Active Brownian particles

From individual to collective stochastic dynamics
  • P. Romanczuk
  • M. Bär
  • W. Ebeling
  • B. Lindner
  • L. Schimansky-Geier
Review

Abstract

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Abaid, M. Porfiri, J.R. Soc. Inter. 7, 1441 (2010)CrossRefGoogle Scholar
  2. 2.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970)Google Scholar
  3. 3.
    J. Adler, Cold Spring Harbor Symposia on Quantitative Biology 30, 289(1965)CrossRefGoogle Scholar
  4. 4.
    J. Adler Science 153, 708(1966)ADSCrossRefGoogle Scholar
  5. 5.
    J. Adler Annual Rev. Biochem. 44, 341(1975)CrossRefGoogle Scholar
  6. 6.
    M.S. Alber, M.A. Kiskowski, Y. Jiang Phys. Rev. Lett. 93, 068102(2004)ADSCrossRefGoogle Scholar
  7. 7.
    G.P. Alexander, J.M. Yeomans EPL (Europhysics Letters) 83, 34006(2008)ADSCrossRefGoogle Scholar
  8. 8.
    J.L. Anderson Ann. Rev. Fluid Mech. 21, 61(1989)ADSCrossRefGoogle Scholar
  9. 9.
    V.S.Anishchenko, V.V. Astakhov, T.E. Vadivasova, A.B. Neiman, L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems Springer Series in Synergetics. (Springer, Berlin Heidelberg, 2002)Google Scholar
  10. 10.
    I. Aoki, Bulletin of the Japanese Society of Scientific Fisheries (Japan), 1982Google Scholar
  11. 11.
    I.S. Aranson, A. Snezhko, J.S. Olafsen, J.S. Urbach Science 320, 612(2008)ADSCrossRefGoogle Scholar
  12. 12.
    I.S. Aranson, L.S. Tsimring Rev. Mod. Phys. 78, 641(2006)ADSCrossRefGoogle Scholar
  13. 13.
    I.S. Aranson, D. Volfson, L.S. Tsimring Phys. Rev. E 75, 051301(2007)ADSCrossRefGoogle Scholar
  14. 14.
    M. Badoual, F. Jülicher J. Prost, Proc. Natl. Acad. Sci. 99, 6696(2002)ADSCrossRefGoogle Scholar
  15. 15.
    M.D. Baker, P.M. Wolanin, J.B. Stock BioEssays 28, 9(2006)CrossRefGoogle Scholar
  16. 16.
    V. Balakrishnan, Stochastic Processes Formalism and Applications (1983), p. 96Google Scholar
  17. 17.
    M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, V. Zdravkovic Proc. National Acad. Sci. 105, 1232(2008)ADSCrossRefGoogle Scholar
  18. 18.
    E.L. Barnhart, G.M. Allen, F. Jülicher, J.A. Theriot Biophys. J. 98, 933(2010)ADSCrossRefGoogle Scholar
  19. 19.
    F. Bartumeus, J. Catalan, G.M. Viswanathan, E.P. Raposo, M.G.E. da Luz J. Theor. Biol. 252, 43(2008)CrossRefGoogle Scholar
  20. 20.
    F. Bartumeus, M.G.E. da Luz, G.M. Viswanathan J. Catalan, Ecology 86, 3078(2005)Google Scholar
  21. 21.
    A. Baskaran, C.M. Marchetti J. Stat. Mech.: Theory Exper. 2010, P04019(2010)CrossRefGoogle Scholar
  22. 22.
    A. Baskaran, M.C. Marchetti Phys. Rev. E 77, 011920(2008)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    A. Baskaran, M.C. Marchetti Proc. National Acad. Sci. USA 106, 15567(2009)ADSCrossRefGoogle Scholar
  24. 24.
    A. Baule, K.V. Kumar, S. Ramaswamy J. Stat. Mech. P11008 (2008)Google Scholar
  25. 25.
    S. Bazazi, J. Buhl, J.J. Hale, M.L. Anstey, G.A. Sword, S.J. Simpson, I.D. Couzin Current Biology 18, 735(2008)CrossRefGoogle Scholar
  26. 26.
    S. Bazazi, C.C. Ioannou, S.J. Simpson, G.A. Sword, C.J. Torney, P.D. Lorch, I.D. Couzin PLoS ONE 5, e15118(2010)ADSCrossRefGoogle Scholar
  27. 27.
    S. Bazazi, P. Romanczuk, S. Thomas, L. Schimansky-Geier, J.J. Hale, G.A. Miller, G.A. Sword, S.J. Simpson, I.D. Couzin Proc. Royal Society B: Biol. Sci. (2010)Google Scholar
  28. 28.
    R. Becker, Theory of Heat. Second Edition, 2nd edn. (Springer, 1967)Google Scholar
  29. 29.
    E. Ben-Jacob, I. Cohen, H. Levine Adv. Phys. 49, 395(2000), review ueber Bakterien u.aeADSCrossRefGoogle Scholar
  30. 30.
    I. Bena International J. Mod. Phys. B 20, 2825(2006)MathSciNetADSMATHCrossRefGoogle Scholar
  31. 31.
    H.C. Berg Nature 254, 389(1975)ADSCrossRefGoogle Scholar
  32. 32.
    H.C. Berg, Random Walks in Biology (Princeton University Press, Princeton, 1993)Google Scholar
  33. 33.
    H.C. Berg Proc. National Acad. Sci. USA 93, 14225(1996)ADSCrossRefGoogle Scholar
  34. 34.
    H.C. Berg, D.A. Brown Nature 239, 500(1972)ADSCrossRefGoogle Scholar
  35. 35.
    H.C. Berg, E.M. Purcell Biophys. J. 20, 193(1977)ADSCrossRefGoogle Scholar
  36. 36.
    E. Bertin, M. Droz, G. Grégoire Phys. Rev. E 74, 022101(2006)ADSCrossRefGoogle Scholar
  37. 37.
    E. Bertin, M. Droz, G. Grégoire Phys. Rev. E 74, 022101(2006)ADSCrossRefGoogle Scholar
  38. 38.
    E. Bertin, M. Droz, G. Grégoire J. Phys. A: Math. Theor. 42, 445001(2009)ADSCrossRefGoogle Scholar
  39. 39.
    K. Bhattacharya, T. Vicsek New J. Phys. 12, 093019(2010)ADSCrossRefGoogle Scholar
  40. 40.
    I.I. Blekhman, Vibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications (World Scientific, 2000)Google Scholar
  41. 41.
    N.W.F. Bode, D.W. Franks, A. Jamie Wood J. Theor. Biol. 267, 292(2010)CrossRefGoogle Scholar
  42. 42.
    H.U. Bödeker, C. Beta, T.D. Frank, E. Bodenschatz EPL (Europhysics Letters) 90, 28005(2010)ADSCrossRefGoogle Scholar
  43. 43.
    U. Börner, A. Deutsch, H. Reichenbach, M. Bär Phys. Rev. Lett. 89, 078101(2002)ADSCrossRefGoogle Scholar
  44. 44.
    D. Bray Proc. National Acad. Sci. USA 99, 7(2002)Google Scholar
  45. 45.
    N.V. Brilliantov, T. Pöschel, Kinetic Theory of Granular Gases (Oxford Univ. Press, 2004)Google Scholar
  46. 46.
    C. Van Den Broeck J. Stat. Phys. 31, 467((1983)ADSMATHCrossRefGoogle Scholar
  47. 47.
    D.A. Brown, H.C. Berg Proc. National Acad. Sci. USA 71, 1388(1974)ADSCrossRefGoogle Scholar
  48. 48.
    R. Brown Philosophical Mag. Ser. 4, 161(1828)Google Scholar
  49. 49.
    E.O. Budrene, H.C. Berg Nature 349, 630(1991)ADSCrossRefGoogle Scholar
  50. 50.
    E.O. Budrene, H.C. Berg Nature 376, 49(1995)ADSCrossRefGoogle Scholar
  51. 51.
