Table of Contents
ISRN Mathematical Physics
Volume 2013, Article ID 149169, 28 pages
http://dx.doi.org/10.1155/2013/149169
Review Article

Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

Center for the Ecological Study of Perception and Action, Department of Psychology, University of Connecticut, 406 Babbidge Road, Storrs, CT 06269, USA

Received 14 January 2013; Accepted 13 February 2013

Academic Editors: G. Cleaver, G. Goldin, and D. Singleton

Copyright © 2013 T. D. Frank. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. P. McKean Jr., “A class of Markov processes associated with nonlinear parabolic equations,” Proceedings of the National Academy of Sciences of the United States of America, vol. 56, pp. 1907–1911, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. H. P. McKean Jr., “Propagation of chaos for a class of nonlinear parabolic equations,” in Lectures in Differential Equations, A. K. Aziz, Ed., vol. 2, pp. 177–193, Van Nostrand Reinhold Company, New York, NY, USA, 1969. View at Google Scholar
  3. J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford, UK, 1975. View at MathSciNet
  4. C. W. Gardiner, Handbook of Stochastic Methods, Springer, Berlin, Germany, 1985. View at MathSciNet
  5. S. Karlin and H. M. Taylor, A first course in stochastic processes, Academic Press, New York, NY, USA, 1975. View at MathSciNet
  6. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Springer, Berlin, Germany, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  7. S. Méléard, “Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models,” in Probabilistic Models for Nonlinear Partial Differential Equations, C. Graham, T. G. Kurtz, S. Meleard, P. E. Potter, M. Pulvirenti, and D. Talay, Eds., vol. 1627, pp. 42–95, Springer, Berlin, Germany, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. F. Wehner and W. G. Wolfer, “Numerical evaluation of path-integral solutions to Fokker-Planck equations. III. Time and functionally dependent coefficients,” Physical Review A, vol. 35, no. 4, pp. 1795–1801, 1987. View at Publisher · View at Google Scholar · View at Scopus
  9. T. D. Frank, Nonlinear Fokker-Planck Equations, Springer, Berlin, Germany, 2005. View at MathSciNet
  10. P. J. Diggle, Time Series: A Biostatistical Introduction, The Clarendon Press, Oxford, UK, 1990. View at MathSciNet
  11. M. B. Priestley, Nonlinear and Nonstationary Time Series Analysis, Academic Press, London, UK, 1988. View at MathSciNet
  12. T. D. Frank, “Markov chains of nonlinear Markov processes and an application to a winner-takes-all model for social conformity,” Journal of Physics A, vol. 41, no. 28, Article ID 282001, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. T. D. Frank, “Nonlinear Markov processes,” Physics Letters A, vol. 372, no. 25, pp. 4553–4555, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. E. M. F. Curado and F. D. Nobre, “Derivation of nonlinear Fokker-Planck equations by means of approximations to the master equation,” Physical Review E, vol. 67, no. 2, Article ID 021107, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  15. T. D. Frank, “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations,” Physica A, vol. 331, no. 3-4, pp. 391–408, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  16. D. S. Zhang, G. W. Wei, D. J. Kouri, and D. K. Hoffman, “Numerical method for the nonlinear Fokker-Planck equation,” Physical Review E, vol. 62, no. 1, pp. 3167–3172, 1997. View at Google Scholar · View at Scopus
  17. T. D. Frank, “Deterministic and stochastic components of nonlinear Markov models with an application to decision making during the bailout votes 2008 (USA),” European Physical Journal B, vol. 70, no. 2, pp. 249–255, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. T. D. Frank, “Chaos from nonlinear Markov processes: why the whole is different from the sum of its parts,” Physica A, vol. 388, no. 19, pp. 4241–4247, 2009. View at Publisher · View at Google Scholar · View at Scopus
  19. R. M. May, “Simple mathematical models with very complicated dynamics,” Nature, vol. 261, no. 5560, pp. 459–467, 1976. View at Google Scholar · View at Scopus
  20. H. G. Schuster, Deterministic Chaos: An Introduction, VCH Verlagsgesellschaft, Weinheim, Germany, 1988. View at MathSciNet
  21. S. Galam and J.-D. Zucker, “From individual choice to group decision-making,” Physica A, vol. 287, no. 3-4, pp. 644–659, 2000. View at Publisher · View at Google Scholar
  22. B. Latane and A. Nowak, “Measuring emergent social phenomena: dynamism, polarization, and clustering as order parameters of social systems,” Behavioral Science, vol. 39, pp. 1–24, 1994. View at Google Scholar
  23. F. Schweitzer and J. A. Hołyst, “Modelling collective opinion formation by means of active Brownian particles,” European Physical Journal B, vol. 15, no. 4, pp. 723–732, 2000. View at Google Scholar
  24. M. Takatsuji, “An information theoretical approach to a system of interacting elements,” Biological Cybernetics, vol. 17, no. 4, pp. 207–210, 1975. View at Google Scholar
  25. W. A. Bousfield and C. H. Sedgewick, “An analysis of restricted associative responses,” Journal of General Psychology, vol. 30, pp. 149–165, 1944. View at Google Scholar
  26. J. Wixted and D. Rohrer, “Analyzing the dynamics of free recall: an integrative review of the empirical literature,” Psychonomic Bulletin and Review, vol. 1, pp. 89–106, 1994. View at Google Scholar
  27. T. D. Frank and T. Rhodes, “Micro-dynamic associated with two-state nonlinear Markov processes: with an application to free recall,” Fluctuation and Noise Letters, vol. 10, no. 1, pp. 41–58, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  28. T. Rhodes and M. T. Turvey, “Human memory retrieval as Lévy foraging,” Physica A, vol. 385, no. 1, pp. 255–260, 2007. View at Publisher · View at Google Scholar · View at Scopus
  29. J. P. Crutchfield and O. Görnerup, “Objects that make objects: the population dynamics of structural complexity,” Journal of the Royal Society Interface, vol. 3, no. 7, pp. 345–349, 2006. View at Publisher · View at Google Scholar · View at Scopus
  30. O. Gonerup and J. P. Crutchfield, “Hierachical self-organization in the finitary process soup,” Artificial Life, vol. 14, pp. 245–254, 2008. View at Google Scholar
  31. T. D. Frank, “Stochastic processes and mean field systems defined by nonlinear Markov chains: an illustration for a model of evolutionary population dynamics,” Brazilian Journal of Physics, vol. 41, no. 2, pp. 129–134, 2011. View at Publisher · View at Google Scholar · View at Scopus
  32. T. D. Frank, “Nonlinear Markov processes: deterministic case,” Physics Letters A, vol. 372, no. 41, pp. 6235–6239, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. D. Brogioli, “Marginally stable chemical systems as precursors of life,” Physical Review Letters, vol. 105, no. 5, Article ID 058102, 2010. View at Publisher · View at Google Scholar · View at Scopus
