Free Access
Issue
Math. Model. Nat. Phenom.
Volume 6, Number 5, 2011
Complex Fluids
Page(s) 184 - 262
DOI https://doi.org/10.1051/mmnp/20116509
Published online 10 August 2011
  1. G.I. Barenblatt. On some unsteady motions of a liquid or a gas in a porous medium. Prikl. Mat. Mekh., 16 (1952), 67–78. [Google Scholar]
  2. G.I. Barenblatt, Y.B. ZelŠdovich. Self-similar solutions as intermediate asymptotics. Ann. Rev. Fluid Mech., 4 (1972), 285–312. [CrossRef] [Google Scholar]
  3. L. Bertini, C. Landim, S. Olla. Derivation of Cahn–Hilliard Equations from Ginzburg-Landau Models. J. Stat. Phys., 88 (1997), Nos. 1/2, 365–381. [CrossRef] [Google Scholar]
  4. T. Blesgen, U. Weikard. Multi-component Allen-Cahn equation for elastically stressed solids. Electron. J. Diff. Eqns., 89 (2005), 1–17. [Google Scholar]
  5. M. Boudart. From the century of the rate equation to the century of the rate constants: a revolution in catalytic kinetics and assisted catalyst design. Catal. Lett., 65 (2000), 1–3. [CrossRef] [Google Scholar]
  6. L. Boltzmann. Lectures on gas theory. U. of California Press, Berkeley, CA, 1964. [Google Scholar]
  7. G.E. Briggs, J.B.S. Haldane. A note on the kinetics of enzyme action. Biochem. J., 19 (1925), 338–339. [PubMed] [Google Scholar]
  8. R.A. Brownlee, A.N. Gorban, J. Levesley. Nonequilibrium entropy limiters in lattice Boltzmann methods. Physica A, 387 (2008), 385–406. [CrossRef] [MathSciNet] [Google Scholar]
  9. V.I. Bykov, S.E. Gilev, A.N. Gorban, G.S. Yablonskii. Imitation modeling of the diffusion on the surface of a catalyst. Dokl. Akad. Nauk SSSR, 283 (1985), 1217–1220. [Google Scholar]
  10. V.I. Bykov, A.N. Gorban, G.S. Yablonskii. Description of non-isothermal reactions in terms of Marcelin-De-Donder Kinetics and its generalizations. React. Kinet. Catal. Lett. 20 (1982), 261–265. [CrossRef] [Google Scholar]
  11. J.W. Cahn. Free energy of a nonuniform system. II. Thermodynamic basis. J. Chem. Phys., 30 (1959), 1121–1124. [CrossRef] [Google Scholar]
  12. J.W. Cahn, J.E. Hilliard. Free energy of a nonuniform system. I. Interfacial energy. J. Chem. Phys., 28 (1958), 258–266. [CrossRef] [Google Scholar]
  13. J.W. Cahn, J.E. Hilliard. Spinodal decomposition: A reprise. Acta Metallurgica, 19 (1971), 151–161. [CrossRef] [Google Scholar]
  14. H.B. Callen. Thermodynamics and an introduction to themostatistics (2nd ed.). John Wiley & Sons, NY, 1985. [Google Scholar]
  15. C. Cercignani, M. Lampis. On the H-theorem for polyatomic gases. J. Stat. Phys., 26 (1981), 795–801. [CrossRef] [Google Scholar]
  16. B. Chopard, M. Droz. Cellular automata modeling of physical systems. Cambridge University Press, Cambridge, UK, 1998. [Google Scholar]
  17. R. Clausius. Über verschiedene für die Anwendungen bequeme Formen der Hauptgleichungen der Wärmetheorie. Poggendorffs Annalen der Physic und Chemie, 125 (1865), 353–400. [CrossRef] [Google Scholar]
  18. A.J. Chorin, O.H. Hald, R. Kupferman. Optimal prediction with memory. Physica D, 166 (2002), 239–257. [CrossRef] [MathSciNet] [Google Scholar]
  19. F. Coester. Principle of detailed balance. Phys. Rev., 84, 1259 (1951) [Google Scholar]
  20. S.R. De Groot, P. Mazur. Non-equilibrium Thermodynamics. North-Holland, Amsterdam, 1962. [Google Scholar]
  21. K. Denbigh. The principles of chemical equilibrium. Cambridge University Press, Cambridge, UK, 1981. [Google Scholar]
  22. S. Dushman, I. Langmuir. The diffusion coefficient in solids and its temperature coefficient. Phys. Rev., 20 (1922), 113. [Google Scholar]
  23. P. Ehrenfest, T. Ehrenfest-Afanasyeva. Begriffliche Grundlagen der statistischen Auffassung in der Mechanik. In: Mechanics Enziklopädie der Mathematischen Wissenschaften, Vol. 4. Leipzig, 1911. (Reprinted in: Ehrenfest, P., Collected Scientific Papers. North–Holland, Amsterdam, 1959, pp. 213–300.) [Google Scholar]
