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.
  2. G.I. Barenblatt, Y.B. ZelŠdovich. Self-similar solutions as intermediate asymptotics. Ann. Rev. Fluid Mech., 4 (1972), 285–312. [CrossRef]
  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]
  4. T. Blesgen, U. Weikard. Multi-component Allen-Cahn equation for elastically stressed solids. Electron. J. Diff. Eqns., 89 (2005), 1–17.
  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]
  6. L. Boltzmann. Lectures on gas theory. U. of California Press, Berkeley, CA, 1964.
  7. G.E. Briggs, J.B.S. Haldane. A note on the kinetics of enzyme action. Biochem. J., 19 (1925), 338–339. [PubMed]
  8. R.A. Brownlee, A.N. Gorban, J. Levesley. Nonequilibrium entropy limiters in lattice Boltzmann methods. Physica A, 387 (2008), 385–406. [CrossRef] [MathSciNet]
  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.
  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]
  11. J.W. Cahn. Free energy of a nonuniform system. II. Thermodynamic basis. J. Chem. Phys., 30 (1959), 1121–1124. [CrossRef]
  12. J.W. Cahn, J.E. Hilliard. Free energy of a nonuniform system. I. Interfacial energy. J. Chem. Phys., 28 (1958), 258–266. [CrossRef]
  13. J.W. Cahn, J.E. Hilliard. Spinodal decomposition: A reprise. Acta Metallurgica, 19 (1971), 151–161. [CrossRef]
  14. H.B. Callen. Thermodynamics and an introduction to themostatistics (2nd ed.). John Wiley & Sons, NY, 1985.
  15. C. Cercignani, M. Lampis. On the H-theorem for polyatomic gases. J. Stat. Phys., 26 (1981), 795–801. [CrossRef]
  16. B. Chopard, M. Droz. Cellular automata modeling of physical systems. Cambridge University Press, Cambridge, UK, 1998.
  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]
  18. A.J. Chorin, O.H. Hald, R. Kupferman. Optimal prediction with memory. Physica D, 166 (2002), 239–257. [CrossRef] [MathSciNet]
  19. F. Coester. Principle of detailed balance. Phys. Rev., 84, 1259 (1951)
  20. S.R. De Groot, P. Mazur. Non-equilibrium Thermodynamics. North-Holland, Amsterdam, 1962.
  21. K. Denbigh. The principles of chemical equilibrium. Cambridge University Press, Cambridge, UK, 1981.
  22. S. Dushman, I. Langmuir. The diffusion coefficient in solids and its temperature coefficient. Phys. Rev., 20 (1922), 113.
  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.)
  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]
  25. A. Einstein. Strahlungs-Emission und -Absorption nach der Quantentheorie. Verhandlungen der Deutschen Physikalischen Gesellschaft, 18 (1916), No. 13/14, Braunschweig, Vieweg, 318–323.
  26. C.M. Elliott, Z. Songmu. On the Cahn-Hilliard equation. Arch. Rat. Mechan. Anal., 96 (1986), 339–357.
  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]
