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Methodological notes


Green's function method in the theory of Brownian motors

 a,  b,  c, d
a Chuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine, Ghenerala Naumova str. 17, Kiev, 03164, Ukraine
b Belarussian State University, prosp. Nezavisimosti 4, Minsk, 220030, Belarus
c Semenov Federal Research Center for Chemical Physics, Russian Academy of Sciences, ul. Kosygina, 4, Moscow, 119991, Russian Federation
d Lomonosov Moscow State University, Vorobevy Gory, Moscow, 119991, Russian Federation

We present the main results of the theory of Brownian motors obtained using the author approach in which a Brownian particle moving in a slightly fluctuating potential profile is considered. By the Green's function method, the perturbation theory in small fluctuations of potential energy is constructed. This approach allows obtaining an analytical expression for the average particle velocity that is valid for two main types of Brownian motors (flashing and rocking ratchets) and any (stochastic or deterministic) time dependence of the fluctuations. The advantage of the proposed approach lies in the compactness of the description and, at the same time, in the variety of motor systems analyzed with its help: the overwhelming majority of known analytical results in the theory of Brownian motors follow from this expression. The mathematical derivations and analysis of those results forms the main content of these methodological notes.

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Fulltext is also available at DOI: 10.3367/UFNe.2018.04.038347
Keywords: Brownian motors, ratchets, driven diffusive systems, Green's functions
PACS: 05.40.−a, 05.60.Cd (all)
DOI: 10.3367/UFNe.2018.04.038347
URL: https://ufn.ru/en/articles/2019/5/e/
000477641200005
2-s2.0-85070662806
2019PhyU...62..496R
Citation: Rozenbaum V M, Shapochkina I V, Trakhtenberg L I "Green's function method in the theory of Brownian motors" Phys. Usp. 62 496–509 (2019)
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Received: 7th, November 2017, revised: 19th, April 2018, 23rd, April 2018

Оригинал: Розенбаум В М, Шапочкина И В, Трахтенберг Л И «Метод функций Грина в теории броуновских моторов» УФН 189 529–543 (2019); DOI: 10.3367/UFNr.2018.04.038347

References (85) ↓ Cited by (25) Similar articles (20)