    J. Buhl, D.J.T. Sumpter, I.D. Couzin, J.J. Hale, E. Despland, E.R. Miller, S.J. Simpson Science 312, 1402(2006)ADSCrossRefGoogle Scholar
  52. 52.
    T. Butt, T. Mufti, A. Humayun, P.B. Rosenthal, S. Khan, S. Khan, J.E. Molloy J. Biol. Chem. 285, 4964(2010)CrossRefGoogle Scholar
  53. 53.
    M.O. Cáceres, A.A Budini J. Phys. A: Math. Gen. 30, 8427(1997)ADSMATHCrossRefGoogle Scholar
  54. 54.
    D. Campos, V. Mendez, I. Llopis J. Theor. Biol. 267, 526(2010)CrossRefGoogle Scholar
  55. 55.
    A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini, M. Viale Proc. National Acad. Sci. 107, 11865(2010)ADSCrossRefGoogle Scholar
  56. 56.
    A. Cavagna, I. Giardina, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, V. Zdravkovic Animal Behaviour 76, 217(2008)CrossRefGoogle Scholar
  57. 57.
    G. Cebiroglu, C. Weber, L. Schimansky-Geier Chem. Phys. 375, 439(2010)ADSCrossRefGoogle Scholar
  58. 58.
    S. Chandrasekhar Rev. Mod. Phys. 15, 1(1943)MathSciNetADSMATHCrossRefGoogle Scholar
  59. 59.
    H. Chaté, F. Ginelli, G. Grégoire, F. Peruani, F. Raynaud The Euro. Phys. J. B 64, 451(2008)ADSCrossRefGoogle Scholar
  60. 60.
    H. Chaté, F. Ginelli, G. Grégoire, F. Raynaud Phys. Rev. E 77, 046113(2008)ADSCrossRefGoogle Scholar
  61. 61.
    H. Chaté, F. Ginelli, R. Montagne Phys. Rev. Lett. 96, 180602(2006)ADSCrossRefGoogle Scholar
  62. 62.
    A. Chetverikov, J. Dunkel Eur. Phys. J. B 35, 239(2003)ADSCrossRefGoogle Scholar
  63. 63.
    S. Cilla, F. Falo, L.M. Floria Phys. Rev. E 63, 031110(2001)ADSCrossRefGoogle Scholar
  64. 64.
    E.A. Codling, M.J. Plank, S. Benhamou J. Roy. Soc. Inter. 5, 813(2008)CrossRefGoogle Scholar
  65. 65.
    T.C. O’Keefe, M.C. Brewer, S.I. Dodson J. Plankton Research 20, 973(1998)CrossRefGoogle Scholar
  66. 66.
    C.A. Condat, G.J. Sibona. Phys. D 168, 235(2002)MathSciNetCrossRefGoogle Scholar
  67. 67.
    C.A. Condat, J. Jäckle, S.A. Menchón, Phys. Rev. E 72, 021909(2005)ADSCrossRefGoogle Scholar
  68. 68.
    C.A. Condat, G.J. Sibona Physica D: Nonlinear Phenomena 168, 235(2002)MathSciNetADSCrossRefGoogle Scholar
  69. 69.
    I.D. Couzin Trends Cognitive Sci. 13, 36(2009)CrossRefGoogle Scholar
  70. 70.
    I.D. Couzin, Ch.C. Ioannou, G. Demirel, T. Gross, C.J. Torney, A. Hartnett, L. Conradt, S.A. Levin, N.E. Leonard Science 334, 1578(2011)ADSCrossRefGoogle Scholar
  71. 71.
    I.D. Couzin, J. Krause, N.R. Franks, S.A. Levin Nature 433, 513(2005)ADSCrossRefGoogle Scholar
  72. 72.
    I.D. Couzin, J. Krause, R. James, G.D. Ruxton, N.R. Franks J. Theoretical Biol. 218, 1(2002)MathSciNetCrossRefGoogle Scholar
  73. 73.
    T. Czernik, J. Luczka Annal. Phys. 9, 721(2000)ADSMATHCrossRefGoogle Scholar
  74. 74.
    A. Czirók, E. Ben-Jacob, I. Cohen, T. Vicsek Phys. Rev. E 54, 1791(1996)ADSCrossRefGoogle Scholar
  75. 75.
    A. Czirók, H.E. Stanley, T. Vicsek J. Physics A: Math. General 30, 1375(1997)ADSCrossRefGoogle Scholar
  76. 76.
    P.-G. de Gennes European Biophys. J. 33, 691(2004)ADSCrossRefGoogle Scholar
  77. 77.
    N.D. Dees, S. Bahar, R. Garcia, F. Moss J. Theoretical Biol. 252, 69(2008)CrossRefGoogle Scholar
  78. 78.
    N.D. Dees, S. Bahar F. Moss Phys. Biol. 5, 044001(2008)ADSCrossRefGoogle Scholar
  79. 79.
    N.D. Dees, M. Hofmann, S. Bahar Animal Behaviour 79, 603(2010)CrossRefGoogle Scholar
  80. 80.
    C. Van den Broeck Physica A 168, 677(1990)MathSciNetADSCrossRefGoogle Scholar
  81. 81.
    S. Denisov Phys. Lett. A 296, 197(2002)ADSMATHCrossRefGoogle Scholar
  82. 82.
    J. Deseigne, O. Dauchot , H. Chaté Phys. Rev. Lett. 105, 098001(2010)ADSCrossRefGoogle Scholar
  83. 83.
    P. Dieterich, R. Klages, R. Preuss, A. Schwab Proc. National Acad. Sci. 105, 459(2008)ADSCrossRefGoogle Scholar
  84. 84.
    E.J. Doedel Congr. Numer 30, 265(1981)MathSciNetGoogle Scholar
  85. 85.
    M. Doi, S.F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1988)Google Scholar
  86. 86.
    M.R. D’Orsogna, Y.L. Chuang, A.L. Bertozzi, L.S. Chayes Phys. Rev. Lett. 96, 104302(2006)ADSCrossRefGoogle Scholar
  87. 87.
    M.T. Downton, H. Stark J. Physics: Condensed Matter 21, 204101(2009)ADSCrossRefGoogle Scholar
  88. 88.
    K. Drescher, J. Dunkel, L.H. Cisneros, S. Ganguly, R.E. Goldstein Proc. National Acad. Sci. 108, 10940(2011)CrossRefGoogle Scholar
  89. 89.
    K. Drescher, R.E. Goldstein, N. Michel, M. Polin, I. Tuval Phys. Rev. Lett. 105, 168101(2010)ADSCrossRefGoogle Scholar
  90. 90.
    R. Dreyfus, J. Baudry, M.L. Roper, M. Fermigier, H.A. Stone, J. Bibette Nature 437, 862(2005)ADSCrossRefGoogle Scholar
  91. 91.
    J. Dunkel, W. Ebeling, U. Erdmann Eur. Phys. J. B 24, 511(2001)ADSCrossRefGoogle Scholar
  92. 92.
    J. Dunkel, W. Ebeling, U. Erdmann, V.A. Makarov Int. J. Bif. Chaos 12, 2359(2002)MathSciNetMATHCrossRefGoogle Scholar
  93. 93.
    J. Dunkel, P. Hänggi Physics Reports 471, 1(2009)MathSciNetADSCrossRefGoogle Scholar
  94. 94.
    D.B. Dusenbery Proc. National Acad. Sci. USA 94, 10949(1997)ADSCrossRefGoogle Scholar
  95. 95.
    M. Dworkin, D. Kaiser, Myxobacteria II (ASM Press, 1993)Google Scholar
  96. 96.
    S.J. Ebbens, J.R. Howse Soft Matter 6, 726(2010)ADSCrossRefGoogle Scholar
  97. 97.
    S.J. Ebbens, J.R. Howse Langmuir 27, 12293(2011)CrossRefGoogle Scholar
  98. 98.
    W. Ebeling Cond. Mat. Phys. 7, 539(2004)Google Scholar
  99. 99.
    W. Ebeling, U. Erdmann Complexity 8, 23(2003)MathSciNetCrossRefGoogle Scholar
  100. 100.