  24. A. Einstein. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys., 17 (1905), 549–560. [NASA ADS] [CrossRef] [Google Scholar]
  25. A. Einstein. Strahlungs-Emission und -Absorption nach der Quantentheorie. Verhandlungen der Deutschen Physikalischen Gesellschaft, 18 (1916), No. 13/14, Braunschweig, Vieweg, 318–323. [Google Scholar]
  26. C.M. Elliott, Z. Songmu. On the Cahn-Hilliard equation. Arch. Rat. Mechan. Anal., 96 (1986), 339–357. [Google Scholar]
  27. C.M. Elliott, A.M. Stuart. The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal., 30 (1993), 1622–1663. [CrossRef] [MathSciNet] [Google Scholar]
  28. H. Eyring. The activated complex in chemical reactions. J. Chem. Phys., 3 (1935), 107–115. [CrossRef] [Google Scholar]
  29. M. Feinberg. On chemical kinetics of a certain class. Arch. Rat. Mechan. Anal., 46 (1972), 1–41. [Google Scholar]
  30. M. Feinberg. Complex balancing in general kinetic systems. Arch. Rat. Mechan. Anal., 49 (1972), 187–194. [Google Scholar]
  31. R.F. Feynman. Simulating physics with computers. Internat. J. Theor. Phys., 21 (1982), 467–488. [CrossRef] [Google Scholar]
  32. A. Fick. Über Diffusion. Poggendorff’s Annalen der Physik und Chemie, 94 (1855), 59–86. [NASA ADS] [CrossRef] [Google Scholar]
  33. R.A. Fisher. The genetical theory of natural selection. Oxford University Press, Oxford, 1930. [Google Scholar]
  34. F.C. Frank, D. Turnbull. Mechanism of diffusion of copper in Germanium. Phys. Rev., 104 (1956), 617–618. [CrossRef] [Google Scholar]
  35. A. Fratzl, O. Penrose, J.L. Lebowitz. Modelling of phase separation in alloys with coherent elastic misfit. J. Stat. Phys., 95 (1999), 1429–1503. [CrossRef] [MathSciNet] [Google Scholar]
  36. J. Frenkel. Theorie der Adsorption und verwandter Erscheinungen. Zeitschrift für Physik, 26 (1924), 117–138 [Google Scholar]
  37. J. Frenkel. Über die Wärmebewegung in festen und flüssigen Körpern. Zeitschrift für Physik, 35 (1925), 652–669. [Google Scholar]
  38. G.F. Gause. The struggle for existence. Williams & Wilkins, Baltimore, 1934. [Google Scholar]
  39. J.W. Gibbs. On the equilibrium of heterogeneous substance. Trans. Connect. Acad., 1875–1876, 108–248; 1877–1878, 343–524. [Google Scholar]
  40. D.T. Gillespie. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81 (1977), 2340–2361. [CrossRef] [Google Scholar]
  41. D.T. Gillespie. Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem., 58 (2007), 35–55. [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
  42. A.N. Gorban. Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis. Nauka, Novosibirsk, 1984. [Google Scholar]
  43. A.N. Gorban. Singularities of transition processes in dynamical systems: qualitative theory of critical delays. Electron. J. Diff. Eqns., Monograph 05, 2004. E-print: http://arxiv.org/abs/chao-dyn/9703010, 1997. [Google Scholar]
  44. A.N. Gorban. Basic types of coarse-graining. In: Model reduction and coarse–graining approaches for multiscale phenomena, Ed. by A.N. Gorban, N. Kazantzis, I.G. Kevrekidis, H.C. Öttinger, C. Theodoropoulos. Springer, Berlin-Heidelberg-New York, 2006, 117–176. E-print: http://arxiv.org/abs/cond-mat/0602024, 2006. [Google Scholar]
  45. A.N. Gorban, V.I. Bykov, G.S. Yablonskii. Macroscopic clusters induced by diffusion in catalytic Oxidation Reactions. Chem. Eng. Sci., 35 (1980), 2351–2352. [CrossRef] [Google Scholar]
  46. A.N. Gorban, V.I. Bykov, G.S. Yablonskii. Essays on chemical relaxation. Novosibirsk, Nauka Publ., 1986. [Google Scholar]
  47. A.N. Gorban, P.A. Gorban, G. Judge. Entropy: The Markov ordering approach. Entropy, 12 (2010), 1145–1193. E-print: http://arxiv.org/abs/1003.1377, 2010. [CrossRef] [MathSciNet] [Google Scholar]