  1. Vladimirov V S Uravneniya Matematicheskoi Fiziki (M.: Nauka, 1981); Per. na angl. yaz., Vladimirov V S Equations Of Mathematical Physics (Moscow: Mir, 1984)
  2. Lifshits E M, Pitaevskii L P Statisticheskaya Fizika Ch. 2 Teoriya Kondensirovannogo Sostoyaniya (M.: Nauka, 1978); Per. na angl. yaz., Lifshitz E M, Pitaevskii L P Statistical Physics Pt. 2 Theory Of The Condensed State (Oxford: Pegramon Press, 1980)
  3. Riskin H The Fokker - Plank Equation. Methods Of Solution And Applications (Berlin: Springer-Verlag, 1989)
  4. Eyges L The Classical Electromagnetic Field (New York: Dover Publ., 1972)
  5. Zubarev D N Usp. Fiz. Nauk 71 71 (1960); Zubarev D N Sov. Phys. Usp. 3 320 (1960)
  6. Bonch-Bruevich V L, Tyablikov S V Metod Funktsii Grina v Statisticheskoi Mekhanike (M.: Fizmatlit, 1961); Per. na angl. yaz., Bonch-Bruevich V L, Tyablikov S V The Green Function Method In Statistical Mechanics (Amsterdam: Interscience Publ., 1962)
  7. Abrikosov A A, Gor’kov L P, Dzyaloshinskii I E Metody Kvantovoi Teorii Polya v Statisticheskoi Fizike (M.: Fizmatgiz, 1962); Per. na angl. yaz., Abrikosov A A, Gorkov L P, Dzyaloshinski I E Methods Of Quantum Field Theory In Statistical Physics (Englewood Cliffs, N.J.: Prentice-Hall, 1963)
  8. Ludwig W Dynamics Of Crystal Lattice With Defects (Prague: Academics, 1966)
  9. Gol’danskii V I, Trakhtenberg L I, Flerov V N Tunnel’nye Yavleniya v Khimicheskoi Fizike (M.: Nauka, 1986); Per. na angl. yaz., Goldanskii V I, Trakhtenberg L I, Fleurov V N Tunneling Phenomena In Chemical Physics (New York: Gordon and Breach, 1989)
  10. Vetchinkin S I, Kchristenko S V Chem. Phys. Lett. 1 437 (1967)
  11. Zon B A, Rapoport L I Pis’ma ZhETF 7 70 (1968); Zon B A, Rapoport L I JETP Lett. 7 52 (1968)
  12. Zon B A, Manakov N L, Rapoport L I Zh. Eksp. Teor. Fiz. 55 924 (1968); Zon B A, Manakov N L, Rapoport L I Sov. Phys. JETP 28 480 (1969)
  13. Rapoport L I, Zon B A, Manakov N L Zh. Eksp. Teor. Fiz. 56 400 (1969); Rapoport L I, Zon B A, Manakov N L Sov. Phys. JETP 29 220 (1969)
  14. Lifshits I M Zh. Eksp. Teor. Fiz. 17 1017 (1947)
  15. Lifshits I M Zh. Eksp. Teor. Fiz. 17 1075 (1947)
  16. Dyadyusha G G, Rozenbaum V M, Dekhtyar’ M L Zh. Eksp. Teor. Fiz. 100 1051 (1991); Dyadyusha G G, Rozenbaum V M, Dekhtyar M L Sov. Phys. JETP 73 581 (1991)
  17. Reimann P Phys. Rep. 361 57 (2002)
  18. Hänggi P, Marchesoni F Rev. Mod. Phys. 81 387 (2009)
  19. Schadschneider A, Chowdhury D, Nishinari K Stochastic Transport In Complex Systems: From Molecules To Vehicles (Amsterdam: Elsevier, 2010)
  20. Cubero D, Renzoni F Brownian Ratchets: From Statistical Physics To Bio And Nano-motors (Cambridge: Cambridge Univ. Press, 2016)
  21. Bressloff P C, Newby J M Rev. Mod. Phys. 85 135 (2013)
  22. Kolomeisky A B Motor Proteins And Molecular Motors (Boca Raton, FL: CRS Press, 2015)
  23. Howard J Mechanics Of Motor Proteins And The Cytoskeleton (Sunderland, MA: Sinauer Associates, 2001)
  24. Romanovskii Yu L, Tikhonov A N Usp. Fiz. Nauk 180 931 (2010); Romanovsky Yu M, Tikhonov A N Phys. Usp. 53 893 (2010)
  25. Romanovskii Yu L, Tikhonov A N Usp. Fiz. Nauk 186 125 (2016); Romanovsky Yu M, Tikhonov A N Phys. Usp. 59 121 (2016)
  26. Zakharov P N i dr Usp. Fiz. Nauk 186 853 (2016); Zakharov P N et al Phys. Usp. 59 773 (2016)
  27. Chowdhury D Phys. Rep. 529 1 (2013)
  28. Goychuk I Beilstein J. Nanotechnol. 7 328 (2016)
  29. Browne W R, Feringa B L Nature Nanotechnol. 1 25 (2006)
  30. Cheng C, Stoddart J F Chem. Phys. Chem. 17 1780 (2016)
  31. Kay E R, Leigh D A, Zerbetto F Angew. Chem. Int. Ed. 46 72 (2007)
  32. Saha S, Stoddart J F Chem. Soc. Rev. 36 77 (2007)
  33. Zheng Y B et al J. Nanophoton. 4 042501 (2010)
  34. Michl J, Sykes E C H ACS Nano 3 1042 (2009)
