Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations

Physica A: Statistical Mechanics and its Applications

Volumn 331, Issue 3-4, 2004, Pages 391-408

  Frank, T D ^a  

^a UNIVERSITY OF MÜNSTER   (Germany)

Author keywords

Linear and nonlinear families of Markov processes; Nonlinear Fokker Planck equations; Stochastic feedback

Indexed keywords

DIFFUSION; FEEDBACK; NONLINEAR EQUATIONS; PROBABILITY DENSITY FUNCTION; QUANTUM THEORY;

FOKKER-PLANCK EQUATIONS;

MARKOV PROCESSES;

EID: 0242522252     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physa.2003.09.056     Document Type: Article

References (86)

1. Nicholson D.R. Introduction to Plasma Theory. 1983;Wiley, New York.
2. Takai M., Akiyama H., Takeda S. J. Phys. Soc. Japan. 50:1981;1716.
3. Soler M., Martinez F.C., Donoso J.M. J. Stat. Phys. 69:1992;813.
4. Spohn H. J. Phys. I France. 3:1993;69.
5. Marsili M., Bray A.J. Phys. Rev. Lett. 76:1996;2750.
6. Giada L., Marsili M. Phys. Rev. E. 62:2000;6015.
7. Okubo A. Diffusion and Ecological Problems: Mathematical Models. 1980;Springer, Berlin.
8. Winfree A.T. J. Theor. Biol. 16:1967;15.
9. Winfree A.T. The Geometry of Biological Time. 2nd Edition:2001;Springer, Berlin.
10. Strogatz S.H. Physica D. 143:2000;1.
11. Strogatz S.H., Stewart I. Sci. Am. 269(6):1993;68.
12. Daido H. Phys. Rev. Lett. 87:2001;048 101.
13. Schweitzer F., Ebeling W., Tilch B. Phys. Rev. E. 64:2001;021 110.
14. Primak S. Phys. Rev. E. 61:2000;100.
15. Primak S., Lyandres V., Kontorovich V. Phys. Rev. E. 63:2001;061 103.
16. W. Gerstner, J.L. van Hemmen, in: E. Domany, J.L. van Hemmen, K. Schulten (Eds.), Models of Neural Networks II: Temporal Aspects of Coding and Information Processing in Biological Systems, Springer, Berlin, 1994, pp. 1-93.
17. Kuramoto Y. Chemical Oscillations, Waves, and Turbulence. 1984;Springer, Berlin.
18. M.G. Kuzmina, E.A. Manykin, I.I. Surina, in: J. Mira, F. Sandoval (Eds.), From Natural to Artificial Neural Computation, Springer, Berlin, 1995, pp. 246-251.
19. Schuster H.G., Wagner P. Biol. Cybern. 64:1990;77.
20. Sompolinsky H., Golomb D., Kleinfeld D. Phys. Rev. A. 43:1991;6990.
21. Tass P.A. Phys. Rev. E. 66:2002;036 226.
22. Tass P.A. Phase Resetting in Medicine and Biology - Stochastic Modelling and Data Analysis. 1999;Springer, Berlin.
23. Yamana M., Shiino M., Yoshioka M. J. Phys. A: Math. Gen. 32:1999;3525.
24. Yoshioka M., Shiino M. Phys. Rev. E. 61:2000;4732.
25. D.G. Aronson, in: A. Dobb, B. Eckmann (Eds.), Nonlinear Diffusion Problems - Lecture Notes in Mathematics, Vol. 1224, Springer, Berlin, 1986, pp. 1-46.
26. Barenblatt G.I., Entov V.M., Ryzhik V.M. Theory of Fluid Flows Through Natural Rocks. 1990;Kluwer Academic Publishers, Dordrecht.
27. Crank J. The Mathematics of Diffusion. 1975;Clarendon Press, Oxford.
28. Logan J.D. Transport Modeling in Hydrogeochemical Systems. 2001;Springer, Berlin.
29. L.A. Peletier, in: H. Amann, N. Bazley, K. Kirchgässner (Eds.), Applications of Nonlinear Analysis in the Physical Science, Pitman Advanced Publishing Program, Boston, 1981, pp. 229-241.
30. Öttinger H.C. Stochastic Processes in Polymeric Fluids. 1996;Springer, Berlin.
31. Wedgewood L.E. J. Non-Newtonian Fluid Mech. 31:1989;127.
32. Zylka W., Öttinger H.C. J. Chem. Phys. 90:1989;474.
33. Kozyreff G., Vladimirov A.G., Mandel P. Phys. Rev. Lett. 85:2000;3809.
34. Garcia-Ojalvo J., Sancho J.M. Noise in Spatially Extended Systems. 1999;Springer, New York.