  48. A.N. Gorban, I.V. Karlin, H.C. Öttinger, L.L. Tatarinova. Ehrenfest’s argument extended to a formalism of nonequilibrium thermodynamics. Phys. Rev. E, 63 (2001), 066124. [CrossRef] [Google Scholar]
  49. A.N. Gorban, I.V. Karlin. Uniqueness of thermodynamic projector and kinetic basis of molecular individualism. Physica A, 336 (2004), 391–432. E-print: http://arxiv.org/abs/cond-mat/0309638, 2003. [CrossRef] [Google Scholar]
  50. A.N. Gorban, I.V. Karlin. Method of invariant manifold for chemical kinetics. Chem. Eng. Sci., 58 (2003), 4751–4768. [CrossRef] [Google Scholar]
  51. A.N. Gorban, I.V. Karlin. Invariant manifolds for physical and chemical kinetics. Lect. Notes Phys. 660, Springer, Berlin, Heidelberg, 2005. [Google Scholar]
  52. A.N. Gorban, I.V. Karlin, P. Ilg, H.C. Öttinger. Corrections and enhancements of quasi-equilibrium states. J. Non-Newtonian Fluid Mech., 96 (2001), 203–219. [CrossRef] [Google Scholar]
  53. A.N. Gorban, H.P. Sargsyan. Mass action law for nonlinear multicomponent diffusion and relations between its coefficients. Kinetics and Catalysis, 27 (1986), 527. [Google Scholar]
  54. A.N. Gorban, M. Shahzad. QE+QSS for derivation of kinetic equations and stiffness removing. E-print: http://arxiv.org/abs/1008.3296, 2010. [Google Scholar]
  55. T. Graham. The Bakerian lecture: on the diffusion of liquids. Phil. Trans. R. Soc. Lond., 140 (1) (1850), 1–46; doi: 10.1098/rstl.1850.0001. [CrossRef] [Google Scholar]
  56. M. Grmela, H.C. Öttinger. Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E, 56 (1997), 6620–6632. [CrossRef] [MathSciNet] [Google Scholar]
  57. W.S.C. Gurney, R.M. Nisbet. A note on nonlinear population transport. J. Theor. Biol., 56 (1976), 249–251. [CrossRef] [PubMed] [Google Scholar]
  58. M.E. Gurtin. Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D, 92 (1996), 178–192. [CrossRef] [MathSciNet] [Google Scholar]
  59. I. Gyarmati. Non-equilibrium thermodynamics. Field theory and variational principles. Springer, Berlin, 1970. [Google Scholar]
  60. W. Heitler. Quantum Theory of Radiation. Oxford University Press, London, 1944. [Google Scholar]
  61. R. Hengeveld. Dynamics of biological invasions. Chapman and Hall, London, 1989. [Google Scholar]
  62. F. Horn, R. Jackson. General mass action kinetics. Arch. Rat. Mechan. Anal., 47 (1972), 81–116. [Google Scholar]
  63. W.G. Hoover. Computational statistical mechanics. Elsevier, Amsterdam, 1991. [Google Scholar]
  64. I.V. Karlin, A.N. Gorban, S. Succi, V. Boffi. Maximum entropy principle for lattice kinetic equations. Phys. Rev. Lett., 81 (1998), 6–9. [CrossRef] [Google Scholar]
  65. L.B. Kier, P.G. Seybold, Ch-K. Cheng. Modeling chemical systems using cellular automata. Dordrecht, The Netherlands, 2005. [Google Scholar]
  66. J.F. Kincaid, H. Eyring, A.E. Stearn. The theory of absolute reaction rates and its application to viscosity and diffusion in the liquid State. Chem. Rev., 28 (1941), 301–365. [CrossRef] [Google Scholar]
  67. E.O. Kirkendall. Diffusion of zinc in alpha brass. Trans. Am. Inst. Min. Metall. Eng., 147 (1942), 104–110. [Google Scholar]
  68. A.B. Kudryavtsev, R.F. Jameson, W. Linert. The law of mass action. Springer, Berlin – Heidelberg – New York, 2001. [Google Scholar]
  69. K.J. Laidler, A. Tweedale. The current status of Eyring’s rate theory. In: Advances in Chemical Physics: Chemical dynamics: Papers in honor of Henry Eyring, Volume 21 (eds J. O. Hirschfelder and D. Henderson). John Wiley & Sons, Inc., Hoboken, NJ, USA, 2007. [Google Scholar]
  70. L.D. Landau, E.M. Lifshitz. Fluid mechanics: Volume 6 (Course of theoretical physics). Butterworth-Heinemann, Oxford–Woburn, 1987. [Google Scholar]
