On the mathematical transport theory in microporous media: The billiard approach

Nonlinear Analysis: Hybrid Systems

Volumn 4, Issue 4, 2010, Pages 699-735

  Bianca, C ^a  

^a POLITECNICO DI TORINO   (Italy)

Author keywords

Billiards; Decay of correlations; Ergodicity; Hyperbolicity; Kolmogorov Sinai entropy; Lyapunov exponents; Membranes; Thermostats

Indexed keywords

BILLIARDS; DECAY OF CORRELATIONS; ERGODICITY; HYPERBOLICITY; KOLMOGOROV-SINAI ENTROPY; LYAPUNOV EXPONENT;

COMPUTATIONAL COMPLEXITY; DIFFERENTIAL EQUATIONS; ENTROPY; HYBRID SYSTEMS; LYAPUNOV FUNCTIONS; LYAPUNOV METHODS; MEDICAL APPLICATIONS; MICROPOROSITY; STATISTICAL MECHANICS; THERMOSTATS;

CHAOS THEORY;

EID: 77956232342     PISSN: 1751570X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nahs.2010.04.007     Document Type: Article

References (134)

1. Desai T.A., Hansford D.J., Kulinsky L., Nashat A.H., Rasi G., Tu J., Wang Y., Zhang M., Ferrari M. Nanopore technology for biomedical applications. Biomedical Microdevices 1999, 2:11-13.
2. Davis H.T. Fundamentals of Inhomogeneous Fluids 1992, Marcel Dekker, New York. D. Henderson (Ed.).
3. Pozhar L.A. Transport Theory of Inhomogeneous Fluids 1994, World Scientific, Singapore.
4. Travis K.P., Gubbins K.E. Poiseuille flow of Lennard-Jones fluids in narrow slit pores. J. Chem. Phys. 2000, 112:1984-1994.
5. Jepps O., Bianca C., Rondoni L. Onset of diffusive behaviour in confined transport systems. Chaos 2008, 18:013127.
6. Bellomo N., Bianca C., Delitala M. Complexity analysis and mathematical tools towards the modelling of living systems. Phys. Life Rev. 2009, 6:144-175.
7. Bianca C. On the modelling of space dynamics in the kinetic theory for active particles. Math. Comput. Modelling 2010, 51:72-83.
8. C. Bianca, Modelling aggregation-fragmentation phenomena by the kinetic theory for active particles, preprint.
9. Bellomo N., Bellouquid A. On the derivation of macroscopic tissue equations from hybrid models of the kinetic theory of multicellular growing systems - the effect of global equilibrium. Nonlinear Anal.: Hybrid Systems 2009, 3:215-224.
10. Gallavotti G., Cohen E.G.D. Dynamical esembles in stationary states. J. Stat. Phys. 1995, 80:931-970.
11. Gallavotti G., Cohen E.G.D. Dynamical esembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 1995, 94:2694-2697.
12. Gaspard P. Chaos, Scattering and Statistical Mechanics 1998, Cambridge University Press, Cambridge.
13. Bertini L., De Sole A., Gabrielli D., Jona-Lasinio G., Landin C. Current fluctuations in stochastic lattice gases. Phys. Rev. Lett. 2005, 94.
14. Evans D.J., Cohen E.G.D., Morris G.P. Probability of second law violations in shearing steady flows. Phys. Rev. Lett. 1993, 71:2401.
15. Gallavotti G. Reversible Anosov diffeomorphisms and large deviations. Mat. Phys. Electron. J. 1995, 1:1-12.
16. Gentile G. Large deviations for Anosov flows. Forum math. 1998, 10:89.
17. Arya G., Chang H.-C., Maginn E.J. Molecular simulations of Knudsen wall-slip: effect of wall morphology. Mol. Simul. 2003, 29:697-709.
18. Arya G., Chang H.-C., Maginn E.J. Knudsen diffusivity of a hard sphere in a rough slit pore. Phys. Rev. Lett. 2003, 91. 026102-1.
19. Sinai Ya.G. Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russian Math. Surveys 1970, 25:137-189.