  34. D. Brogioli, “Marginally stability in chemical systems and its relevance in the origin of life,” Physical Review E, vol. 84, Article ID 031931, 2011. View at Google Scholar
  35. T. D. Frank, “Nonlinear physics approach to DNA cross-replication: marginal stability, generalized logistic growth, and impacts of degradation,” Physics Letters A, vol. 375, pp. 3851–3857, 2011. View at Google Scholar
  36. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, Germany, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  37. H. Daido, “Discrete-time population dynamics of interacting self-oscillators,” Progress of Theoretical Physics, vol. 75, no. 6, pp. 1460–1463, 1986. View at Google Scholar
  38. H. Daido, “Population dynamics of randomly interacting self-oscillators. I. Tractable models without frustration,” Progress of Theoretical Physics, vol. 77, no. 3, pp. 622–634, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  39. H. Daido, “Scaling behaviour at the onset of mutual entrainment in a population of interacting oscillators,” Journal of Physics A, vol. 20, no. 10, article 002, pp. L629–L636, 1987. View at Publisher · View at Google Scholar · View at Scopus
  40. J. N. Teramae and Y. Kuramoto, “Strong desynchronizing effects of weak noise in globally coupled systems,” Physical Review E, vol. 63, no. 3, Article ID 36210, 2001. View at Google Scholar · View at Scopus
  41. H. Hara, “Path integrals for Fokker-Planck equation described by generalized random walks,” Zeitschrift für Physik B, vol. 45, no. 2, pp. 159–166, 1981. View at Publisher · View at Google Scholar · View at MathSciNet
  42. E. W. Montroll and B. J. West, “Models of population growth, diffusion, competition, and rearrangement,” in Synergetics: Cooperative Phenomena in Multi-Component Systems, Haken, Ed., pp. 143–156, Springer, Berlin, Germany, 1973. View at Google Scholar
  43. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems, John Wiley & Sons, New York, NY, USA, 1977. View at MathSciNet
  44. F. D. Nobre, E. M. F. Curado, and G. Rowlands, “A procedure for obtaining general nonlinear Fokker-Planck equations,” Physica A, vol. 334, no. 1-2, pp. 109–118, 2004. View at Publisher · View at Google Scholar · View at Scopus
  45. T. D. Frank, “On a nonlinear master equation and the Haken-Kelso-Bunz model,” Journal of Biological Physics, vol. 30, no. 2, pp. 139–159, 2004. View at Publisher · View at Google Scholar · View at Scopus
  46. D. G. Aronson, “The porous medium equation,” in Nonlinear Diffusion Problems: Lecture Notes in Mathematics, A. Dobb and B. Eckmann, Eds., vol. 1224, pp. 1–46, Springer, Berlin, Germany, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. L. A. Peletier, “The porous media equation,” in Applications of Nonlinear Analysis in the Physical Sciences, H. Amann, N. Bazley, and K. Kirchgassner, Eds., vol. 6, pp. 229–241, Pitman Advanced Publishing Program, Boston, Mass, USA, 1981. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, vol. 888, North-Holland Publishing, Amsterdam, The Netherlands, 1981. View at MathSciNet
  49. S. Kullback, Information Theory and Statistics, Dover Publications, Mineola, NY, USA, 1968. View at MathSciNet
  50. F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill Book Company, New York, NY, USA, 1965.
  51. E. T. Jaynes, “Information theory and statistical mechanics,” Physical Review, vol. 106, no. 4, pp. 620–630, 1957. View at Publisher · View at Google Scholar · View at Scopus
  52. A. Wehrl, “General properties of entropy,” Reviews of Modern Physics, vol. 50, no. 2, pp. 221–260, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
  53. T. D. Frank, “Nonextensive cutoff distributions of postural sway for the old and the young,” Physica A, vol. 388, no. 12, pp. 2503–2510, 2009. View at Publisher · View at Google Scholar · View at Scopus
  54. C. Tsallis, “Possible generalization of Boltzmann-Gibbs statistics,” Journal of Statistical Physics, vol. 52, no. 1-2, pp. 479–487, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  55. C. Tsallis, “Non-extensive thermostatistics: brief review and comments,” Physica A, vol. 221, no. 1–3, pp. 277–290, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  56. S. Abe and Y. Okamoto, Nonextensive Statistical Mechanics and Its Applications, Springer, Berlin, Germany, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  57. H. Haken, J. A. S. Kelso, and H. Bunz, “A theoretical model of phase transitions in human hand movements,” Biological Cybernetics, vol. 51, no. 5, pp. 347–356, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  58. T. D. Frank, A. Daffertshofer, C. E. Peper, P. J. Beek, and H. Haken, “Towards a comprehensive theory of brain activity: coupled oscillator systems under external forces,” Physica D, vol. 144, no. 1-2, pp. 62–86, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  59. G. Schoner, H. Haken, and J. A. S. Kelso, “A stochastic theory of phase transitions in human hand movement,” Biological Cybernetics, vol. 53, no. 4, pp. 247–257, 1986. View at Google Scholar · View at Scopus
  60. T. D. Frank, “Linear and nonlinear Fokker-Planck equations,” in Encyclopedia of Complexity and Systems Science, R. A. Meyers, Ed., vol. 5, pp. 5239–5265, Springer, Berlin, Germany, 2009. View at Google Scholar
  61. J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto model: a simple paradigm for synchronization phenomena,” Reviews of Modern Physics, vol. 77, no. 1, pp. 137–185, 2005. View at Publisher · View at Google Scholar · View at Scopus
  62. H. Haken, Synergetics: Introduction and Advanced Topics, Springer, Berlin, Germany, 2004. View at MathSciNet
  63. P. W. Andersen, “More is different,” Science, vol. 177, no. 4047, pp. 393–396, 1972. View at Google Scholar · View at Scopus
  64. A. Ichiki, H. Ito, and M. Shiino, “Chaos-nonchaos phase transitions induced by multiplicative noise in ensembles of coupled two-dimensional oscillators,” Physica E, vol. 40, no. 2, pp. 402–405, 2007. View at Publisher · View at Google Scholar · View at Scopus
  65. T. Kanamaru, “Blowout bifurcation and on-off intermittency in pulse neural networks with multiple modules,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 11, pp. 3309–3321, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  66. M. Shiino and K. Yoshida, “Chaos-nonchaos phase transitions induced by external noise in ensembles of nonlinearly coupled oscillators,” Physical Review E, vol. 63, no. 2, Article ID 026210, pp. 1–6, 2001. View at Publisher · View at Google Scholar · View at Scopus
  67. M. A. Zaks, X. Sailer, L. Schimansky-Geier, and A. B. Neiman, “Noise induced complexity: from subthreshold oscillations to spiking in coupled excitable systems,” Chaos, vol. 15, no. 2, Article ID 026117, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  68. R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley & Sons, New York, NY, USA, 1975. View at MathSciNet