    W. Ebeling, U. Erdmann, J. Dunkel, M. Jenssen J. Stat. Phys. 101, 443-457(2000)ADSCrossRefGoogle Scholar
  101. 101.
    W. Ebeling, E. Gudowska-Nowak, A. Fiasconaro Acta Phys. Polonica B 39, 1251(2008)ADSGoogle Scholar
  102. 102.
    W. Ebeling, G. Röpke Physica D 187, 268(2004)MathSciNetADSMATHCrossRefGoogle Scholar
  103. 103.
    W. Ebeling, L. Schimansky-Geier The Eur. Phys. J. Special Topics 157, 17(2008)ADSCrossRefGoogle Scholar
  104. 104.
    W. Ebeling, L. Schimansky-Geier, A. Neiman, A. Scharnhorst Fluctuation Noise Lett. 5, 185(2005)CrossRefGoogle Scholar
  105. 105.
    W. Ebeling, F. Schweitzer Theory Biosci. 120, 207(2001)Google Scholar
  106. 106.
    W. Ebeling, F. Schweitzer, B. Tilch BioSystems 49, 17(1999)CrossRefGoogle Scholar
  107. 107.
    W. Ebeling, I.M. Sokolov, Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems (World Scientific, 2005)Google Scholar
  108. 108.
    A Einstein Annal. Phys. 17, 549(1905)ADSMATHCrossRefGoogle Scholar
  109. 109.
    A. Einstein Annal. Phys. 324, 371(1906)ADSCrossRefGoogle Scholar
  110. 110.
    M. Eisenbach, Chemotaxis (World Scientific Publishing, 2004)Google Scholar
  111. 111.
    I. Eliazar, J. Klafter Proc. National Acad. Sci. USA 102 13779(2005)MathSciNetADSMATHCrossRefGoogle Scholar
  112. 112.
    M. Enculescu, M. Sabouri-Ghomi, G. Danuser, M. Falcke Biophys. J. 98, 1571(2010)ADSCrossRefGoogle Scholar
  113. 113.
    M. Enculescu, H. Stark Phys. Rev. Lett. 107, 058301(2011)ADSCrossRefGoogle Scholar
  114. 114.
    S.A. Endow, H. Higuchi Nature 406, 913(2000)ADSCrossRefGoogle Scholar
  115. 115.
    U. Erdmann, Kollektive Bewegung: Komplexe Strukturen in 2D-Systemen aktiver Brown’scher Teilchen fernab vom Gleichgewicht, volume 10 of Nichtlineare und Stochastische Physik (Logos-Verlag, Berlin, 2003)Google Scholar
  116. 116.
    U. Erdmann, W. Ebeling Fluctuation Noise Lett. 3, 145(2003)CrossRefGoogle Scholar
  117. 117.
    U. Erdmann, W. Ebeling Inter. J. Bifurcation Chaos 15, 3623(2005)MathSciNetMATHCrossRefGoogle Scholar
  118. 118.
    U. Erdmann, W. Ebeling, V.S. Anishchenko Phys. Rev. E 65, 061106(2002)MathSciNetADSCrossRefGoogle Scholar
  119. 119.
    U. Erdmann, W. Ebeling, A.S. Mikhailov Phys. Rev. E 71, 051904(2005)ADSCrossRefGoogle Scholar
  120. 120.
    U. Erdmann, W. Ebeling, L. Schimansky-Geier, A. Ordemann, F. Moss (2004) http://arxiv.org/abs/q-bio.PE/0404018Google Scholar
  121. 121.
    U. Erdmann, W. Ebeling, L. Schimansky-Geier, F. Schweitzer Eur. Phys. J. B 15, 105(2000)ADSCrossRefGoogle Scholar
  122. 122.
    B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to Xppaut for Researchers and Students (Software, Environments, Tools), 1st edn. (Society for Industrial Mathematics, 2002)Google Scholar
  123. 123.
    A. Fiasconaro, W. Ebeling, E. Gudowska-Nowak Eur. Phys. J. B 65, 403(2008)ADSCrossRefGoogle Scholar
  124. 124.
    A. Fiasconaro, E. Gudowska-Nowak, W. Ebeling J. Statistical Mech. Theory Exper. 2009, P01029(2009)CrossRefGoogle Scholar
  125. 125.
    S. Flach, O. Yevtushenko, Y. Zolotaryuk Phys. Rev. Lett. 84, 2358(2000)ADSCrossRefGoogle Scholar
  126. 126.
    B. Flaherty, J.P. McGarry, P.E. McHugh Cell Biochem. Biophys. 49, 14(2007)CrossRefGoogle Scholar
  127. 127.
    S. Fournier-Bidoz, A.C. Arsenault, I. Manners, G.A. Ozin, Chem. Comm. 441 (2005)Google Scholar
  128. 128.
    B.M. Friedrich, F. Julicher Proc. National Acad. Sci. 104, 13256(2007)ADSCrossRefGoogle Scholar
  129. 129.
    B.M. Friedrich, F. Jülicher New J. Phys. 10, 123025(2008)ADSCrossRefGoogle Scholar
  130. 130.
    A. Fulinski Phys. Rev. E 50, 2668(1994)MathSciNetADSMATHCrossRefGoogle Scholar
  131. 131.
    R. Fürth Z. Phys. 2, 244(1920)ADSCrossRefGoogle Scholar
  132. 132.
    M. Garcia, S. Berti, P. Peyla, S. Rafai Phys. Rev. E 83, 035301(2011)ADSCrossRefGoogle Scholar
  133. 133.
    R. Garcia, F. Moss, A. Nihongi, J.R. Strickler, S. Göller, U. Erdmann, Lutz Schimansky-Geier, I.M. Sokolov Math. Biosci. 207, 165(2007)MathSciNetMATHCrossRefGoogle Scholar
  134. 134.
    V. Garcia, M. Birbaumer, F. Schweitzer Eur. Phys. J. B 82, 235(2011)ADSCrossRefGoogle Scholar
  135. 135.
    C.W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1985)Google Scholar
  136. 136.
    C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, volume 13 of Springer Series in Synergetics, 2nd edn. (Springer, Berlin Heidelberg New York, 1985)Google Scholar
  137. 137.
    J. Gautrais, C. Jost, M. Soria, A. Campo, S. Motsch, R. Fournier, S. Blanco, G. Theraulaz J. Math. Biol. 58, 429(2008)MathSciNetCrossRefGoogle Scholar
  138. 138.
    S.A. Gerasimov J. Appl. Mech. Tech. Phys. 43, 90(2002)CrossRefGoogle Scholar
  139. 139.
    A. Gholami, M. Falcke, E. Frey New J. Phys. 10 033022(2008)ADSCrossRefGoogle Scholar
  140. 140.
    J. Gibbs, Y. Zhao Frontiers Mater. Sci. 5, 25(2011)ADSCrossRefGoogle Scholar
  141. 141.
    F. Ginelli, F. Peruani, M. Bär, H. Chaté Phys. Rev. Lett. 104, 184502(2010)ADSCrossRefGoogle Scholar
  142. 142.
    S. Goldstein The Quarterly J. Mech. Appl. Math. 4, 129(1951)MATHCrossRefGoogle Scholar
  143. 143.
    R. Golestanian, T.B. Liverpool, A. Ajdari Phys. Rev. Lett. 94, 220801(2005)ADSCrossRefGoogle Scholar
  144. 144.
    R. Golestanian, T.B. Liverpool, A. Ajdari New J. Phys. 9, 126(2007)ADSCrossRefGoogle Scholar
  145. 145.
    M. Gouy J. Phys. Théor. Appliquée 7, 4(1888)Google Scholar
  146. 146.
    I. Goychuk, P. Jung, S. Kohler, G. Schmid, P. Talkner (ed.), Chem. Phys. 375 (2010)Google Scholar
  147. 147.
    G. Grégoire, H. Chaté Phys. Rev. Lett. 92, 025702(2004)ADSCrossRefGoogle Scholar
  148. 148.
    D. Grossman, I.S. Aranson, E.B. Jacob New J. Phys. 10, 023036(2008)ADSCrossRefGoogle Scholar
  149. 149.