20. Arnold V.I., Avez A. Ergodic Problems of Classical Mechanics 1968, W.A. Benjamin, New York.
21. Dettman C.P., Cohen E.G.D., Van Beijeren H. Microscopic chaos from Brownian motion?. Nature 1999, 401:875.
22. ~ P., Ehrenfest T. The Conceptual Fondations of the Statistical Approach in Mechanics (M.J. Moravicsik, Trans.) 1959, 10-13. Cornell University Press, Itacha NY.
23. Dettmann C.P., Morriss G.P. Hamiltonian formulation of the gaussian isokinetic thermostat. Phys. Rev. E 1996, 54:2495.
24. Chernov N.I. Sinai billiards under small external forces. Ann. Henri Poincaré 2001, 2:197-236.
25. Lepri S., Rondoni L., Benettin G. The Gallavotti-Cohen fluctuation theorem for a nonchaotic model. J. Stat. Phys. 2000, 99:857.
26. C. Bianca, Chaotic and polygonal billiards as models of mass transport in microporous media, Ph.D. Thesis, Politecnico di Torino, 2008.
27. Rondoni L. Deterministic thermostats and fluctuation relations. Dynamics of Dissipation, vol. 597 2002, 35-61. Springer Verlag, Berlin. P. Garbaczewski, R. Olkiewicz (Eds.).
28. Klages R. Microscopic Chaos, Fractals and Transport in Nonequilibrium, Statistical Mechanics 2007, World Scientific, Singapore.
29. G. Gallavotti, Modern theory of billiards-an introduction. Global analysis and its applications, Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, (1972), vol. 2, 193-202; Internat. Atomic Energy Agency, Vienna, (1974).
30. G. Gallavotti, Lectures on the billiard in Dynamical systems, theory and applications Rencontres, Battelle Res. Inst., Seattle., Wash., (1974), 23-6-295. Lecture Notes in Phys., vol. 38, Springer, Berlin, (1975).
31. Sinai Ya.G., Chernov N.I. Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls. Russian Math. Surveys 1987, 42:101-207.
32. Bunimovich L.A. Many-dimensional nowhere dispersing billiards with chaotic behaviour. Physica D 1988, 33:58-64.
33. Bunimovich L.A., Rehacek J. Nowhere dispersing 3D billiards with non-vanishing Lyapunov exponents. Comm. Math. Phys. 1997, 189:729-758.
34. Bunimovich L.A., Rehacek J. How high-dimensional stadia look like. Comm. Math. Phys. 1998, 197:277-301.
35. Bunimovich L.A., Casati G., Guarneri I. Chaotic focusing billiards in higher dimensions. Phys. Rev. Lett. 1996, 77:2941-2944.
36. Wojtkowski M.P. Linearly stable orbits in 3 dimensional billiards. Comm. Math. Phys. 1990, 129:319-327.
37. Bálint P., Chernov N., Szász D., Tóth I. Geometry of multi-dimensional disperding billiards. Astérisque 2003, 286:37-59. Geometric methods in dynamics. I.
38. Bálint P., Tóth I. Hyperbolicity in multi-dimensional Hamiltonian systems with applications to soft billiards. Discrete Contin. Dyn. Syst. 2006, 15:37-59.
39. Anosov D.V. Geodesics flows in Riemannian manifolds with negative curvature. Proc. Steklov. Inst. 1967, 90.
40. Wojtkowski M.P., Liverani C. Conformally symplectic dynamics and symmetry of the Laypunov spectrum. Comm. Math. Phys. 1998, 194:47-60.
41. Donnay V.J. Geodesic flow on the two sphere. I: positive measure entropy. Ergodic Theory Dynam. Systems 1988, 8:531-553.
42. Chernov N.I., Markarian R. Introduction to The Ergodic Theory of Chaotic Billiards 2003, IMPA, Rio de Janeiro, Brasil. 2nd ed.
43. Katok A., Strelcyn J.M. Invariant Manifolds, Entropy and Billiard; Smooth Map with Singulatiries. Lect. Notes Math. 1986, vol. 1222. Springer, New York.
44. Santalo L.A. Integral Geometry and Geometric Probability 1976, Addison Wesley, Reading, Mass.
45. Wojtkowski M.P. Measure theoretic entropy of the system of hard spheres. Ergodic Theory Dynam. Systems 1988, 8:133-153.
46. Birkhoff G.D. Proof of the ergodic theorem. Proc. Natl. Acad. Sci. USA 1931, 17:656-660.
47. Katok A.B., Sinai Ya.G., Stepin A.M. Theory of dynamical systems and general transformation groups with invariant measure. J. Soviet Math. 1977, 7:974-1065.
48. Peterson K. Ergodic Theory 1983, Cambridge Univ. Press.
49. Katok A. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. I.H.E.S. 1980, 51:137-173.
50. Rondoni L., Morriss G.P. Applications of periodic orbit theory to N-particle systems. J. Stat. Phys. 1997, 86:991.
51. Lloyd J., Niemeyer M., Rondoni L., Morriss G.P. The nonequilibrium Lorentz gas. Chaos 1995, 5:536.
52. Rondoni L., Cohen E.G.D. Orbital measures in statistical mechanics. Nonlinearity 1998, 11:1395-1406.
53. Morriss G.P., Rondoni L. Periodic orbit expansions for the Lorentz gas. J. Stat. Phys. 1994, 75:553.
54. Parry W. Synchronisation of canonical measures for hyperbolic attractors. Comm. Math. Phys. 1986, 106:267.
55. Wojtkowski M. Principles for the design of billiards with nonvanishing Lyapunov exponents. Comm. Math. Phys. 1986, 105:391-414.
56. Pesin Ya.B. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys D 1977, 32:55-114.
57. Benettin G., Galgani L., Giorgilli A., Strelcyn J.M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, part 1: theory. Meccanica 1980, 15:9-20.
58. Benettin G., Galgani L., Giorgilli A., Strelcyn J.M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, part 2: numerical applications. Meccanica 1980, 15:21-30.
59. Dellago Ch., Glatz L., Posch H.A. Lyapunov spectrum of the driven Lorentz gas. Phys. Rev. E 1995, 52:4817-4826.
60. Chernov N. Decay of correlations and dispersing billiards. J. Stat. Phys. 1999, 94:513-556.
61. Chernov N. Advanced statistical properties of dispersing billiards. J. Stat. Phys. 2006, 122.
62. Chernov N., Zhang H.-K. Billiards with polynomial mixing rates. Nonlinearity 2005, 18:1527-1553.
63. Chernov N., Zhang H.-K. A family of chaotic billiards with variable mixing rates. Stoch. Dyn. 2005, 5:535-553.
64. Markarian R. Billiards with polynomial decay of correlations. Erg. Th. Dyn. Sys. 2004, 24:177-197.
65. Melbourne I., Torok A. Statistical limit theorems for suspension flows. Israel J. Math. 2004, 144:191-209.
66. Young L.-S. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 1998, 147:585-650.
67. Young L.-S. Recurrence times and rates of mixing. Israel J. Math. 1999, 110:153-188.
68. Markarian R. Non-uniformly hyperbolic billiards. Ann. Fac. Sci. Toulose Math. 1994, 3:223-257.
69. Wojtkowski M. Invariant families of cones and Lyapunov exponents. Ergodic Theory Dynam. Systems 1985, 5:145-161.
70. Tabachnikov S. Billiards. Panor. Synth. 1995, vol. 1. SMF, Paris.
71. Birkoff G.D. Dynamical Systems 1927, A.M.S. Colloquium Publications, New York.
72. Amiran E. Caustics and evolutes for convex planar domains. J. Differential Geometry 1988, 28:345-357.
73. Lazutkin V.F. On the existence of caustics for the billiard ball problem in a convex domain. Math. USSR Izv. 1973, 7:185-215.
74. Gutkin E. Billiards in polygons. Physica D 1986, 19:311-333.