  69. R. A. Cairns, Plasma Physics, Blackie, Philadelphia, Pa, USA, 1985.
  70. J.-L. Delcroix, Introduction to the Theory of Ionized Gases, Interscience Publishers, London, UK, 1960. View at MathSciNet
  71. Y. N. Dnestrovskii and D. P. Kostomarov, Numerical Simulations of Plasmas, Springer, Berlin, Germany, 1986.
  72. E. H. Holt and R. E. Haskell, Plasma Dynamics, MacMillian, New York, NY, USA, 1965.
  73. Yu. L. Klimontovich, Statistical Physics, Harwood Academic Publishers, New York, NY, USA, 1986. View at MathSciNet
  74. D. R. Nicholson, Introduction to Plasma Theory, John Wiley and Sons, New York, NY, USA, 1983.
  75. M. Soler, F. C. Martínez, and J. M. Donoso, “Integral kinetic method for one dimension: the spherical case,” Journal of Statistical Physics, vol. 69, no. 3-4, pp. 813–835, 1992. View at Publisher · View at Google Scholar · View at Scopus
  76. E. J. Allen and H. D. Victory, “A computational investigation of the random particle method for numerical solution of the kinetic Vlasov-Poisson-Fokker-Planck equations,” Physica A, vol. 209, no. 3-4, pp. 318–346, 1994. View at Google Scholar · View at Scopus
  77. W. M. MacDonald, M. N. Rosenbluth, and W. Chuck, “Relaxation of a system of particles with coulomb interactions,” Physical Review, vol. 107, no. 2, pp. 350–353, 1957. View at Publisher · View at Google Scholar · View at Scopus
  78. M. N. Rosenbluth, W. M. MacDonald, and D. L. Judd, “Fokker-Planck equation for an inverse-square force,” vol. 107, pp. 1–6, 1957. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  79. M. Takai, H. Akiyama, and S. Takeda, “Stabilization of drift-cyclotron loss-cone instability of plasmas by high frequency field,” Journal of the Physical Society of Japan, vol. 50, no. 5, pp. 1716–1722, 1981. View at Google Scholar · View at Scopus
  80. B. M. Boghosian, “Thermodynamic description of the relaxation of two-dimensional turbulence using Tsallis statistics,” Physical Review E, vol. 53, no. 5, pp. 4754–4763, 1996. View at Google Scholar · View at Scopus
  81. P. H. Chavanis, “Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations,” European Physical Journal B, vol. 62, no. 2, pp. 179–208, 2008. View at Publisher · View at Google Scholar · View at Scopus
  82. P. H. Chavanis and C. Sire, “Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions,” Physical Review E, vol. 69, no. 1, Article ID 016116, 2004. View at Google Scholar · View at Scopus
  83. A. R. Plastino and A. Plastino, “Stellar polytropes and Tsallis' entropy,” Physics Letters A, vol. 174, no. 5-6, pp. 384–386, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  84. A. Plastino and A. R. Plastino, “Tsallis Entropy and Jaynes' information theory formalism,” Brazilian Journal of Physics, vol. 29, no. 1, pp. 50–60, 1999. View at Google Scholar · View at Scopus
  85. J. Binney and S. Tremaine, Galatic Dynamics, Princeton University Press, Princeton, NJ, USA, 1987.
  86. C. Lancellotti and M. Kiessling, “Self-similar gravitational collapse in stellar dynamics,” Astrophysical Journal, vol. 549, pp. L93–L96, 2001. View at Google Scholar
  87. A. Chao, Physics of Collective Instabilities in High Energy Accelerators, John Wiley and Sons, New York, NY, USA, 1993.