  71. J.S. Langer, M. Bar-on, H.D. Miller. New computational method in the theory of spinodal decomposition. Phys. Rev. A, 11 (1975), 1417–1429. [CrossRef] [Google Scholar]
  72. G. Lebon, D. Jou, J. Casas-Vázquez. Understanding non-equilibrium thermodynamics: Foundations, applications, Frontiers. Springer, Berlin, 2008. [Google Scholar]
  73. A.J. Lotka. Elements of physical biology. Williams and Wilkins, Baltimore, 1925. [Google Scholar]
  74. R.J.P. Lyon. Time aspects of geothermometry. Mining Eng., 11 (1959), 1145–1151. [Google Scholar]
  75. B.H. Mahan. Microscopic reversibility and detailed balance. An analysis. J. Chem. Educ., 52 (1975), 299–302. [CrossRef] [Google Scholar]
  76. S. Maier-Paape, B. Stoth, T. Wanner. Spinodal decomposition for multicomponent cahnŰhilliard systems. J. Stat. Phys., 98 (2000), 871–896. [CrossRef] [Google Scholar]
  77. E. McLaughlin. The Thermal conductivity of liquids and dense gases. Chem. Rev., 64 (1964), 389–428. [CrossRef] [Google Scholar]
  78. H. Mehrer. Diffusion in solids – fundamentals, methods, materials, diffusion-controlled processes. Textbook, Springer Series in Solid-State Sciences, Vol. 155, Springer, Berlin – Heidelberg – New York, 2007. [Google Scholar]
  79. H. Mehrer, N.A. Stolwijk. Heroes and highlights in the history of diffusion. Diffusion Fundamentals, 11 (2009), 1–32. [Google Scholar]
  80. L. Michaelis, M. Menten. Die Kinetik der Intervintwirkung. Biochemistry Zeitung, 49 (1913), 333–369. [Google Scholar]
  81. H. Nakajima. The discovery and acceptance of Kirkendall effect: The result of a short research career. JOM, 49 (1997), 15–19. [CrossRef] [Google Scholar]
  82. T.N. Narasimhan. Energetics of the Kirkendall effect. Current Science, 93 (2007), 1257–1264. [Google Scholar]
  83. L. Onsager. Reciprocal relations in irreversible processes. I. Phys. Rev., 37 (1931), 405–426. [NASA ADS] [CrossRef] [Google Scholar]
  84. L. Onsager. Reciprocal relations in irreversible processes. II. Phys. Rev., 38 (1931), 2265–2279. [CrossRef] [Google Scholar]
  85. H.C. Öttinger. Beyond equilibrium thermodynamics. Wiley-Blackwell, Hoboken, NJ, 2005. [Google Scholar]
  86. H.C. Öttinger. Constraints in nonequilibrium thermodynamics: General framework and application to multicomponent diffusion. J. Chem. Phys., 130 (2009), 114904. [CrossRef] [PubMed] [Google Scholar]
  87. H.C. Öttinger, M. Grmela. Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys. Rev. E, 56 (1997), 6633–6655. [CrossRef] [MathSciNet] [Google Scholar]
  88. K. Oura, V.G. Lifshits, A.A. Saranin, A.V. Zotov, M. Katayama. Surface science: An introduction. Springer, Berlin – Heidelberg, 2003. [Google Scholar]
  89. S.V. Petrovskii, B.-L. Li. Exactly solvable models of biological invasion. Chapman & Hall / CRC Press, Boca–Raton–London–New York–Washington D.C., 2006. [Google Scholar]
  90. J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2 (2005), 1.1–1.10. [Google Scholar]
  91. W.C. Roberts-Austen. Bakerian lecture on the diffusion in metals. Phil. Trans. Roy. Soc. A 187, (1896). Part I: Diffusion of Molten Metals. 383–403; Part II: Diffusion of Solid Metals. 404–415. [Google Scholar]
  92. D. Rothman, S. Zaleski. Lattice-gas models of phase separation: interfaces, phase transitions and multiphase flow. Rev. Mod. Phys., 66 (1994), 1417–1480. [CrossRef] [Google Scholar]
  93. P.K. Schelling, S.R. Phillpot, P. Keblinski. Comparison of atomic-level simulation methods for computing thermal conductivity. Phys. Rev. B, 65 (2002), 144306. [CrossRef] [Google Scholar]
  94. N.N. Semenov. Some problems relating to chain reactions and to the theory of combustion. Nobel Lecture, December 11, 1956. In: Nobel lectures in chemistry 1942–1962. World Scientific, Hackensack, NJ, 1999. [Google Scholar]