    T. Guerin, J. Prost, P. Martin, J.-F. Joanny Curr. Opin. Cell Biol. 22, 14(2010)CrossRefGoogle Scholar
  150. 150.
    V. Guttal, I.D. Couzin Proc. National Acad. Sci. 107, 16172(2010)ADSCrossRefGoogle Scholar
  151. 151.
    V. Guttal, P. Romanczuk, I.D. Couzin, S.J. Simpson, G.A. Sword (submitted) (2012)Google Scholar
  152. 152.
    L. Haeggqwist, L. Schimansky-Geier, I.M. Sokolov, F. Moss The Eur. Phys. J. Special Topics 157, 33(2008)ADSCrossRefGoogle Scholar
  153. 153.
    P. Hänggi Adv. Chem. Phys. 89 (1995)Google Scholar
  154. 154.
    P. Hänggi, F. Marchesoni (ed.), Chaos 15 (2005)Google Scholar
  155. 155.
    P. Hänggi, F. Marchesoni Rev. Mod. Phys. 81, 387(2009)ADSCrossRefGoogle Scholar
  156. 156.
    P. Hänggi, P. Talkner, M. Borkovec Rev. Mod. Phys. 62, 251(1990)ADSCrossRefGoogle Scholar
  157. 157.
    P. Hänggi, H. Thomas Phys. Rep. 88, 207(1982)MathSciNetADSCrossRefGoogle Scholar
  158. 158.
    T. Harada Europhys. Lett. (EPL) 70, 49(2005)ADSCrossRefGoogle Scholar
  159. 159.
    D. Helbing Rev. Mod. Phys. 73, 1067(2001)ADSCrossRefGoogle Scholar
  160. 160.
    D. Helbing, I.J. Farkas, T. Vicsek Phys. Rev. Lett. 84, 1240(2000)ADSCrossRefGoogle Scholar
  161. 161.
    D. Helbing, F. Schweitzer, J. Keltsch, P. Molnár Phys. Rev. E 56, 2527(1997)ADSCrossRefGoogle Scholar
  162. 162.
    J.E. Herbert-Read, A. Perna, R.P. Mann, T.M. Schaerf, D.J.T. Sumpter, A.J.W. Ward, Proc. National Acad. Sci. (2011)Google Scholar
  163. 163.
    J.P. Hernandez-Ortiz, C.G. Stoltz, M.D. Graham Phys. Rev. Lett. 95, 204501(2005)ADSCrossRefGoogle Scholar
  164. 164.
    J.P. Hernandez-Ortiz, P.T. Underhill, M.D. Graham J. Phys.: Condensed Matt. 21, 204107(2009)ADSCrossRefGoogle Scholar
  165. 165.
    T. Hillen, H.G. Othmer SIAM J. Appl. Math. 61, 751(2000)MathSciNetMATHCrossRefGoogle Scholar
  166. 166.
    T. Hillen, K.J. Painter J. Math. Biol. 58, 183(2008)MathSciNetCrossRefGoogle Scholar
  167. 167.
    W. Horsthemke, R. Lefever, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology (Springer Verlag, 1984)Google Scholar
  168. 168.
    J. Howard, Mechanics of Motor Proteins and the Cytoskeleton (Sinauer Associates, Sunderland, Mass, 2001)Google Scholar
  169. 169.
    J.R. Howse, R.A.L. Jones, A.J. Ryan, T. Gough, R. Vafabakhsh, R. Golestanian Phys. Rev. Lett. 99, 048102(2007)ADSCrossRefGoogle Scholar
  170. 170.
    C. Huepe, M. Aldana Phys. Rev. Lett. 92, 168701(2004). Copyright (C) 2008. The American Physical Society; Please report any problems to prola@aps.orgADSCrossRefGoogle Scholar
  171. 171.
    C. Huepe, M. Aldana, Physica A (2008)Google Scholar
  172. 172.
    A. Huth, C. Wissel J. Theor. Biol. 156, 365(1992)CrossRefGoogle Scholar
  173. 173.
    H. i Wu, B.-L. Li, T.A. Springer, W.H. Neill Ecological Modell. 132, 115(2000)CrossRefGoogle Scholar
  174. 174.
    P.A. Iglesias, P.N. Devreotes Current Opinion Cell Biol. 20, 35(2008)CrossRefGoogle Scholar
  175. 175.
    O.A. Igoshin, R. Welch, D. Kaiser, G. Oster Proc. National Acad. Sci. USA 101, 4256(2004)ADSCrossRefGoogle Scholar
  176. 176.
    T. Ihle Phys. Rev. E 83, 030901(2011)ADSCrossRefGoogle Scholar
  177. 177.
    J. Ingen-Housz, Verm. Schriften Physisch-Medicinischen Inhalts (1784)Google Scholar
  178. 178.
    R.F. Ismagilov, A. Schwartz, N. Bowden, G.M. Whitesides Angewandte Chemie 114, 674(2002)CrossRefGoogle Scholar
  179. 179.
    L. Jelsbak Proc. National Acad. Sci. 99, 2032(2002)ADSCrossRefGoogle Scholar
  180. 180.
    J.-F. Joanny, F. Jülicher J. Prost, Phys. Rev. Lett. 90, 168102(2003)ADSCrossRefGoogle Scholar
  181. 181.
    F. Jülicher, A. Ajdari, J. Prost Rev. Mod. Phys. 69, 1269(1997)ADSCrossRefGoogle Scholar
  182. 182.
    F. Jülicher, A. Ajdari, J. Prost Rev. Mod. Phys. 69, 1269(1997)ADSCrossRefGoogle Scholar
  183. 183.
    F. Jülicher, K. Kruse, J. Prost, J.-F. Joanny Phys. Rep. 449, 3(2007)MathSciNetADSCrossRefGoogle Scholar
  184. 184.
    P. Jung, H. Risken Z. Phys. B 59, 469(1985)MathSciNetADSCrossRefGoogle Scholar
  185. 185.
    M. Kac Math. J 4, 497(1974)MATHGoogle Scholar
  186. 186.
    D. Kaiser Nat. Rev. Micro. 1, 45(2003)MathSciNetCrossRefGoogle Scholar
  187. 187.
    P.M. Kareiva, N. Shigesada Oecologia 56, 234(1983)CrossRefGoogle Scholar
  188. 188.
    Y. Katz, K. Tunstrøm, C.C. Ioannou, C. Huepe, I.D. Couzin, Proc. National Acad. Sci. (2011)Google Scholar
  189. 189.
    R.F. Kayser, H.J., Raveché Phys. Rev. A 17, 2067(1978)ADSCrossRefGoogle Scholar
  190. 190.
    H. Ke, S. Ye, R.L. Carroll, K. Showalter The J. Phys. Chemistry A 114, 5462(2010)CrossRefGoogle Scholar
  191. 191.
    D.B. Kearns Nat. Rev. Micro. 8, 634(2010)CrossRefGoogle Scholar
  192. 192.
    E.F. Keller, L.A. Segel J. Theor. Biol. 30, 225(1971)CrossRefGoogle Scholar
  193. 193.
    E.F. Keller, L.A. Segel J. Theor. Biol. 30, 235(1971)CrossRefGoogle Scholar
  194. 194.
    R. Kemkemer, V. Teichgräber, S. Schrank-Kaufmann, D. Kaufmann, H. Gruler Eur. Phys. J. E 3, 101(2000)CrossRefGoogle Scholar
  195. 195.
    C. Kim, E.K. Lee, P. Hänggi, P. Talkner Phys. Rev. E 76, 011109(2007)ADSCrossRefGoogle Scholar
  196. 196.
    Y.L. Klimontovich Physics-Uspekhi 37, 737(1994)ADSCrossRefGoogle Scholar
  197. 197.
    Yu.L. Klimontovich, Statistical Theory of Open Systems (Kluwer Academic Publishers, Dordrecht, Boston, London, 1995)Google Scholar
  198. 198.
    S. Köhler, V. Schaller, A.R. Bausch Nat. Mater. 10, 462(2011)ADSCrossRefGoogle Scholar
  199. 199.
    G.V. Kolmakov, A. Schaefer, I. Aranson, A.C. Balazs Soft Matter 8, 180(2011)ADSCrossRefGoogle Scholar
  200. 200.
    N. Komin, U. Erdmann, L. Schimansky-Geier Fluctuation Noise Lett. 4, L151(2004)CrossRefGoogle Scholar
  201. 201.
    J.M. Kosterlitz, D.J. Thouless J. Phys. C 6, 10(1973)CrossRefGoogle Scholar
  202. 202.
    J. Krause, G.D. Ruxton, Living in groups (Oxford University Press, 2002)Google Scholar
  203. 203.
    K. Kruse, S. Camalet, F. Jülicher Phys. Rev. Lett. 87, 138101(2001)ADSCrossRefGoogle Scholar
  204. 204.
    K. Kruse, J.F. Joanny, F. Jülicher J. Prost, Phys. Biol. 3, 130(2006)Google Scholar
  205. 205.
    R. Kubo Rep. Prog. Phys. 29, 255(1966)ADSCrossRefGoogle Scholar
  206. 206.
    A. Kudrolli Phys. Rev. Lett. 104, 088001(2010)ADSCrossRefGoogle Scholar
  207. 207.
    A. Kudrolli, G. Lumay, D. Volfson, L.S. Tsimring Phys. Rev. Lett. 100, 058001(2008)ADSCrossRefGoogle Scholar
  208. 208.
    K.V. Kumar, S. Ramaswamy, M. Rao Phys. Rev. E 77, 020102(2008)ADSCrossRefGoogle Scholar
  209. 209.
    P. Langevin Comptes Rendus Hebdomadaires des Séances de L’Academie des Sciences 146, 530(1908)MATHGoogle Scholar
  210. 210.
    E. Lauga, W.R. DiLuzio, G.M. Whitesides, H.A. Stone Biophys. J. 90, 400(2006)ADSCrossRefGoogle Scholar
  211. 211.
    E. Lauga, T.R. Powers Reports Progr. Phys. 72, 096601(2009)MathSciNetADSCrossRefGoogle Scholar
  212. 212.
    P.A. Lebwohl, G. Lasher Phys. Rev. A 6, 426(1972)ADSCrossRefGoogle Scholar
  213. 213.
    C.F. Lee Phys. Rev. E 81, 031125(2010)ADSCrossRefGoogle Scholar
  214. 214.
    K. Lekkas, L. Schimansky-Geier, H. Engel-Herbert Z. Phys. B Cond. Matter 70, 517(1988)CrossRefGoogle Scholar
  215. 215.
    H. Levine, W.-J. Rappel, I. Cohen Phys. Rev. E 63, 017101(2000)ADSCrossRefGoogle Scholar
  216. 216.
    L. Li, E. C. Cox, H. Flyvbjerg Phys. Biol. 8, 046006(2011)ADSCrossRefGoogle Scholar
  217. 217.
    L. Li, S.F. Nørrelykke, Edward C. Cox PLoS ONE 3, e2093(2008)ADSCrossRefGoogle Scholar
  218. 218.
    B. Lindner, in Physik Irreversibler Prozesse und Selbstorganisation, edited by T. Pöschel, H. Malchow, L. Schimansky-Geier (Logos-Verlag, Berlin, 2006)Google Scholar
  219. 219.
    B. Lindner New J. Phys. 9, 136(2007)ADSCrossRefGoogle Scholar
  220. 220.
    B. Lindner J. Stat. Phys. 130, 523(2008)MathSciNetADSMATHCrossRefGoogle Scholar
  221. 221.
    B. Lindner New J. Phys. 12, 063026(2010)ADSCrossRefGoogle Scholar
  222. 222.
    B. Lindner, U. Erdmann, I. Sokolov (eds.), Eur. Phys. J. Special Topics, 187 (2010)Google Scholar
  223. 223.
    B. Lindner, E.M. Nicola Phys. Rev. Lett. 101, 190603(2008)ADSCrossRefGoogle Scholar
  224. 224.
    B. Lindner, E.M. Nicola Eur. Phys. J. Special Topics 157, 43(2008)ADSCrossRefGoogle Scholar
  225. 225.
    R. Lukeman, Y.-X. Li, L. Edelstein-Keshet Proc. National Acad. Sci. 107, 12576(2010)ADSCrossRefGoogle Scholar
  226. 226.
    V.A. Makarov, E. del Río, W. Ebeling, M.G. Velarde Phys. Rev. E 64, 036601(2001)ADSCrossRefGoogle Scholar
  227. 227.
    V.A. Makarov, W. Ebeling, M.G. Velarde Intern. J. Bifurcation Chaos 10, 1075(2000)MathSciNetMATHGoogle Scholar
  228. 228.
    M.D. Manson Adv. Microbial Physiol. 33, 277(1992)ADSCrossRefGoogle Scholar
  229. 229.
    J. Masoliver Phys. Rev. A 35, 3918(1987)MathSciNetADSCrossRefGoogle Scholar
  230. 230.
    J. Masoliver, K. Lindenberg, G.H. Weiss Physica A: Stat. Mech. Appl. 157, 891(1989)MathSciNetADSCrossRefGoogle Scholar
  231. 231.
    M. Matsushita, J. Wakita, H. Itoh, K. Watanabe, T. Arai, T. Matsuyama, H. Sakaguchi, M. Mimura Physica A 274, 190(1999), unterstreicht, dass Zellen sich aktiv bewegenADSCrossRefGoogle Scholar
  232. 232.
    S. Menden-Deuer PLoS Comput. Biol. 6, e1000942(2010)CrossRefGoogle Scholar
  233. 233.
    N.D. Mermin, H. Wagner Phys. Rev. Lett. 17, 1133(1966)ADSCrossRefGoogle Scholar
  234. 234.
    R. Metzler, J. Klafter Phys. Rep. 339, 1(2000)MathSciNetADSMATHCrossRefGoogle Scholar
  235. 235.
    A.S. Mikhailov, V. Calenbuhr, From Cells to Societies: Models of Complex Coherent Action (Springer, 2002)Google Scholar
  236. 236.
    A.S. Mikhailov, D. Meinköhn, in Stochastic Dynamics, edited by L. Schimansky-Geier, Thorsten Pöschel, editors, vol. 484 of Lecture Notes in Physics (Springer Berlin, 1997), p. 334Google Scholar
  237. 237.
    A.S. Mikhailov, D.H. Zanette Phys. Rev. E 60, 4571(1999)ADSCrossRefGoogle Scholar
  238. 238.
    T. Mirkovic, N.S. Zacharia, G.D. Scholes, G.A. Ozin Small 6, 159(2010)CrossRefGoogle Scholar
  239. 239.
    S. Mishra, A. Baskaran, M.C. Marchetti Phys. Rev. E 81, 061916(2010)ADSCrossRefGoogle Scholar
  240. 240.
    S. Mishra, S. Ramaswamy Phys. Rev. Lett. 97, 090602(2006)ADSCrossRefGoogle Scholar
  241. 241.
    T.J. Mitchison, L.P. Cramer Cell 84, 371(1996)CrossRefGoogle Scholar
  242. 242.
    N. Mittal E.O., amd Brenner Budrene, A. van Oudenaarden Proc. National Acad. Sci. USA 100, 13259(2003)ADSCrossRefGoogle Scholar
  243. 243.
    T. Mora, W. Bialek J. Stat. Phys. 144, 268(2011)MathSciNetADSMATHCrossRefGoogle Scholar
  244. 244.
    H. Mori Prog. Theor. Phys. 33, 423(1965)ADSMATHCrossRefGoogle Scholar
  245. 245.
    J.D. Murray, Spatial models and biomedical applications (Springer, 2003)Google Scholar
  246. 246.
    (a) H. Reichenbach Ber. Deutsch. Bot. Ges. 78, 102(1965); (b) Myxobacteriales: Schwarmentfaltung und Bildung von Protocysten, Institut für den wissenschaftlichen Film, Göttingen, 1966Google Scholar
  247. 247.