  88. H. Wiedemann, Particle Accelerator Physics II: Nonlinear and Higher Order Dynamics, Springer, Berlin, Germany, 1993.
  89. R. C. Davidson, H. Qin, and T. S. F. Wang, “Vlasov-Maxwell description of electron-ion two-stream instability in high-intensity linacs and storage rings,” Physics Letters A, vol. 252, no. 5, pp. 213–221, 1999. View at Google Scholar · View at Scopus
  90. R. C. Davidson, H. Qin, and P. J. Channell, “Periodically-focused solutions to the nonlinear Vlasov-Maxwell equations for intense charged particle beams,” Physics Letters A, vol. 258, no. 4–6, pp. 297–304, 1999. View at Google Scholar · View at Scopus
  91. S. Heifets, “Microwave instability beyond threshold,” Physical Review Special Topics—Accelerators and Beams, vol. 54, pp. 2889–2898, 1996. View at Google Scholar
  92. S. Heifets, “Saturation of the coherent beam-beam instability,” Physical Review Special Topics—Accelerators and Beams, vol. 4, no. 4, Article ID 044401, pp. 98–105, 2001. View at Publisher · View at Google Scholar · View at Scopus
  93. S. Heifets, “Single-mode coherent synchrotron radiation instability of a bunched beam,” Physical Review Special Topics—Accelerators and Beams, vol. 6, no. 8, Article ID 080701, pp. 10–21, 2003. View at Publisher · View at Google Scholar · View at Scopus
  94. S. Heifets and B. Podobedov, “Single bunch stability to monopole excitation,” Physical Review Special Topics—Accelerators and Beams, vol. 2, no. 4, Article ID 044402, pp. 43–49, 1999. View at Google Scholar · View at Scopus
  95. Y. Shobuda and K. Hirata, “The existence of a static solution of the Haissinski equation with purely inductive wake force,” Part Accel, vol. 62, pp. 165–177, 1999. View at Google Scholar
  96. Y. Shobuda and K. Hirata, “Proof of the existence and uniqueness of a solution for the Haissinski equation with a capacitive wake function,” Physical Review E, vol. 64, no. 6, Article ID 067501, 2001. View at Google Scholar · View at Scopus
  97. G. V. Stupakov, B. N. Breizman, and N. S. Pekker, “Nonlinear dynamics of microwave instability in accelerators,” Physical Review E, vol. 55, pp. 5976–5984, 1997. View at Google Scholar
  98. M. Venturini and R. Warnock, “Bursts of coherent synchrotron radiation in electron storage rings: a dynamical model,” Physical Review Letters, vol. 89, no. 22, Article ID 224802, 2002. View at Google Scholar · View at Scopus
  99. J. Haïssinski, “Exact longitudinal equilibrium distribution of stored electrons in the presence of self-fields,” Nuovo Cimento Della Società Italiana Di Fisica B, vol. 18, no. 1, pp. 72–82, 1973. View at Google Scholar
  100. T. D. Frank, “Smoluchowski approach to nonlinear Vlasov-Fokker-Planck equations: Stability analysis of beam dynamics and Haïssinski theory,” Physical Review Special Topics—Accelerators and Beams, vol. 9, no. 8, Article ID 084401, 2006. View at Publisher · View at Google Scholar · View at Scopus
  101. T. D. Frank and S. Mongkolsakulvong, “A nonextensive thermostatistical approach to the Haïssinski theory of accelerator beams,” Physica A, vol. 387, no. 19-20, pp. 4828–4838, 2008. View at Publisher · View at Google Scholar · View at Scopus
  102. T. D. Frank and S. Mongkolsakulvong, “Parametric solution method for self-consistency equations and order parameter equations derived from nonlinear Fokker-Planck equations,” Physica D, vol. 238, no. 14, pp. 1186–1196, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  103. C. Thomas, R. Bartolini, J. I. M. Botman, G. Dattoli, L. Mezi, and M. Migliorati, “An analytical solution for the Haissinski equation with purely inductive wake fields,” Europhysics Letters, vol. 60, no. 1, pp. 66–71, 2002. View at Publisher · View at Google Scholar · View at Scopus
  104. M. Venturini, “Stability analysis of longitudinal beam dynamics using noncanonical Hamiltonian methods and energy principles,” Physical Review Special Topics—Accelerators and Beams, vol. 5, no. 5, pp. 43–50, 2002. View at Publisher · View at Google Scholar · View at Scopus
  105. S. Chandrasekhar, Liquid Crystals, Cambridge University Press, Cambridge, UK, 1977.
  106. P. De Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, UK, 1974.
  107. M. Plischke and B. Bergersen, Equilibrium Statistical Physics, World Scientifric, Singapore, 1994.
  108. G. Strobl, Condensed Matter Physics: Crystals, Liquids, Liquid Crystals, and Polymers, Springer, Berlin, Germany, 2004.
  109. W. Maier and A. Saupe, “Eine einfache molekulare Theorie des namatischen kristallinflussigen Zustandes,” Zeitschrift Fur Naturforschung A, vol. 13, pp. 564–566, 1958 (German). View at Google Scholar
  110. W. Maier and A. Saupe, “Eine einfache molekulare Theorie des namatischen kristallinflussigen Zustandes. Teil II,” Zeitschrift Fur Naturforschung A, vol. 15, pp. 287–292, 1960 (German). View at Google Scholar
  111. A. Saupe, “Liquid crystals,” Annual Review of Physical Chemistry, vol. 24, pp. 441–471, 1973. View at Google Scholar
  112. C. V. Hess, “Fokker-Planck equation approach to flow alignment in liquid crystals,” Zeitschrift Fur Naturforschung, vol. 31, pp. 1034–1037, 1976. View at Google Scholar
  113. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, UK, 1988.