  28. H. Eyring. The activated complex in chemical reactions. J. Chem. Phys., 3 (1935), 107–115. [CrossRef]
  29. M. Feinberg. On chemical kinetics of a certain class. Arch. Rat. Mechan. Anal., 46 (1972), 1–41.
  30. M. Feinberg. Complex balancing in general kinetic systems. Arch. Rat. Mechan. Anal., 49 (1972), 187–194.
  31. R.F. Feynman. Simulating physics with computers. Internat. J. Theor. Phys., 21 (1982), 467–488. [CrossRef]
  32. A. Fick. Über Diffusion. Poggendorff’s Annalen der Physik und Chemie, 94 (1855), 59–86. [NASA ADS] [CrossRef]
  33. R.A. Fisher. The genetical theory of natural selection. Oxford University Press, Oxford, 1930.
  34. F.C. Frank, D. Turnbull. Mechanism of diffusion of copper in Germanium. Phys. Rev., 104 (1956), 617–618. [CrossRef]
  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]
  36. J. Frenkel. Theorie der Adsorption und verwandter Erscheinungen. Zeitschrift für Physik, 26 (1924), 117–138
  37. J. Frenkel. Über die Wärmebewegung in festen und flüssigen Körpern. Zeitschrift für Physik, 35 (1925), 652–669.
  38. G.F. Gause. The struggle for existence. Williams & Wilkins, Baltimore, 1934.
  39. J.W. Gibbs. On the equilibrium of heterogeneous substance. Trans. Connect. Acad., 1875–1876, 108–248; 1877–1878, 343–524.
  40. D.T. Gillespie. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81 (1977), 2340–2361. [CrossRef]
  41. D.T. Gillespie. Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem., 58 (2007), 35–55. [NASA ADS] [CrossRef] [PubMed]
  42. A.N. Gorban. Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis. Nauka, Novosibirsk, 1984.
  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.
  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.
  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]
  46. A.N. Gorban, V.I. Bykov, G.S. Yablonskii. Essays on chemical relaxation. Novosibirsk, Nauka Publ., 1986.
  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]
  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]
  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]
  50. A.N. Gorban, I.V. Karlin. Method of invariant manifold for chemical kinetics. Chem. Eng. Sci., 58 (2003), 4751–4768. [CrossRef]
  51. A.N. Gorban, I.V. Karlin. Invariant manifolds for physical and chemical kinetics. Lect. Notes Phys. 660, Springer, Berlin, Heidelberg, 2005.
  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]
  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.
  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.
  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]
  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]
  57. W.S.C. Gurney, R.M. Nisbet. A note on nonlinear population transport. J. Theor. Biol., 56 (1976), 249–251. [CrossRef] [PubMed]
  58. M.E. Gurtin. Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D, 92 (1996), 178–192. [CrossRef] [MathSciNet]
  59. I. Gyarmati. Non-equilibrium thermodynamics. Field theory and variational principles. Springer, Berlin, 1970.
  60. W. Heitler. Quantum Theory of Radiation. Oxford University Press, London, 1944.
  61. R. Hengeveld. Dynamics of biological invasions. Chapman and Hall, London, 1989.
  62. F. Horn, R. Jackson. General mass action kinetics. Arch. Rat. Mechan. Anal., 47 (1972), 81–116.
  63. W.G. Hoover. Computational statistical mechanics. Elsevier, Amsterdam, 1991.
  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]
  65. L.B. Kier, P.G. Seybold, Ch-K. Cheng. Modeling chemical systems using cellular automata. Dordrecht, The Netherlands, 2005.
  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]
  67. E.O. Kirkendall. Diffusion of zinc in alpha brass. Trans. Am. Inst. Min. Metall. Eng., 147 (1942), 104–110.
  68. A.B. Kudryavtsev, R.F. Jameson, W. Linert. The law of mass action. Springer, Berlin – Heidelberg – New York, 2001.
  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.
  70. L.D. Landau, E.M. Lifshitz. Fluid mechanics: Volume 6 (Course of theoretical physics). Butterworth-Heinemann, Oxford–Woburn, 1987.
  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]
  72. G. Lebon, D. Jou, J. Casas-Vázquez. Understanding non-equilibrium thermodynamics: Foundations, applications, Frontiers. Springer, Berlin, 2008.
  73. A.J. Lotka. Elements of physical biology. Williams and Wilkins, Baltimore, 1925.
  74. R.J.P. Lyon. Time aspects of geothermometry. Mining Eng., 11 (1959), 1145–1151.
  75. B.H. Mahan. Microscopic reversibility and detailed balance. An analysis. J. Chem. Educ., 52 (1975), 299–302. [CrossRef]
  76. S. Maier-Paape, B. Stoth, T. Wanner. Spinodal decomposition for multicomponent cahnŰhilliard systems. J. Stat. Phys., 98 (2000), 871–896. [CrossRef]
  77. E. McLaughlin. The Thermal conductivity of liquids and dense gases. Chem. Rev., 64 (1964), 389–428. [CrossRef]
  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.