35. Parrondo J.M.R., van den Broeck C., Buceta J., de la Rubia F.J. Physica A. 224:1996;153.
36. Zaikin A.A., Schimansky-Geier L. Phys. Rev. E. 58:1998;4355.
37. T.D. Frank, P.J. Beek, in: W. Tschacher, J.P. Dauwalder (Eds.), The Dynamical Systems Approach to Cognition, World Scientific, Singapore, 2003, pp. 159-179.
38. Borland L. Phys. Rev. Lett. 89:2002;098 701.
39. Borland L. Phys. Rev. E. 57:1998;6634.
40. Borland L., Pennini F., Plastino A.R., Plastino A. Euro. Phys. J. B. 12:1999;285.
41. Curado E.M.F., Nobre F.D. Phys. Rev. E. 67:2003;021 107.
42. Frank T.D., Daffertshofer A. Physica A. 272:1999;497.
43. Frank T.D. Phys. Lett. A. 267:2000;298.
44. Frank T.D., Daffertshofer A. Physica A. 285:2000;351.
45. Frank T.D., Daffertshofer A. Physica A. 295:2001;455.
46. Frank T.D. Physica A. 301:2001;52.
47. Frank T.D. Phys. Lett. A. 290:2001;93.
48. Frank T.D. J. Math. Phys. 43:2002;344.
49. Giordano C., Plastino A.R., Casas M., Plastino A. Eur. Phys. J. B. 22:2001;361.
50. Kadanoff L.P. Statistical Physics: Statics, Dynamics and Renormalization. 2000;World Scientific, Singapore.
51. Kaniadakis G., Quarati P. Phys. Rev. E. 48:1993;4263.
52. Kaniadakis G., Quarati P. Phys. Rev. E. 49:1994;5103.
53. Kaniadakis G., Lavagno A., Quarati P. Nucl. Phys. B. 466:1996;527.
54. Kaniadakis G. Physica A. 296:2001;405.
55. Kaniadakis G. Phys. Lett. A. 288:2001;283.
56. Lapenta G., Kaniadakis G., Quarati P. Physica A. 225:1996;323.
57. Lenzi E.K., Anteneodo C., Borland L. Phys. Rev. E. 63:2001;051 109.
58. Malacarne L.C., Mendes R.S., Pedron I.T., Lenzi E.K. Phys. Rev. E. 63:2001;030 101.
59. Malacarne L.C., Mendes R.S., Pedron I.T., Lenzi E.K. Phys. Rev. E. 65:2002;052 101.
60. Martinez S., Plastino A.R., Plastino A. Physica A. 259:1998;183.
61. Plastino A.R., Plastino A. Physica A. 222:1995;347.
62. Shiino M. J. Math. Phys. 42:2001;2540.
63. Shiino M. J. Math. Phys. 43:2002;2654.
64. Shiino M. J. Korean Phys. Soc. 40:2002;1037.
65. Tsallis C., Bukman D.J. Phys. Rev. E. 54:1996;R2197.
66. Tsallis C. J. Stat. Phys. 52:1988;479.
67. Tsallis C. Braz. J. Phys. 29:1999;1.
68. Abe S., Okamoto Y. Nonextensive Statistical Mechanics and its Applications. 2001;Springer, Berlin.
69. Frank T.D. Phys. Lett. A. 305:2002;150.
70. Haken H. Synergetics. An Introduction. 1977;Springer, Berlin.
71. Risken H. The Fokker-Planck Equation - Methods of Solution and Applications. 1989;Springer, Berlin.
72. T.D. Frank, A. Daffertshofer, P.J. Beek, Stochastic Processes with Statistical Feedback, 1998, unpublished manuscript.
73. Frank T.D., Daffertshofer A., Beek P.J. J. Biol. Phys. 28:2002;39.
74. Wehner M.F., Wolfer W.G. Phys. Rev. A. 35:1987;1795.
75. Wehner M.F., Wolfer W.G. Phys. Rev. A. 27:1983;2663.
76. Graham R. Z. Physik B. 26:1976;281.
77. Haken H. Z. Physik B. 24:1976;321.
78. Horsthemke W., Bach A. Z. Physik B. 22:1975;189.
79. Frank T.D. Physica A. 320:2003;204.
80. Horsthemke W., Lefever R. Noise-Induced Transitions. 1984;Springer, Berlin.
81. M. Iosifescu, P. Tautu, Stochastic Processes and its Applications in Biology and Medicine, Vol. I, Springer, Berlin, 1973.
82. E.B. Dynkin, Markov Processes, Vol. I, Springer, Berlin, 1965.
83. Shimizu H. Prog. Theor. Phys. 52:1974;329.
84. Shimizu H., Yamada T. Prog. Theor. Phys. 47:1972;350.
85. Zhang D.S., Wei G.W., Kouri D.J., Hoffman D.K. Phys. Rev. E. 56:1997;1197.
86. Hänggi P., Thomas H. Z. Physik B. 26:1977;85.