  114. T. D. Frank, “Maier-Saupe model of liquid crystals: Isotropic-nematic phase transitions and second-order statistics studied by Shiino's perturbation theory and strongly nonlinear Smoluchowski equations,” Physical Review E, vol. 72, no. 4, Article ID 041703, 2005. View at Publisher · View at Google Scholar · View at Scopus
  115. P. Ilg, I. V. Karlin, and H. C. Öttinger, “Generating moment equations in the Doi model of liquid-crystalline polymers,” Physical Review E, vol. 60, pp. 5783–5787, 1999. View at Google Scholar · View at Scopus
  116. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer, Berlin, Germany, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  117. B. U. Felderhof, “Orientational relaxation in the Maier-Saupe model of nematic liquid crystals,” Physica A, vol. 323, no. 1–4, pp. 88–106, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  118. B. U. Felderhof and R. B. Jones, “Mean field theory of the nonlinear response of an interacting dipolar system with rotational diffusion to an oscillating field,” Journal of Physics Condensed Matter, vol. 15, no. 23, pp. 4011–4024, 2003. View at Publisher · View at Google Scholar · View at Scopus
  119. P. Ilg, M. Kroger, S. Hess, and A. Y. Zubarev, “Dynamics of colloidal suspensions of ferromagnetic particles in plane Couette flow: comparison of approximate solutions with Brownian dynamics simulations,” Physical Review E, vol. 67, Article ID 061401, 2003. View at Google Scholar
  120. R. G. Larson and H. C. Öttinger, “Effect of molecular elasticity on out-of-plane orientations in shearing flows of liquid-crystalline polymers,” Macromolecules, vol. 24, no. 23, pp. 6270–6282, 1991. View at Google Scholar · View at Scopus
  121. H. C. Öttinger, Stochastic Processes in Polymeric Fluids, Springer, Berlin, Germany, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  122. G. Kaniadakis, “H-theorem and generalized entropies within the framework of nonlinear kinetics,” Physics Letters A, vol. 288, no. 5-6, pp. 283–291, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  123. G. Kaniadakis and P. Quarati, “Kinetic equation for classical particles obeying an exclusion principle,” Physical Review E, vol. 48, no. 6, pp. 4263–4270, 1993. View at Publisher · View at Google Scholar · View at Scopus
  124. G. Kaniadakis and P. Quarati, “Classical model of bosons and fermions,” Physical Review E, vol. 49, no. 6, pp. 5103–5110, 1994. View at Publisher · View at Google Scholar · View at Scopus
  125. E. A. Uehling and G. E. Uhlenbeck, “Transport phenomena in Einstein-Bose or Fermi-Dirac gas I,” Physical Review, vol. 43, pp. 552–561, 1933. View at Google Scholar
  126. T. D. Frank, “Classical Langevin equations for the free electron gas and blackbody radiation,” Journal of Physics A, vol. 37, no. 11, pp. 3561–3567, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  127. T. D. Frank, “Modelling the stochastic single particle dynamics of relativistic fermions and bosons using nonlinear drift-diffusion equations,” Mathematical and Computer Modelling, vol. 42, no. 9-10, pp. 1057–1062, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  128. T. D. Frank and A. Daffertshofer, “Nonlinear Fokker-Planck equations whose stationary solutions make entropy-like functionals stationary,” Physica A, vol. 272, no. 3-4, pp. 497–508, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  129. T. D. Frank and A. Daffertshofer, “Multivariate nonlinear Fokker-Planck equations and generalized thermostatistics,” Physica A, vol. 292, no. 1–4, pp. 392–410, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  130. L. P. Kadanoff, Statistical Physics: Statics, Dynamics and Renormalization, World Scientific Publishing, River Edge, NJ, USA, 2000. View at MathSciNet
  131. J. Sopik, C. Sire, and P.-H. Chavanis, “Dynamics of the Bose-Einstein condensation: analogy with the collapse dynamics of a classical self-gravitating Brownian gas,” Physical Review E, vol. 74, no. 1, Article ID 011112, p. 15, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  132. R. Tsekov, “Dissipation in quantum systems,” Journal of Physics A, vol. 28, no. 21, pp. L557–L561, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  133. R. Tsekov, “A quantum theory of thermodynamic relaxation,” International Journal of Molecular Sciences, vol. 2, no. 2, pp. 66–71, 2001. View at Google Scholar · View at Scopus
  134. W. Ebeling, “Canonical nonequilibrium statistics and applications to Fermi-Bose systems,” Condensed Matter Physics, vol. 3, pp. 285–293, 2000. View at Google Scholar
  135. W. M. Ni, L. A. Peletier, and J. Serrin, Nonlinear Diffusion Equations and Their Equilibrium States, Springer, Berlin, Germany, 1988.
  136. G. I. Barenblatt, V. M. Entov, and V. M. Ryzhik, Theory of Fluid Flows through Natural Rocks, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. View at Scopus
  137. R. E. Pattle, “Diffusion from an instantaneous point source with a concentration-dependent coefficient,” The Quarterly Journal of Mechanics and Applied Mathematics, vol. 12, pp. 407–409, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  138. L. Borland, “Microscopic dynamics of the nonlinear Fokker-Planck equation,” Physical Review E, vol. 57, no. 6, pp. 6634–6642, 1996. View at Google Scholar · View at Scopus
  139. A. Compte and D. Jou, “Non-equilibrium thermodynamics and anomalous diffusion,” Journal of Physics A, vol. 29, no. 15, pp. 4321–4329, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  140. C. Tsallis and D. J. Bukman, “Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis,” Physical Review E, vol. 54, no. 3, pp. R2197–R2200, 1996. View at Google Scholar · View at Scopus
  141. T. D. Frank, “Autocorrelation functions of nonlinear Fokker-Planck equations,” European Physical Journal B, vol. 37, no. 2, pp. 139–142, 2004. View at Publisher · View at Google Scholar · View at Scopus
  142. J. S. Andrade, G. F. T. Da Silva, A. A. Moreira, F. D. Nobre, and E. M. F. Curado, “Thermostatistics of overdamped motion of interacting particles,” Physical Review Letters, vol. 105, no. 26, Article ID 260601, 2010. View at Publisher · View at Google Scholar · View at Scopus
  143. M. S. Ribeiro, F. D. Nobre, and E. M. F. Curado, “Time evolution of interacting vortices under overdamped motion,” Physical Review E, vol. 85, Article ID 021146, 2012. View at Google Scholar
