J. Phys. A: Math. Theor. 43 375003 (2010)
  64. Arenas Z G, Barci D G Phys. Rev. E 81 (5) (2010)
  65. Dubkov A A Chemical Physics 375 364 (2010)
  66. Sokolov I M Chemical Physics 375 359 (2010)
  67. Rauscher M J. Phys.: Condens. Matter 22 364109 (2010)
  68. KHOMENKO ALEXEI V, LYASHENKO IAKOV A, BORISYUK VADIM N Fluct. Noise Lett. 09 19 (2010)
  69. Dubkov A A, Hänggi P, Goychuk I J. Stat. Mech. 2009 P01034 (2009)
  70. Chetrite R Phys. Rev. E 80 (5) (2009)
  71. Lubashevsky I, Friedrich R, Heuer A Phys. Rev. E 80 (3) (2009)
  72. Vinogradov E A, Dorofeyev I A Uspekhi Fizicheskikh Nauk 179 449 (2009)
  73. Dunkel J, Hänggi P Physics Reports 471 1 (2009)
  74. Ao P Commun. Theor. Phys. 49 1073 (2008)
  75. Plyukhin A V, Froese A M Phys. Rev. E 76 (3) (2007)
  76. Dunkel J, Hänggi P Physica A: Statistical Mechanics And Its Applications 374 559 (2007)
  77. Fröbrich P Nuclear Physics A 787 170 (2007)
  78. Arkhipov B V, Koterov V N et al Comput. Math. And Math. Phys. 47 280 (2007)
  79. Lindner B New J. Phys. 9 136 (2007)
  80. Arkhipov B, Koterov V et al 120 87 (2007)
  81. Antonov V A, Baranov A S, Gnedin Yu N Sol Syst Res 40 251 (2006)
  82. Dunkel J, Hänggi P Phys. Rev. E 74 (5) (2006)
  83. Dunkel J, Hänggi P Phys. Rev. E 74 (6) (2006)
  84. CHETVERIKOV A P, EBELING W, VELARDE M G Int. J. Bifurcation Chaos 16 1613 (2006)
  85. Muzychuk O V Radiophys Quantum Electron 49 645 (2006)
  86. Ebeling W, Dunkel J et al J. Phys.: Conf. Ser. 11 89 (2005)
  87. Dunkel J, Hänggi P Phys. Rev. E 71 (1) (2005)
  88. Chetverikov A P, Ebeling W, Velarde M G Eur. Phys. J. B 44 509 (2005)
  89. ERDMANN UDO, EBELING WERNER Int. J. Bifurcation Chaos 15 3623 (2005)
  90. Kun W, Yu-Gang M et al Chinese Phys. Lett. 22 53 (2005)
  91. Dunkel J, Hänggi P Phys. Rev. E 72 (3) (2005)
  92. Ma Y G, Wang K et al Phys. Rev. C 72 (6) (2005)
  93. ERDMANN UDO, EBELING WERNER Fluct. Noise Lett. 03 L145 (2003)
  94. Erdmann U, Ebeling W, Anishchenko V S Phys. Rev. E 65 (6) (2002)
  95. El-Wakil S A, Elhanbaly A, Elgarayhi A Chaos, Solitons & Fractals 12 1385 (2001)
  96. KOLEVA MARIA K, COVACHEV VALÉRY C Fluct. Noise Lett. 01 R131 (2001)
  97. Ebeling W, Schweitzer F Theory Biosci. 120 207 (2001)
  98. Ebeling W, Schweitzer F Integrative Systems Approaches to Natural and Social Dynamics Chapter 9 (2001) p. 119
  99. Gettinger A E, Gontchar I I J. Phys. G: Nucl. Part. Phys. 26 347 (2000)
  100. Muzychuk O V Radiophys Quantum Electron 43 422 (2000)
  101. Koleva M K, Eiyias A E, Petrov L A Metal-Ligand Interactions in Chemistry, Physics and Biology Chapter 15 (2000) p. 353
  102. Dubkov A A, Malakhov A N, Saichev A I Radiophys Quantum Electron 43 335 (2000)
  103. Garbaczewski P Physica A: Statistical Mechanics And Its Applications 285 187 (2000)
  104. Klimontovich Yu L Uspekhi Fizicheskikh Nauk 169 39 (1999)
  105. Prokop’ev E P Tech. Phys. Lett. 25 650 (1999)
  106. Vanag V K Uspekhi Fizicheskikh Nauk 169 481 (1999)
  107. MAMONTOV Y V, WILLANDER M Math. Models Methods Appl. Sci. 09 1247 (1999)
  108. Allahverdyan A E, Saakian D B Phys. Rev. E 58 1148 (1998)
  109. Olemskoi A I Uspekhi Fizicheskikh Nauk 168 287 (1998) [Olemskoi A I Phys.-Usp. 41 269 (1998)]
  110. Niemiec M, Gadomski A et al Physica A: Statistical Mechanics And Its Applications 248 365 (1998)
  111. Bellotti U, Bornatici M, Engelmann F Riv. Nuovo Cim. 20 1 (1997)
  112. Gontchar I, Litnevsky L A, Fröbrich P Computer Physics Communications 107 223 (1997)
  113. Klimontovich Yu L Phys.-Usp. 40 21 (1997)

© 1918–2020 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions