Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

European Physical Journal B

Volumn 62, Issue 2, 2008, Pages 179-208

  Chavanis, P H ^a  

^a Lab de Physique Théorique   (France)

Author keywords

[No Author keywords available]

Indexed keywords

BACTERIAL POPULATIONS;

BACTERIA; BOSE-EINSTEIN CONDENSATION; ELECTROLYTES; MATHEMATICAL MODELS; POROUS MATERIALS; THERMODYNAMICS;

NONLINEAR EQUATIONS;

EID: 42149190237     PISSN: 14346028     EISSN: 14346036     Source Type: Journal    
DOI: 10.1140/epjb/e2008-00142-9     Document Type: Article

References (93)

1. A.D. Fokker, Ann. Physik 43, 810 (1914)
2. M. Planck, Sitzber. Preuss. Akad. Wiss. 324 (1917)
3. A. Einstein, Ann. Physik 17, 549 (1905)
4. M. von Smoluchowski, Ann. Physik 48, 1103 (1915)
5. H.A. Kramers, Physica A 7, 284 (1940)
6. O. Klein, Arkiv for Mathematik, Astronomi, och Fysik 16, 5 (1921)
7. H. Risken, The Fokker-Planck equation (Springer, 1989)
8. P.H. Chavanis, C.R. Physique 7, 318 (2006)
9. C. Tsallis, J. Stat. Phys. 52, 479 (1988)
10. S. Abe, Phys. Lett. A 224, 326 (1997)
11. E.P. Borges, I. Roditi, Phys. Lett. A 246, 399 (1998)
12. G. Kaniadakis, Physica A 296, 405 (2001)
13. J. Naudts, Physica A 340, 32 (2004)
14. G. Kaniadakis, M. Lissia, A.M. Scarfone, Phys. Rev. E 71, 046128 (2005)
15. J. Harvda, F. Charvat, Kybernetica 3, 30 (1967)
16. D.P. Mittal, Metrika 22, 35 (1975)
17. B.D. Sharma, I.J. Taneja, Metrika 22, 205 (1975)
18. A. Reyni, Probability Theory (North-Holland, Amsterdam, 1970)
19. B.D. Sharma, D.P. Mittal, J. Math. Sci. 1, 28 (1975)
20. G. Kaniadakis, M. Lissia, Physica A 340, xv-xix (2004)
21. A.R. Plastino, A. Plastino, Physica A 222, 347 (1995)
22. H. Spohn, J. Phys. France 3, 69 (1993)
23. C. Tsallis, D.J. Bukman, Phys. Rev. E 54, R2197 (1996)
24. D. Stariolo, Phys. Rev. E 55, 4806 (1997)
25. L. Borland, Phys. Rev. E 57, 6634 (1998)
26. F. Nobre, E. Curado, G. Rowlands, Physica A 334, 109 (2004)
27. G. Kaniadakis, P. Quarati Phys. Rev. E 49, 5103 (1994)
28. P.H. Chavanis, J. Sommeria, R. Robert, Astrophys. J. 471, 385 (1996)
29. D. Lynden-Bell, Mon. Not. Roy. Astr. Soc. 136, 101 (1967)
30. J. Sopik, C. Sire, P.H. Chavanis, Phys. Rev. E 74, 011112 (2006)
31. S. Martinez, A.R. Plastino, A. Plastino, Physica A 259, 183 (1998)
32. T.D. Frank, Physica A 310, 397 (2002)
33. P.H. Chavanis, Phys. Rev. E 68, 036108 (2003)
34. P.H. Chavanis, Physica A 332, 89 (2004)
35. E. Curado, F. Nobre, Phys. Rev. E 67, 021107 (2003)
36. T.D. Frank, Nonlinear Fokker-Planck Equations: Fundamentals and Applications (Springer-Verlag, 2005)
37. Dynamics and thermodynamics of systems with long range interactions, edited by T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens, Lect. Not. in Phys. 602 (Springer, 2002)
38. Y. Kuramoto, Chemical oscillations, waves, and turbulence (Springer, Berlin, 1984)
39. N. Martzel, C. Aslangul, J. Phys. A 34, 11225 (2001)
40. P.H. Chavanis, Physica A 361, 55 (2006); P.H. Chavanis, Physica A 361, 81 (2006)
41. P.H. Chavanis, C. Rosier, C. Sire, Phys. Rev. E 66, 036105 (2002)
42. C. Sire, P.H. Chavanis, Phys. Rev. E 66, 046133 (2002)
43. P.H. Chavanis, J. Vatteville, F. Bouchet, Eur. Phys. J. B 46, 61 (2005)
44. M. Antoni, S. Ruffo, Phys. Rev. E 52, 2361 (1995)
45. P.H. Chavanis, C. Sire, Phys. Rev. E 69, 016116 (2004)
46. M. Shiino, Phys. Rev. E 67, 056118 (2003)
47. J.D. Murray, Mathematical Biology (Springer, Berlin, 1991)
48. C.S. Patlak, Bull. of Math. Biophys. 15, 311 (1953)
49. E.F. Keller, L.A. Segel, J. Theor. Biol. 30, 225 (1971)
50. H. Othmer, A. Stevens, SIAM J. Appl. Math. 57, 1044 (1997)
51. P.H. Chavanis, C. Sire, Physica A 384, 199 (2007)
52. D. Horstmann, Jahresber. Deutsch. Math. Verein. 106, 51 (2004)
53. P.H. Chavanis, C. Sire, Physica A 387, 1999 (2008)
54. P.H. Chavanis, P. Laurençot, M. Lemou, Physica A 341, 145 (2004)
55. P.H. Chavanis, [arXiv:0803.0263]
56. C. Sire, P.H. Chavanis, Phys. Rev. E 69, 066109 (2004)
57. P.H. Chavanis, C. Sire, Phys. Rev. E 70, 026115 (2004)
58. P.H. Chavanis, C. Sire, Phys. Rev. E 73, 066103 (2006)
59. P.H. Chavanis, C. Sire, Phys. Rev. E 73, 066104 (2006)
60. P. Debye, E. Hückel, Phys. Z. 24, 305 (1923)
61. R. Robert, J. Sommeria, Phys. Rev. Lett. 69, 2776 (1992)
62. P.H. Chavanis, Physica D 200, 257 (2005)
63. J.M. Burgers, The Nonlinear Diffusion Equation (Riedel, Boston, 1974)
64. A. Stevens, SIAM J. Appl. Math. 61, 183 (2000)
65. T.J. Newman, R. Grima, Phys. Rev. E 70, 051916 (2004)
66. P.H. Chavanis, M. Ribot, C. Rosier, C. Sire, Banach Center Publ. 66, 103 (2004)
67. P.H. Chavanis, C. Sire, Physica A 384, 199 (2007)
68. R. Grima, Curr. Topics. Dev. Bio 81, 435 (2008)
69. P.H. Chavanis, Physica A 359, 177 (2006)
70. D.E. McLaughlin, R.E. Pudritz, Astrophys. J. 476, 750 (1997)
71. P.H. Chavanis, C. Sire, Physica A 375, 140 (2007)
72. T. Hillen, K. Painter, Adv. Appl. Math. 26, 280 (2001)
73. P.H. Chavanis, Eur. Phys. J. B 54, 525 (2006)
74. P.H. Chavanis, Banach Center Publ. 66, 79 (2004)
75. J. Paret, P. Tabeling, Phys. Fluids 10, 3126 (1998)
76. M. Rivera, X.L. Wu, Phys. Rev. Lett. 85, 976 (2000)
77. P.H. Chavanis, A&A 356, 1089 (2000)
78. A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. di Talia, E. Giraudo, G. Serini, L. Preziosi, F. Bussolino, Phys. Rev. Lett. 90, 118101 (2003)
79. F. Filbet, P. Laurençot, B. Perthame, J. Math. Biol. 50, 189 (2005)
80. P.H. Chavanis, C. Sire, Physica A 356, 419 (2005)
81. Du Jiulin, Astrophys. Space Sci. 306, 247 (2006)
82. P.H. Chavanis, Phys. Rev. E 69, 066126 (2004)
83. P. Biler, P. Laurençot, T. Nadzieja, Adv. Differential Equations 9, 563 (2004)
84. A. Einstein, Ann. Physik 33, 1275 (1910)
85. E.G.D. Cohen, Physica A 305, 19 (2002)
86. J. Dolbeault, P. Markowich, D. Ölz, C. Schmeiser, Arch. Rational Mech. Anal. 1, 133 (2007)
87. P.H. Chavanis, C. Sire, [arXiv:0708.3163]
88. P.H. Chavanis, Eur. Phys. J. B 52, 433 (2006)
89. F. Bouchet, [arXiv:0710.5094]
90. P.H. Chavanis, AIP Conf. Proc. 970, 39 (2008)
91. D.D. Holm, J.E. Marsden, T. Ratiu, A. Weinstein, Phys. Rep. 123, 1 (1985)
92. P.H. Chavanis, A&A 451, 109 (2006)
93. A.S. Eddington, MNRAS 76, 572 (1916)