    M. Nagy, I. Daruka, T. Vicsek Physica A: Stat. Theor. Phys. 373, 445(2007)ADSCrossRefGoogle Scholar
  248. 248.
    V. Narayan, N. Menon, S. Ramaswamy J. Stat. Mech.: Theory Exper. 2006, P01005(2006)CrossRefGoogle Scholar
  249. 249.
    V. Narayan, S. Ramaswamy, N. Menon Science 317, 105(2007)ADSCrossRefGoogle Scholar
  250. 250.
    H.-S. Niwa J. Theor. Biol. 171, 123(1994)CrossRefGoogle Scholar
  251. 251.
    B. Nordén, Y. Zolotaryuk, P.L. Christiansen, A.V. Zolotaryuk Phys. Rev. E 65, 011110(2001)ADSCrossRefGoogle Scholar
  252. 252.
    A. Okubo, S.A. Levin, Diffusion and Ecological Problems: Modern Perspectives, 2nd edn., volume 14 of Interdisciplinary Applied Mathematics (Springer, New York, 2001)Google Scholar
  253. 253.
    L. Onsager Ann. N. Y. Acad. Sci. 51, 627(1949)ADSCrossRefGoogle Scholar
  254. 254.
    A. Ordemann, G. Balazsi, F. Moss Physica A: Stat. Mech. Appl. 325, 260(2003)ADSCrossRefGoogle Scholar
  255. 255.
    A. Ordemann, F. Moss, G. Balazsi, in Nonlinear Dynamics and the Spatiotemporal Principles of Biology: Leopoldina Symposium, Vol. 88 (Deutsche Akademie der Naturforscher Leopoldina, 2003), p. 87Google Scholar
  256. 256.
    L.S. Ornstein Proc. Ser. B Phys. Sci. 21, 96(1919)Google Scholar
  257. 257.
    H.G. Othmer, S.R. Dunbar, W. Alt J. Math. Biol. 26, 263(1988)MathSciNetMATHCrossRefGoogle Scholar
  258. 258.
    E. Palsson, H.G. Othmer Proc. National Acad. Sci. USA 97, 10448(2000)ADSCrossRefGoogle Scholar
  259. 259.
    A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd edn. (McGraw-Hill, 1991)Google Scholar
  260. 260.
    J.K. Parrish, L. Edelstein-Keshet Science 284, 99(1999)ADSCrossRefGoogle Scholar
  261. 261.
    J.K. Parrish, W.M. Hamner, Animal Groups in Three Dimensions (Cambridge University Press, 1997)Google Scholar
  262. 262.
    J.K. Parrish, S.V. Viscido, D. Grünbaum Biol. Bull. 202, 296(2002)CrossRefGoogle Scholar
  263. 263.
    R.F. Pawula Phys. Rev. 162, 186(1967)ADSCrossRefGoogle Scholar
  264. 264.
    R.F. Pawula Phys. Rev. A 36, 4996(1987)ADSCrossRefGoogle Scholar
  265. 265.
    W.F. Paxton, K.C. Kistler, C.C. Olmeda, A. Sen, S.K. St. Angelo, Y. Cao, T.E. Mallouk, P.E. Lammert, V.H. Crespi J. Amer. Chem. Soc. 126, 13424(2004)CrossRefGoogle Scholar
  266. 266.
    K. Pearson Nature 72, 294(1905)ADSCrossRefGoogle Scholar
  267. 267.
    K. Pearson Nature 72, 342(1905)ADSCrossRefGoogle Scholar
  268. 268.
    K. Pearson, J. Blakeman, A Mathematical Theory of Random Migration (Dulau and co., 1906)Google Scholar
  269. 269.
    J. Perrin, Brownian Movement and Molecular Reality (Dover Publications, 2005)Google Scholar
  270. 270.
    F. Peruani, From Individual to Collective Motion of Self-Propelled Particles: the Role of Particle Shape, Orientational Ordering and Clustering, Ph.D. thesis, Technische Universität Berlin, Berlin, 2008Google Scholar
  271. 271.
    F. Peruani, A. Deutsch, M. Bär Phys. Rev. E 74, 030904(2006)ADSCrossRefGoogle Scholar
  272. 272.
    F. Peruani, A. Deutsch, M. Bär Eur. Phys. J. Special Topics 157, 111(2008)ADSCrossRefGoogle Scholar
  273. 273.
    F. Peruani, L.G. Morelli Phys. Rev. Lett. 99, 010602(2007)ADSCrossRefGoogle Scholar
  274. 274.
    F. Peruani, L. Schimansky-Geier, M. Bär Eur. Phys. J. Special Topics 191, 173(2011)ADSCrossRefGoogle Scholar
  275. 275.
    F. Peruani, J. Starruss, V. Jakovljevic, L. Søgaard-Andersen, A. Deutsch, M. Bär Phys. Rev. Lett. 108, 098102(2012)ADSCrossRefGoogle Scholar
  276. 276.
    L. Pohlmann, H. Tributsch Electrochim. Acta 42, 2737(1997)CrossRefGoogle Scholar
  277. 277.
    C.M. Pooley, G.P. Alexander, J.M. Yeomans Phys. Rev. Lett. 99, 228103(2007)ADSCrossRefGoogle Scholar
  278. 278.
    K. Przibram Pflüger, Archiv Gesammte Physiol. Mensche. Thiere 153, 401(1913)CrossRefGoogle Scholar
  279. 279.
    K. Przibram Archiv für Entwicklungsmech. Organism. 43, 20(1917)CrossRefGoogle Scholar
  280. 280.
    E.M. Purcell Am. J. Phys. 45, 11(1977)ADSCrossRefGoogle Scholar
  281. 281.
    V.B. Putz, J. Dunkel, J.M. Yeomans Chem. Phys. 375, 557(2010)ADSCrossRefGoogle Scholar
  282. 282.
    S. Ramaswamy Ann. Rev. Condens. Matter Phys. 1, 323(2010)ADSCrossRefGoogle Scholar
  283. 283.
    S. Ramaswamy, R.A. Simha, J. Toner Europhysics Letters (EPL) 62, 196(2003)ADSCrossRefGoogle Scholar
  284. 284.
    J.W.S. Rayleigh, The Theory of Sound, Vol. I, 2nd edn. (MacMillan, London, 1894) erstmalige Erwaehnung des sog. Rayleigh-OszillatorsGoogle Scholar
  285. 285.
    P. Reimann Phys. Rep. 361, 57(2002)MathSciNetADSMATHCrossRefGoogle Scholar
  286. 286.
    C.W. Reynolds Proc. 14th annual conference on Computer graphics and interactive techniques - SIGGRAPH ’87, (1987), p. 25Google Scholar
  287. 287.
    M.J.E. Richardson, W. Gerstner Neural Comput. 17, 923(2005)MathSciNetMATHCrossRefGoogle Scholar
  288. 288.
    M.J.E. Richardson, W. Gerstner Chaos: Interdisciplinary J. Nonlinear Sci. 16, 026106(2006)MathSciNetCrossRefGoogle Scholar
  289. 289.
    A.J. Ridley, M.A. Schwartz, K. Burridge, R.A. Firtel, M.H. Ginsberg, G. Borisy, J.T. Parsons, A.R. Horwitz Science 302, 1704(2003)ADSCrossRefGoogle Scholar
  290. 290.
    I.H. Riedel, K. Kruse, J. Howard Science 309, 300(2005)ADSCrossRefGoogle Scholar
  291. 291.
    T. Riethmüller, L. Schimansky-Geier, D. Rosenkranz, T. Pöschel J. Stat. Phys. 86, 421(1997)ADSMATHCrossRefGoogle Scholar
  292. 292.
    H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer Verlag, 1996)Google Scholar
  293. 293.
    P. Romanczuk, Active Motion and Swarming: From Individual to Collective Dynamics (Logos Verlag, Berlin, 2011)Google Scholar
  294. 294.
    P. Romanczuk, I.D. Couzin, L. Schimansky-Geier Phys. Rev. Lett. 102, 010602(2009)ADSCrossRefGoogle Scholar
  295. 295.