  35. Belinicher V I, Sturman B I Usp. Fiz. Nauk 130 415 (1980); Belinicher V I, Sturman B I Sov. Phys. Usp. 23 199 (1980)
  36. Olemskoi I A Usp. Fiz. Nauk 168 475 (1998); Olemskoi A I Phys. Usp. 41 411 (1998)
  37. Gardiner C W Stochastic Methods. Handbook For The Natural And Social Sciences 4th Ed. (Berlin: Springer-Verlag, 2009); Per. na russk. yaz., Gardiner K V Stokhasticheskie Metody v Estestvennykh Naukakh (M.: Mir, 1986)
  38. Astumian R D, Bier M Phys. Rev. Lett. 72 1766 (1994)
  39. Sokolov I M Phys. Rev. E 63 021107 (2001)
  40. Reimann P Phys. Rev. Lett. 86 4992 (2001)
  41. Rozenbaum V M Zh. Eksp. Teor. Fiz. 137 740 (2010); Rozenbaum V M JETP 110 653 (2010)
  42. Shved N Yu, Shapochkina I V, Rozenbaum V M Vestnik Belorussk. Gos. Un-ta. Ser. 1 (2) 27 (2014)
  43. Rozenbaum V M FNT 40 604 (2014); Rozenbaum V M Low Temp. Phys. 40 469 (2014)
  44. Denisov S, Flach S, Hänggi P Phys. Rep. 538 77 (2014)
  45. Cubero D, Renzoni F Phys. Rev. Lett. 116 010602 (2016)
  46. Rozenbaum V M i dr Pis’ma ZhETF 107 525 (2018); Rozenbaum V M et al JETP Lett. 107 525 (2018)
  47. Van den Broeck C, Hänggi P Phys. Rev. A 30 2730 (1984)
  48. Astumian R D J. Phys. Chem. 100 19075 (1996)
  49. Rozenbaum V M et al J. Chem. Phys. 145 064110 (2016)
  50. Rozenbaum V M et al Phys. Rev. E 83 051120 (2011)
  51. Rozenbaum V M et al Phys. Rev. E 94 052140 (2016)
  52. Rozenbaum V M et al Phys. Rev. E 92 062132 (2015)
  53. Rozenbaum V M et al Phys. Rev. E 89 052131 (2014)
  54. Rozenbaum V M, Shapochkina I V Pis’ma ZhETF 102 275 (2015); Rozenbaum V M, Shapochkina I V JETP Lett. 102 248 (2015)
  55. Rozenbaum V M Pis’ma ZhETF 88 391 (2008); Rozenbaum V M JETP Lett. 88 342 (2008)
  56. Rozenbaum V M, Shapochkina I Phys. Rev. E 84 051101-1 (2011)
  57. Parrondo J M R Phys. Rev. E 57 7297 (1998)
  58. Parrondo J M R et al Europhys. Lett. 43 248 (1998)
  59. Rozenbaum V M et al Phys. Rev. E 85 041116 (2012)
  60. Rozenbaum V M et al Phys. Rev. E 71 041102 (2005)
  61. Rozenbaum V M Pis’ma ZhETF 79 475 (2004); Rozenbaum V M JETP Lett. 79 388 (2004)
  62. Doering C R, Dontcheva L A, Klosek M M Chaos 8 643 (1998)
  63. Reimann P Stochastic Processes In Physics, Chemistry, And Biology (Eds J A Freund, T Poeschel) (Berlin: Springer-Verlag, 2000) p. 50
  64. Mielke A Ann. Physik 4 721 (1995)
  65. Elston T C, Doering C R J. Stat. Phys. 83 359 (1996)
  66. Dekhtyar M L, Ishchenko A A, Rozenbaum V M J. Phys. Chem. B 110 20111 (2006)
  67. Dekhtyar M L, Rozenbaum V M J. Chem. Phys. 134 044136 (2011)
  68. Rozenbaum V M et al Physica A 363 211 (2006)
  69. Ambaye H, Kehr K W Physica A 267 111 (1999)
  70. Dyson F J Phys. Rev. 75 486 (1949)
  71. Rozenbaum V M et al JETP Lett. 105 542 (2017)
  72. Vysotskaya V A i dr Khimiya, Fizika Tekhnologiya Poverkhnosti 8 299 (2017)
  73. Vysotskaya V A i dr Zhurn. Belorussk. Gos. Univ. Fizika (3) 33 (2017)
  74. Magnasco M O Phys. Rev. Lett. 71 1477 (1993)
  75. Lifson S J. Chem. Phys. 36 2410 (1962)
  76. Rozenbaum V M, Shapochkina I V Pis’ma ZhETF 92 124 (2010); Rozenbaum V M, Shapochkina I V JETP Lett. 92 120 (2010)
  77. Rousselet J et al Nature 370 446 (1994)
  78. Gorre-Talini L, Spatz J P, Silberzan P Chaos 8 650 (1998)
  79. De Souza Silva C C et al Nature 440 651 (2006)
  80. Gommers R, Bergamini S, Renzoni F Phys. Rev. Lett. 95 073003 (2005)
  81. Kedem O, Lau B, Weiss E A Nano Lett. 17 5848 (2017)
  82. Pethig R Dielectrophoresis. Theory, Methodology And Biological Applications (Hoboken, NJ: John Wiley and Sons, Inc., 2017)
  83. Pommer M S Electrophoresis 29 1213 (2008)
  84. Mahabadi S, Labeed F H, Hughes M P Electrophoresis 36 1493 (2015)
  85. Rozenbaum V M et al Phys. Rev. E 99 012103 (2019)
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