  144. M. S. Ribeiro, F. D. Nobre, and E. M. F. Curado, “Overdamped motion of interacting particles in general confining potentials: time-dependent and stationary-state analysis,” European Physics Journal B, vol. 85, article 399, 2012. View at Google Scholar
  145. A. R. Plastino and A. Plastino, “Non-extensive statistical mechanics and generalized Fokker-Planck equation,” Physica A, vol. 222, no. 1–4, pp. 347–354, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  146. S. Martinez, A. R. Plastino, and A. Plastino, “Nonlinear Fokker-Planck equations and generalized entropies,” Physica A, vol. 259, no. 1-2, pp. 183–192, 1998. View at Google Scholar · View at Scopus
  147. T. D. Frank, “A Langevin approach for the microscopic dynamics of nonlinear Fokker-Planck equations,” Physica A, vol. 301, no. 1–4, pp. 52–62, 2001. View at Publisher · View at Google Scholar · View at Scopus
  148. G. Drazer, H. S. Wio, and C. Tsallis, “Anomalous diffusion with absorption: exact time-dependent solutions,” Physical Review E, vol. 61, no. 2, pp. 1417–1422, 2000. View at Google Scholar · View at Scopus
  149. A. R. Plastino, M. Casas, and A. Plastino, “A nonextensive maximum entropy approach to a family of nonlinear reaction-diffusion equations,” Physica A, vol. 280, pp. 289–303, 2000. View at Google Scholar
  150. A. Rigo, A. R. Plastino, M. Casas, and A. Plastino, “Anomalous diffusion coupled with Verhulst-like growth dynamics: exact time-dependent solutions,” Physics Letters A, vol. 276, no. 1–4, pp. 97–102, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  151. E. K. Lenzi, C. Anteneodo, and L. Borland, “Escape time in anomalous diffusive media,” Physical Review E, vol. 63, no. 5 I, Article ID 51109, 2001. View at Google Scholar · View at Scopus
  152. A. Compte, D. Jou, and Y. Katayama, “Anomalous diffusion in linear shear flows,” Journal of Physics A, vol. 30, no. 4, pp. 1023–1030, 1997. View at Publisher · View at Google Scholar · View at Scopus
  153. L. C. Malacarne, R. S. Mendes, and I. T. Pedron, “Nonlinear equation for anomalous diffusion: unified power-law and stretched exponential exact solution,” Physical Review E, vol. 63, no. 3 I, Article ID 030101, 2001. View at Google Scholar · View at Scopus
  154. I. T. Pedron, R. S. Mendes, L. C. Malacarne, and E. K. Lenzi, “Nonlinear anomalous diffusion equation and fractal dimension: exact generalized Gaussian solution,” Physical Review E, vol. 65, no. 4, Article ID 041108, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  155. L. Borland, F. Pennini, A. R. Plastino, and A. Plastino, “The nonlinear Fokker-Planck equation with state-dependent diffusion: a nonextensive maximum entropy approach,” European Physical Journal B, vol. 12, no. 2, pp. 285–297, 1999. View at Google Scholar · View at Scopus
  156. T. D. Frank and A. Daffertshofer, “Exact time-dependent solutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharma and Mittal,” Physica A, vol. 285, no. 3, pp. 129–144, 2000. View at Google Scholar · View at Scopus
  157. P. H. Chavanis, “Generalized thermodynamics and Fokker-Planck equations: applications to stellar dynamics and two-dimensional turbulence,” Physical Review E, vol. 68, no. 3, Article ID 036108, 2003. View at Google Scholar · View at Scopus
  158. P.-H. Chavanis, “Generalized Fokker-Planck equations and effective thermodynamics,” Physica A, vol. 340, no. 1–3, pp. 57–65, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  159. T. D. Frank and A. Daffertshofer, “H-theorem for nonlinear Fokker-Planck equations related to generalized thermostatistics,” Physica A, vol. 295, no. 3-4, pp. 455–474, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  160. M. S. Ribeiro, F. D. Nobre, and E. M. F. Curado, “Classes of N-dimensional nonlinear Fokker-Planck equations associated to Tsallis entropy,” Entropy, vol. 13, no. 11, pp. 1928–1944, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  161. V. Schwämmle, F. D. Nobre, and E. M. F. Curado, “Consequences of the H-theorem for nonlinear Fokker-Planck equations,” Physical Review E, vol. 76, Article ID 041123, 2007. View at Google Scholar
  162. V. Schwämmle, E. M. F. Curado, and F. D. Nobre, “A general nonlinear Fokker-Planck equation and its associated entropy,” The European Physical Journal B, vol. 58, no. 2, pp. 159–165, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  163. V. Schwämmle, E. M. F. Curado, and F. D. Nobre, “Dynamics of normal and anomalous diffusion in nonlinear Fokker-Planck equations,” The European Physical Journal B, vol. 70, no. 1, pp. 107–116, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  164. M. Shiino, “Free energies based on generalized entropies and H-theorems for nonlinear Fokker-Planck equations,” Journal of Mathematical Physics, vol. 42, no. 6, pp. 2540–2553, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  165. S. Mongkolsakulvong and T. D. Frank, “Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space,” Condensed Matter Physics, vol. 13, no. 1, Article ID 13001, pp. 1–18, 2010. View at Google Scholar
  166. W. Ebeling and I. M. Sokolov, Statistical thermodynamics and stochastic theory of nonequilibrium systems, World Scientific Publishing, Singapore, 2004. View at MathSciNet
  167. H. Haken, “Distribution function for classical and quantum systems far from thermal equilibrium,” Zeitschrift für Physik, vol. 263, no. 4, pp. 267–282, 1973. View at Google Scholar · View at MathSciNet
  168. F. Schweitzer, Brownian Agents and Active Particles, Springer, Berlin, Germany, 2003. View at MathSciNet
  169. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, vol. 12, Cambridge University Press, Cambridge, UK, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  170. T. D. Frank and M. J. Richardson, “On a test statistic for the Kuramoto order parameter of synchronization: an illustration for group synchronization during rocking chairs,” Physica D, vol. 239, no. 23-24, pp. 2084–2092, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  171. M. J. Richardson, R. L. Garcia, T. D. Frank, M. Gregor, and K. L. Marsh, “Measuring group synchrony: a cluster-phase method for analyzing multivariate movement time series,” Frontiers in Physiology, vol. 3, article 405, 2012. View at Google Scholar
  172. S. H. Strogatz, “From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators,” Physica D, vol. 143, no. 1–4, pp. 1–20, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  173. T. D. Frank, A. Daffertshofer, C. E. Peper, P. J. Beek, and H. Haken, “H-theorem for a mean field model describing coupled oscillator systems under external forces,” Physica D, vol. 150, no. 3-4, pp. 219–236, 2001. View at Publisher · View at Google Scholar · View at Scopus