    P. Romanczuk, W. Ebeling, U. Erdmann, L. Schimansky-Geier Chaos: Interdisciplinary J. Nonlinear Sci. 21, 047517(2011)CrossRefGoogle Scholar
  296. 296.
    P. Romanczuk, U. Erdmann Eur. Phy. J. Special Topics 187, 127(2010)ADSCrossRefGoogle Scholar
  297. 297.
    P. Romanczuk, U. Erdmann, H. Engel, L. Schimansky-Geier Eur. Phys. J. Special Topics 157, 61(2008)ADSCrossRefGoogle Scholar
  298. 298.
    P. Romanczuk, L. Schimansky-Geier Phys. Rev. Lett. 106, 230601(2011)ADSCrossRefGoogle Scholar
  299. 299.
    P. Romanczuk, L. Schimansky-Geier, Ecological Complexity (in press) (2011)Google Scholar
  300. 300.
    G. Ruckner, R. Kapral Phys. Rev. Lett. 98, 150603(2007)ADSCrossRefGoogle Scholar
  301. 301.
    J.M. Sancho J. Math. Phys. 25, 354(1984)MathSciNetADSMATHCrossRefGoogle Scholar
  302. 302.
    J.M. Sancho, M.S. Miguel, S.L. Katz, J.D. Gunton Phys. Rev. A 26, 1589(1982)ADSCrossRefGoogle Scholar
  303. 303.
    V. Schaller, C. Weber, C. Semmrich, E. Frey, A.R. Bausch Nature 467, 73(2010)ADSCrossRefGoogle Scholar
  304. 304.
    M. Schienbein, K. Franke, H. Gruler Phys. Rev. E 49, 5462(1994)ADSCrossRefGoogle Scholar
  305. 305.
    M. Schienbein, H. Gruler Bull. Math. Biol. 55, 585(1993)MATHGoogle Scholar
  306. 306.
    L. Schimansky-Geier, W. Ebeling, U. Erdmann Acta Phys. Polonica 36, 1757(2005)ADSGoogle Scholar
  307. 307.
    L. Schimansky-Geier, U. Erdmann, N. Komin Physica A: Stat. Mech. Appl. 351, 51(2005)ADSCrossRefGoogle Scholar
  308. 308.
    L. Schimansky-Geier, M. Mieth, H. Rosè, H. Malchow Phys. Lett. A 207, 140(1995)MathSciNetADSMATHCrossRefGoogle Scholar
  309. 309.
    L. Schimansky-Geier, F. Schweitzer, W. Ebeling, H. Ulbricht,in Self-Organization by Nonlinear Irreversible Processes, edited by W. Ebeling, H. Ulbricht, volume 33 of Springer Series in Synergetics (Springer, Berlin, 1986)Google Scholar
  310. 310.
    T. Schwalger, L. Schimansky-Geier Phys. Rev. E, 77, 031914(2008)ADSCrossRefGoogle Scholar
  311. 311.
    F. Schweitzer, Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences, 1st edn. (Springer, 2003)Google Scholar
  312. 312.
    F. Schweitzer, W. Ebeling, B. Tilch Phys. Rev. Lett. 80, 5044(1998)ADSCrossRefGoogle Scholar
  313. 313.
    F. Schweitzer, W. Ebeling, B. Tilch Phys. Rev. E 64, 021110(2001)ADSCrossRefGoogle Scholar
  314. 314.
    F. Schweitzer, K. Lao, F. Family Biosystems 41, 153(1997)CrossRefGoogle Scholar
  315. 315.
    F. Schweitzer, L. Schimansky-Geier Phys. A 206, 359(1994)CrossRefGoogle Scholar
  316. 316.
    F. Schweitzer, L. Schimansky-Geier, W. Ebeling, H. Ulbricht Physica A: Stat. Mech. Appl. 150, 261(1988)ADSCrossRefGoogle Scholar
  317. 317.
    F. Schweitzer, L. Schimansky-Geier, W. Ebeling, H. Ulbricht Physica A: Stat. Theor. Phys. 153, 573(1988)ADSCrossRefGoogle Scholar
  318. 318.
    D. Selmeczi, L. Li, L. I.I. Pedersen, S.F. Nørrelykke, P.H. Hagedorn, S. Mosler, N.B. Larsen, E.C. Cox, H. Flyvbjergm Eur. Phys. J. Special Topics 157, 1(2008)ADSCrossRefGoogle Scholar
  319. 319.
    D. Selmeczi, S. Mosler, P.H. Hagedorn, N.B. Larsen, H. Flyvbjerg Biophys. J. 89, 912(2005)CrossRefGoogle Scholar
  320. 320.
    V.E. Shapiro, V.M. Loginov Physica A: Stat. Theor. Phys. 91, 563(1978)MathSciNetADSCrossRefGoogle Scholar
  321. 321.
    A.D. Shenderov, M.P. Sheetz Biophys. J. 72, 2382(1997)ADSCrossRefGoogle Scholar
  322. 322.
    R.A. Simha, S. Ramaswamy Phys. A 306, 262(2002)MATHCrossRefGoogle Scholar
  323. 323.
    R. Aditi Simha, Sriram Ramaswamy Phys. Rev. Lett. 89, 058101(2002)ADSCrossRefGoogle Scholar
  324. 324.
    S.J. Simpson, G.A. Sword, P.D. Lorch, I.D. Couzin PNAS 103, 4152(2006)ADSCrossRefGoogle Scholar
  325. 325.
    M. Von Smoluchowski Z. Phys. Chem. 92, 129(1917)Google Scholar
  326. 326.
    A. Sokolov, I.S. Aranson, J.O. Kessler, R.E. Goldstein Phys. Rev. Lett. 98, 158102(2007)ADSCrossRefGoogle Scholar
  327. 327.
    A. Sokolov, R.E. Goldstein, F.I. Feldchtein, I.S. Aranson Phys. Rev. E 80, 031903(2009)ADSCrossRefGoogle Scholar
  328. 328.
    Y. Sowa, R.M. Berry Quart. Rev. Biophys. 41, 103(2008)CrossRefGoogle Scholar
  329. 329.
    A.J. Steele, M.R. Tinsley, K. Showalter, Chaos 18, 026108-1-8 (2008)MathSciNetADSCrossRefGoogle Scholar
  330. 330.
    O. Steuernagel, W. Ebeling, V. Calenbuhr Chaos, Solitons & Fractals 4, 1917(1994)ADSMATHCrossRefGoogle Scholar
  331. 331.
    J. Strefler, W. Ebeling, E. Gudowska-Nowak, L. Schimansky-Geier Eur. Phys. J. B 72, 597(2009)ADSMATHCrossRefGoogle Scholar
  332. 332.
    J. Strefler, U. Erdmann, L. Schimansky-Geier Phys. Rev. E 78, 031927(2008)ADSCrossRefGoogle Scholar
  333. 333.
    J.R. Strickler Philosoph. Trans. Royal Soc. London. Series B: Biol. Sci. 353, 671(1998)CrossRefGoogle Scholar
  334. 334.
    Y. Sumino, N. Magome, T. Hamada, K. Yoshikawa Phys. Rev. Lett. 94, 068301(2005)ADSCrossRefGoogle Scholar
  335. 335.
    D.J.T. Sumpter, Collective Animal Behavior (Princeton University Press, 2010)Google Scholar
  336. 336.
    D.J.T. Sumpter, J. Buhl, D. Biro, I.D. Couzin Theory Biosci. 127, 177(2008)CrossRefGoogle Scholar
  337. 337.
    S. Sundararajan, P.E. Lammert, A.W. Zudans, V.H. Crespi, A. Sen Nano Lett. 8, 1271(2008)ADSCrossRefGoogle Scholar
  338. 338.
    W. Sutherland Philosoph. Mag. Ser. 9, 781(1905)CrossRefGoogle Scholar
  339. 339.
    R. Suzuki, S. Sakai Biophysics 13, 281(1973)CrossRefGoogle Scholar
  340. 340.