  79. H. Mehrer, N.A. Stolwijk. Heroes and highlights in the history of diffusion. Diffusion Fundamentals, 11 (2009), 1–32.
  80. L. Michaelis, M. Menten. Die Kinetik der Intervintwirkung. Biochemistry Zeitung, 49 (1913), 333–369.
  81. H. Nakajima. The discovery and acceptance of Kirkendall effect: The result of a short research career. JOM, 49 (1997), 15–19. [CrossRef]
  82. T.N. Narasimhan. Energetics of the Kirkendall effect. Current Science, 93 (2007), 1257–1264.
  83. L. Onsager. Reciprocal relations in irreversible processes. I. Phys. Rev., 37 (1931), 405–426. [NASA ADS] [CrossRef]
  84. L. Onsager. Reciprocal relations in irreversible processes. II. Phys. Rev., 38 (1931), 2265–2279. [CrossRef]
  85. H.C. Öttinger. Beyond equilibrium thermodynamics. Wiley-Blackwell, Hoboken, NJ, 2005.
  86. H.C. Öttinger. Constraints in nonequilibrium thermodynamics: General framework and application to multicomponent diffusion. J. Chem. Phys., 130 (2009), 114904. [CrossRef] [PubMed]
  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]
  88. K. Oura, V.G. Lifshits, A.A. Saranin, A.V. Zotov, M. Katayama. Surface science: An introduction. Springer, Berlin – Heidelberg, 2003.
  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.
  90. J. Philibert. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2 (2005), 1.1–1.10.
  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.
  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]
  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]
  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.
  95. E. Seneta. Nonnegative matrices and Markov chains. Springer, New York, 1981.
  96. N. Shigesada, K. Kawasaki. Biological invasions: theory and practice. Oxford University Press, Oxford, 1997.
  97. E.C.G. Stueckelberg. Théorème H et unitarité de S. Helv. Phys. Acta, 25 (1952), 577–580. [MathSciNet]
  98. S. Succi. The lattice Boltzmann equation for fluid dynamics and beyond. Clarendon Press, Oxford, UK, 2001.
  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]
  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.
  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]
  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]
  103. T. Toffoli, N. Margolus. Cellular automata machines: A new environment for modeling. MIT Press, Cambridge, MA, 1987.
  104. C. Tuijn. On the history of models for solid–state diffusion. Defect and Diffusion Forum, 143-147 (1997), 11–20. [CrossRef]
  105. N.G. Van Kampen. Nonlinear irreversible processes. Physica, 67 (1973), 1–22. [CrossRef] [MathSciNet]
  106. P. Van Mieghem. Performance analysis of communications networks and systems. Cambridge University Press, Cambridge, 2006.
  107. J.H. Van’t Hoff. Etudes de dynamique chimique. Frederic Muller, Amsterdam, 1884.
  108. J.L. Vázquez. The porous medium equation. Mathematical Theory. Oxford University Press, Oxford, 2007.
  109. A.I. Volpert, S.I. Khudyaev. Analysis in classes of discontinuous functions and equations of mathematical physics. Nijoff, Dordrecht, 1985.
  110. V. Volterra. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei, 2 (1926), 31–113.
  111. J. Von Neumann, A.W. Burks. Theory of self-reproducing automata. University of Illinois Press, Urbana, 1966.
  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]
  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]
  114. D.A. Wolf-Gladrow. Lattice-gas cellular automata and lattice Boltzmann models. Springer, 2000.
  115. S. Wolfram. A new kind of science. Wolfram Media, Champaign, IL, 2002.
  116. W.F.K. Wynne-Jones, H. Eyring. The absolute rate of reactions in condensed phases. J. Chem. Phys., 3 (1935), 492–502. [CrossRef]
  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.
  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.

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.