  174. T. D. Frank, “On a mean field Haken-Kelso-Bunz model and a free energy approach to relaxation processes,” Nonlinear Phenomena in Complex Systems, vol. 5, no. 4, pp. 332–341, 2002. View at Google Scholar · View at MathSciNet
  175. P. A. Tass, Phase Resetting in Medicine and Biology: Stochastic Modelling and Data Analysis, Springer, Berlin, Germany, 1999. View at MathSciNet
  176. A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, New York, NY, USA, 2001. View at MathSciNet
  177. A. Okubo, Diffusion and Ecological Problems: Mathematical Models, Springer, Berlin, Germany, 1980. View at MathSciNet
  178. J. D. Murray, Mathematical Biology, Springer, Berlin, Germany, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  179. T. P. Witelski, “Segregation and mixing in degenerate diffusion in population dynamics,” Journal of Mathematical Biology, vol. 35, no. 6, pp. 695–712, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  180. F. J. Boster, J. E. Hunter, M. E. Mayer, and J. L. Hale, “Expanding the persuasive arguments explanation of the polarity shift: a linear discrepancy model,” in Communication Yearbook 4, D. Nimmo, Ed., pp. 165–176, Transaction Books, New Brunswick, Canada, 1980. View at Google Scholar
  181. F. J. Boster, J. E. Fryrear, P. A. Mongeau, and J. E. Hunter, “An unequal speaking linear discrepancy model: implications for polarity shift,” in Communication Yearbook 6, M. Burgoon, Ed., pp. 395–418, Sage Publishers, Beverly Hills, Calif, USA, 1982. View at Google Scholar
  182. F. J. Boster, J. E. Hunter, and J. L. Hale, “An information processing model of jury decision making,” Communication Research, vol. 18, pp. 524–547, 1991. View at Google Scholar
  183. T. D. Frank, “On the linear discrepancy model and risky shifts in group behavior: a nonlinear Fokker-Planck perspective,” Journal of Physics A, vol. 42, no. 15, Article ID 155001, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  184. L. Borland, “Option pricing formulas based on a non-Gaussian stock price model,” Physical Review Letters, vol. 89, no. 9, pp. 987011–987014, 2002. View at Google Scholar · View at Scopus
  185. L. Borland, “Non-Gaussian option pricing. Successes, limitations and perspectives,” in Anomalous Fluctuations in Complex Systems: Plasmas, Fluids and Financial Markets, C. Riccardi and H. E. Roman, Eds., pp. 311–334, 2008. View at Google Scholar
  186. L. Borland and J.-P. Bouchaud, “A non-Gaussian option pricing model with skew,” Quantitative Finance, vol. 4, no. 5, pp. 499–514, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  187. C. F. Lo and C. H. Hui, “A simple analytical model for dynamics of time-varying target leverage ratios,” European Physics Journal B, vol. 85, article 102, 2012. View at Google Scholar
  188. C. H. Hui, C. F. Lo, and M. X. Huang, “Are corporates' target leverage ratios time-dependent?” International Review of Financial Analysis, vol. 15, no. 3, pp. 220–236, 2006. View at Publisher · View at Google Scholar · View at Scopus
  189. S. Primak, “Generation of compound non-Gaussian random processes with a given correlation function,” Physical Review E, vol. 61, no. 1, pp. 100–103, 2000. View at Google Scholar · View at Scopus
  190. S. Primak, V. Lyandres, and V. Kontorovich, “Markov models of non-Gaussian exponentially correlated processes and their applications,” Physical Review E, vol. 63, no. 6 I, Article ID 061103, 2001. View at Google Scholar · View at Scopus
  191. H. Shimizu, “Muscle contraction mechanism as a hard mode instability,” Progress of Theoretical Physics, vol. 52, pp. 329–330, 1974. View at Google Scholar
  192. H. Shimizu and T. Yamada, “Phenomenological equations of motion of muscular contrations,” Progress of Theoretical Physics, vol. 47, pp. 350–351, 1972. View at Google Scholar
  193. T. D. Frank, “A note on the Markov property of stochastic processes described by nonlinear Fokker-Planck equations,” Physica A, vol. 320, no. 1–4, pp. 204–210, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  194. R. C. Desai and R. Zwanzig, “Statistical mechanics of a nonlinear stochastic model,” Journal of Statistical Physics, vol. 19, no. 1, pp. 1–24, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
  195. D. A. Dawson, “Critical dynamics and fluctuations for a mean-field model of cooperative behavior,” Journal of Statistical Physics, vol. 31, no. 1, pp. 29–85, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  196. M. Shiino, “Dynamical behavior of stochastic systems of infinitely many coupled nonlinear oscillators exhibiting phase transitions of mean-field type: H-theorem on asymptotic approach to equilibrium and critical slowing down of order-parameter fluctuations,” Physical Review A, vol. 36, no. 5, pp. 2393–2412, 1987. View at Publisher · View at Google Scholar
  197. A. Arnold, L. L. Bonilla, and P. A. Markowich, “Liapunov functionals and large-time-asymptotics of mean-field nonlinear Fokker-Planck equations,” Transport Theory and Statistical Physics, vol. 25, no. 7, pp. 733–751, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  198. L. L. Bonilla, C. J. Perez-Vicente, F. Ritort, and J. Soler, “Exactly solvable phase oscillator models with synchronization dynamics,” Physical Review Letters, vol. 81, no. 17, pp. 3643–3646, 1998. View at Google Scholar · View at Scopus
  199. G. Kaniadakis, “Nonlinear kinetics underlying generalized statistics,” Physics Letters A, vol. 296, pp. 405–425, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  200. J. A. S. Lima, R. Silva, and A. R. Plastino, “Nonextensive thermostatistics and the H theorem,” Physical Review Letters, vol. 86, no. 14, pp. 2938–2941, 2001. View at Publisher · View at Google Scholar · View at Scopus
  201. M. Shiino, “Nonlinear Fokker-Planck equation exhibiting bifurcation phenomena and generalized thermostatistics,” Journal of Mathematical Physics, vol. 43, no. 5, pp. 2654–2669, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  202. T. D. Frank, “Dynamic mean field models: H-theorem for stochastic processes and basins of attraction of stationary processes,” Physica D, vol. 195, no. 3-4, pp. 229–243, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  203. L. L. Bonilla, “Stable nonequilibrium probability densities and phase transitions for meanfield models in the thermodynamic limit,” Journal of Statistical Physics, vol. 46, no. 3-4, pp. 659–678, 1987. View at Publisher · View at Google Scholar · View at Scopus