    B. Szabó, G.J. Szöllösi, B. Gönci, Zs. Jurányi, D. Selmeczi, Tamás Vicsek Phys. Rev. E 74, 061908(2006)ADSCrossRefGoogle Scholar
  341. 341.
    Y.-G. Tao, R. Kapral J. Chem. Phys. 128, 164518(2008)ADSCrossRefGoogle Scholar
  342. 342.
    G. Taylor Proc. Royal Soc. London. Series A. Math. Phys. Sci. 219, 186(1953)ADSCrossRefGoogle Scholar
  343. 343.
    G. Taylor Proc. Royal Soc. London. Series A. Math. Phys. Sci. 225, 473(1954)ADSCrossRefGoogle Scholar
  344. 344.
    G. Taylor Proc. Royal Soc. London. Series A. Math. Phys. Sci. 223, 446(1954)ADSCrossRefGoogle Scholar
  345. 345.
    G.I. Taylor Proc. London Math. Soc. 20, 196(1921)MATHCrossRefGoogle Scholar
  346. 346.
    P. Tierno, R. Albalat, F. Sagués Small 6, 1749(2010)CrossRefGoogle Scholar
  347. 347.
    M.R. Tinsley, A.J. Steele, K. Showalter Eur. Phys. J. Special Topics 165, 161(2008)ADSCrossRefGoogle Scholar
  348. 348.
    J. Toner, Y. Tu Phys. Rev. Lett. 75, 4326(1995)ADSCrossRefGoogle Scholar
  349. 349.
    J. Toner, Y. Tu Phys. Rev. E 58, 4828(1998)MathSciNetADSCrossRefGoogle Scholar
  350. 350.
    J. Toner, Y. Tu, S. Ramaswamy Ann. Phys. 318, 170(2005)MathSciNetADSMATHCrossRefGoogle Scholar
  351. 351.
    C. Touya, T. Schwalger, B. Lindner Phys. Rev. E 83, 051913(2011)ADSCrossRefGoogle Scholar
  352. 352.
    T. Toyota, N. Maru, M.M. Hanczyc, T. Ikegami, T. Sugawara J. Amer. Chem. Soc. 131, 5012(2009)CrossRefGoogle Scholar
  353. 353.
    R. Tyson, S.R. Lubkin, J.D. Murray Proc. Royal Soc. London B 266, 299(1999) hat Bilder wie meine selbstkons. Feld-SimulationenCrossRefGoogle Scholar
  354. 354.
    R. Tyson, S.R Lubkin, J.D. Murray J. Math. Biol. 38, 359(1999)MathSciNetMATHCrossRefGoogle Scholar
  355. 355.
    G.E. Uhlenbeck, L.S. Ornstein Phys. Rev. 36, 823(1930)ADSCrossRefGoogle Scholar
  356. 356.
    L.F. Valadares, Y.G. Tao, N.S. Zacharia, V. Kitaev, F. Galembeck, R. Kapral, G.A. Ozin Small 6, 565(2010)CrossRefGoogle Scholar
  357. 357.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry, 2nd edn. (North-Holland, Amsterdam, 1992)Google Scholar
  358. 358.
    S. van Teeffelen, H. Löwen Phys. Rev. E 78, 020101(2008)CrossRefGoogle Scholar
  359. 359.
    F. Vanni, M. Luković, P. Grigolini. Phys. Rev. Lett. 107, 078103(2011)ADSCrossRefGoogle Scholar
  360. 360.
    T. Vicsek, Fluctuations and Scaling in Biology (Oxford University Press, 2001)Google Scholar
  361. 361.
    T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, O. Shochet Phys. Rev. Lett. 75, 1226(1995)ADSCrossRefGoogle Scholar
  362. 362.
    T. Vicsek, A. Zafiris, Arxiv preprint [arXiv:1010.5017], (2010)Google Scholar
  363. 363.
    J. Vollmer, A.G. Vegh, C. Lange, B. Eckhardt Phys. Rev. E 73, 061924(2006)ADSCrossRefGoogle Scholar
  364. 364.
    M. von Smoluchowski Annal. Phys. 326, 756(1906)ADSCrossRefGoogle Scholar
  365. 365.
    G.H. Wadhams, J.P. Armitage Nat. Rev. Mol. Cell. Biol. 5, 1024(2004)CrossRefGoogle Scholar
  366. 366.
    C. Weber, Diffusion of Active Particles in 2D (Master-Thesis, Humboldt Universität zu Berlin, 2010)Google Scholar
  367. 367.
    C. Weber, P.K. Radtke L. Schimansky-Geier, P. Hnggi Phys. Rev. E 84, 011132(2011)ADSCrossRefGoogle Scholar
  368. 368.
    R. Welch, D. Kaiser Proc. National Acad. Sci. USA 98, 14907(2001)ADSCrossRefGoogle Scholar
  369. 369.
    H.H. Wensink, H. Löwen Phys. Rev. E 78, 031409(2008)ADSCrossRefGoogle Scholar
  370. 370.
    C. Wiener Poggendorffs Annal. 118, 79(1863)Google Scholar
  371. 371.
    D.E. Woodward, R. Tyson, M.R. Myerscough, J.D. Murray, E.O. Budrene, H.C. Berg Biophys. J. 68, 2181(1995), mathematisches ModellADSCrossRefGoogle Scholar
  372. 372.
    X.-L. Wu, A. Libchaber Phys. Rev. Lett. 84, 3017(2000)ADSCrossRefGoogle Scholar
  373. 373.
    T.D. Yang, J.-S. Park, Y. Choi, W. Choi, T.-W. Ko, K.J. Lee PLoS ONE 6, e20255(2011)CrossRefGoogle Scholar
  374. 374.
    Y. Yang, J. Elgeti, G. Gompper Phys. Rev. E 78, 061903(2008)ADSCrossRefGoogle Scholar
  375. 375.
    Y. Yang, V. Marceau, G. Gompper Phys. Rev. E 82, 31904(2010)ADSCrossRefGoogle Scholar
  376. 376.
    J.H. Yao, K.R Elder, H. Guo, M. Grant Phys. Rev. B 47, 14110(1993)ADSCrossRefGoogle Scholar
  377. 377.
    C.A. Yates, R. Erban, C. Escudero, I.D. Couzin, J. Buhl, I.G. Kevrekidis, P.K. Maini, D.J.T. Sumpter Proc. Natl. Acad. Sci. USA 106, 5464(2009)ADSCrossRefGoogle Scholar
  378. 378.
    M. Zabicki, W. Ebeling, E. Gudowska-Nowak Chem. Phys. (2010)Google Scholar
  379. 379.
    M. Zabicki, E. Gudowska-Nowak, W. Ebeling Acta Phys. Polonica B 41, 1181(2010)Google Scholar
  380. 380.
    V. Zaburdaev, S. Uppaluri T. Pfohl, M. Engstler, R. Friedrich, H. Stark Phys. Rev. Lett. 106, 208103(2011)ADSCrossRefGoogle Scholar
  381. 381.
    H.P. Zhang, A. Be’er, E.-L. Florin, H.L. Swinney Proc. National Acad. Sci. 107, 13626(2010)ADSCrossRefGoogle Scholar
  382. 382.
    D.R. Zusman, A.E. Scott, Z. Yang, J.R. Kirby Nat. Rev. Micro. 5, 862(2007)CrossRefGoogle Scholar
  383. 383.
    R. Zwanzig, J. Stat. Phys. (1973)Google Scholar
  384. 384.
    R. Zygadlo Phys. Rev. E 47, 4067(1993)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2012

Authors and Affiliations

  • P. Romanczuk
    • 1
  • M. Bär
    • 2
  • W. Ebeling
    • 3
  • B. Lindner
    • 3
  • L. Schimansky-Geier
    • 3
  1. 1.Max-Planck-Institute for the Physics of Complex Systems DresdenDresdenGermany
  2. 2.Physikalisch-Technische BundesanstaltBerlinGermany
  3. 3.Institute of PhysicsHumboldt Universität zu BerlinBerlinGermany

Personalised recommendations