  95. E. Seneta. Nonnegative matrices and Markov chains. Springer, New York, 1981. [Google Scholar]
  96. N. Shigesada, K. Kawasaki. Biological invasions: theory and practice. Oxford University Press, Oxford, 1997. [Google Scholar]
  97. E.C.G. Stueckelberg. Théorème H et unitarité de S. Helv. Phys. Acta, 25 (1952), 577–580. [MathSciNet] [Google Scholar]
  98. S. Succi. The lattice Boltzmann equation for fluid dynamics and beyond. Clarendon Press, Oxford, UK, 2001. [Google Scholar]
  99. S. Succi, I. Karlin, H. Chen. Role of the H theorem in lattice Boltzmann hydrodynamic simulations (Colloquium). Rev. Mod. Phys., 74 (2002), 1203–1220. [CrossRef] [Google Scholar]
  100. S. Succi, “Lattice Boltzmann at all-scales: from turbulence to DNA translocation”, Mathematical Modelling Centre Distinguished Lecture, University of Leicester, Leicester, UK, 15 November 2006. [Google Scholar]
  101. T. Teorell. Studies on the “diffusion effect” upon ionic distribution–I Some theoretical considerations. Proc. N. A. S. USA, 21 (1935), 152–161. [CrossRef] [Google Scholar]
  102. T. Teorell. Studies on the diffusion effect upon ionic distribution–II Experiments on ionic accumulation. The Journal of General Physiology, 21 (1937), 107–122. [CrossRef] [PubMed] [Google Scholar]
  103. T. Toffoli, N. Margolus. Cellular automata machines: A new environment for modeling. MIT Press, Cambridge, MA, 1987. [Google Scholar]
  104. C. Tuijn. On the history of models for solid–state diffusion. Defect and Diffusion Forum, 143-147 (1997), 11–20. [CrossRef] [Google Scholar]
  105. N.G. Van Kampen. Nonlinear irreversible processes. Physica, 67 (1973), 1–22. [CrossRef] [MathSciNet] [Google Scholar]
  106. P. Van Mieghem. Performance analysis of communications networks and systems. Cambridge University Press, Cambridge, 2006. [Google Scholar]
  107. J.H. Van’t Hoff. Etudes de dynamique chimique. Frederic Muller, Amsterdam, 1884. [Google Scholar]
  108. J.L. Vázquez. The porous medium equation. Mathematical Theory. Oxford University Press, Oxford, 2007. [Google Scholar]
  109. A.I. Volpert, S.I. Khudyaev. Analysis in classes of discontinuous functions and equations of mathematical physics. Nijoff, Dordrecht, 1985. [Google Scholar]
  110. V. Volterra. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei, 2 (1926), 31–113. [Google Scholar]
  111. J. Von Neumann, A.W. Burks. Theory of self-reproducing automata. University of Illinois Press, Urbana, 1966. [Google Scholar]
  112. S. Watanabe. Symmetry of physical laws. Part I. Symmetry in space-time and balance theorems. Rev. Mod. Phys., 27 (1955), 26–39. [CrossRef] [MathSciNet] [Google Scholar]
  113. R. Wegscheider. Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme. Monatshefte für Chemie / Chemical Monthly, 32 (1911), 849–906. [CrossRef] [Google Scholar]
  114. D.A. Wolf-Gladrow. Lattice-gas cellular automata and lattice Boltzmann models. Springer, 2000. [Google Scholar]
  115. S. Wolfram. A new kind of science. Wolfram Media, Champaign, IL, 2002. [Google Scholar]
  116. W.F.K. Wynne-Jones, H. Eyring. The absolute rate of reactions in condensed phases. J. Chem. Phys., 3 (1935), 492–502. [CrossRef] [Google Scholar]
  117. G.S. Yablonskii, V.I. Bykov, A.N. Gorban, V.I. Elokhin. Kinetic models of catalytic reactions. Series “Comprehensive Chemical Kinetics", Vol. 32, Compton R.G. (ed.), Elsevier, Amsterdam, 1991. [Google Scholar]
  118. Y.B. Zeldovich. Proof of the uniqueness of the solution of the equations of the law of mass action. In: Selected Works of Yakov Borisovich Zeldovich; Volume 1, Ostriker, J.P., Ed. Princeton University Press, Princeton, NJ, USA, 1996; pp. 144–148. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.