  204. J. D. Crawford and K. T. R. Davies, “Synchronization of globally coupled phase oscillators: singularities and scaling for general couplings,” Physica D, vol. 125, no. 1-2, pp. 1–46, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  205. V. N. Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations, Cambridge University Press, Cambridge, UK, 2010. View at MathSciNet
  206. W. Braun and K. Hepp, “The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles,” Communications in Mathematical Physics, vol. 56, no. 2, pp. 101–113, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  207. T. Funaki, “A certain class of diffusion processes associated with nonlinear parabolic equations,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 67, no. 3, pp. 331–348, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  208. J. Gärtner, “On the McKean-Vlasov limit for interacting diffusions,” Mathematische Nachrichten, vol. 137, pp. 197–248, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  209. S. Méléard and S. Roelly-Coppoletta, “A propagation of chaos result for a system of particles with moderate interaction,” Stochastic Processes and their Applications, vol. 26, no. 2, pp. 317–332, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  210. K. Oelschläger, “A law of large numbers for moderately interacting diffusion processes,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 69, no. 2, pp. 279–322, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  211. A. Arnold, P. Markowich, and G. Toscani, “On large time asymptotics for drift-diffusion-Poisson systems,” Transport Theory and Statistical Physics, vol. 29, no. 3–5, pp. 571–581. View at MathSciNet
  212. A. Arnold, P. Markowich, G. Toscani, and A. Unterreiter, “On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations,” Communications in Partial Differential Equations, vol. 26, no. 1-2, pp. 43–100, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  213. J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani, and A. Unterreiter, “Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities,” Monatshefte für Mathematik, vol. 133, no. 1, pp. 1–82, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  214. D. A. Dawson and J. Gärtner, “Large deviations, free energy functional and quasi-potential for a mean field model of interacting diffusions,” Memoirs of the American Mathematical Society, vol. 78, no. 398, 1989. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  215. B. Djehiche and I. Kaj, “The rate function for some measure-valued jump processes,” The Annals of Probability, vol. 23, no. 3, pp. 1414–1438, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  216. J. Fontbona, “Nonlinear martingale problems involving singular integrals,” Journal of Functional Analysis, vol. 200, no. 1, pp. 198–236, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  217. C. Graham, “Nonlinear limit for a system of diffusing particles which alternate between two states,” Applied Mathematics and Optimization, vol. 22, no. 1, pp. 75–90, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  218. A. Greven, “Renormalization and universality for multitype population models,” in Interacting Stochastic Systems, pp. 209–246, Springer, Berlin, Germany, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  219. L. Overbeck, “Nonlinear superprocesses,” The Annals of Probability, vol. 24, no. 2, pp. 743–760, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  220. B. Jourdain, “Diffusion Processes Associated with Nonlinear Evolution Equations for Signed Measures,” Methodology and Computing in Applied Probability, vol. 2, pp. 69–91, 2000. View at Google Scholar
  221. D. A. Dawson, “Measure-Valued Markov Processes,” in Lecture Notes in Mathematics, D. A. Dawson, B. Maisonneuve, and J. Spencer, Eds., vol. 1541, Springer, Berlin, Germany, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  222. D. W. Stroock, Markov Processes from K. Itô's Perspective, vol. 155, Princeton University Press, Princeton, NJ, USA, 2003. View at MathSciNet
  223. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964. View at MathSciNet
  224. A. Friedman, Partial Differential Equations, Holt, Richart, and Winston, New York, NY, USA, 1969. View at MathSciNet
  225. J. D. Logan, Transport Modeling in Hydrogeochemical Systems, Springer, New York, NY, USA, 2001. View at MathSciNet
  226. G. N. Milstein, “The probability approach to numerical solution of nonlinear parabolic equations,” Numerical Methods for Partial Differential Equations, vol. 18, no. 4, pp. 490–522, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  227. G. N. Milstein and M. V. Tretyakov, “Numerical methods for nonlinear parabolic equations with small parameter based on probability approach,” Mathematics of Computation, vol. 60, pp. 237–267, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  228. G. N. Milstein and M. V. Tretyakov, “Numerical solution of the Dirichlet problem for nonlinear parabolic equations by a probabilistic approach,” IMA Journal of Numerical Analysis, vol. 21, no. 4, pp. 887–917, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  229. D. Huber and L. S. Tsimring, “Dynamics of an ensemble of noisy bistable elements with global time delayed coupling,” Physical Review Letters, vol. 91, no. 26 I, Article ID 260601, 2003. View at Google Scholar · View at Scopus
  230. S. Kim, S. H. Park, and H. B. Pyo, “Stochastic resonance in coupled oscillator systems with time delay,” Physical Review Letters, vol. 82, pp. 1620–1623, 1999. View at Google Scholar
  231. E. Niebur, H. G. Schuster, and D. M. Kammen, “Collective frequencies and metastability in networks of limit-cycle oscillators with time delay,” Physical Review Letters, vol. 67, no. 20, pp. 2753–2756, 1991. View at Publisher · View at Google Scholar · View at Scopus
  232. M. K. S. Yeung and S. H. Strogatz, “Time delay in the kuramoto model of coupled oscillators,” Physical Review Letters, vol. 82, no. 3, pp. 648–651, 1999. View at Google Scholar · View at Scopus
  233. T. D. Frank and P. J. Beek, “Fokker-Planck equations for globally coupled many-body systems with time delays,” Journal of Statistical Mechanics, vol. 10, Article ID P10010, 2005. View at Google Scholar
  234. N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology, Academic Press, New York, NY, USA, 1974. View at MathSciNet