75. Kerckhoff S., Masur H., Smile J. Ergodicity of billiard flows and quadratic differentials. Ann. of Math. 1986, 124:293-311.
76. Galperin G.A. Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons. Comm. Math. Phys. 1983, 91:187-211.
77. Boshernitzan M., Galperin G., Krüger T., Troubetzkoy S. Periodic billiard orbits are dense in rational polygons. Trans. AMS 1998, 350:3523-3535.
78. Troubetzkoy S. Periodic billiards in right triangles. Ann. Inst. Fourier 2005, 55:29-46.
79. Cipra B., Hanson R., Kolan A. Periodic trajectories in right triangle billiards. Phys. Rev. E 1995, 52:2066-2071.
80. Galperin G., Kruger T., Troubetzkoy S. Local instabilityof orbits in polygonal and polyhedral billiards. Comm. Math. Phys. 1995, 169:463-473.
81. Katok A. The growth rate for a number of singular and periodic orbits for a polygonal billiards. Comm. Math. Phys. 1987, 111:151-160.
82. Boldrighini C., Keane M., Marchetti F. Billiards in polygons. Ann. Probab. 1978, 6:532-540.
83. Sinai Ya.G. Introduction to Ergodic Theory 1977, Princeton University Press, Princeton, NJ, p. 169.
84. Zemlyakov A., Katok A. Topological transitivity of billiards in polygons. Math. Notes 1976, 18:760-764.
85. Vorobets Ya.B. Ergodicity of billiards in polygons: explicit examples. Usp. Mat. Nauk. 1996, 51:151-152.
86. Katok A. Interval exchange trasformations and some special flows are not mixing. Israel. J. Math. 1980, 35:301-310.
87. Gutkin E., Katok K. Weakly mixing Billiards holomorphic dynamics. Lecture Notes in Mathematics 1989, Springer, Berlin, pp. 163-176.
88. Troubetzkoy S. Billiards in infinite polygons. Nonlinearity 1999, 12:513-524.
89. Chernov N.I., Troubetzkoy S. Ergodicity of billiards in polygons with pockets. Nonlinearity 1999, 12:513-524.
90. Halpern B. Strange billiard tables. Trans. AMS 1977, 232:297-305.
91. Chernov N.I. A new proof of Sinai's formula for entropy of hyperbolic billiards. Its application to Lorentz gas and Stadium. Funct. Anal. Appl. 1991, 25:204-219.
92. Bunimovich L.A. On the ergodic properties of nowhere dispersing billiards. Comm. Math. Phys. 1979, 65:295-312.
93. Benettin G. Power law behavior of Lyapunov exponent in some conservative dynamical system. Physica D 1984, 13:211-220.
94. Bunimovich L.A. On billiards close to dispersing. Math. USSR Sb. 1974, 95:49-73.
95. Bunimovich L.A. On stocastic dynamics of rays in resonators. Radiofizika 1985, 28:1601-1602. (in Russian).
96. Markarian R. Billiards with Pesin region of measure one. Comm. Math. Phys. 1988, 118:87-97.
97. Bunimovich L.A. On Absolutely Focusing Mirrors, Ergodic Theory and Related Topics, III (Gustrow, 1990). Lectures Notes in Math. 1992, vol. 1514:62-82. Springer, Berlin. U. Krengel (Ed.).
98. Donnay V. Using integrability to produce chaos: billiards with positive entropy. Comm. Math. Phys. 1991, 141:225-257.
99. Moser J. Is the solar system stable?. Math. Intell. 1978, 1:65-71.
100. Boyland Ph. Dual billiards, twist maps and impact oscillators. Nonlinearity 1996, 9:1411-1438.
101. Dogru F., Tabachnikov S. On polygonal dual billiard in the hyperbolic plane. Regul. Chaotic Dynam. 2003, 8:67-82.
102. Gutkin E., Katok A. Caustics for inner and outer billiards. Comm. Math. Phys. 1995, 173:101-134.
103. Gutkin E., Simanyi N. Dual polygonal billiards and necklace dynamics. Comm. Math. Phys. 1991, 143:431-450.
104. Tabachnikov S. Outer billiards. Russian Math. Surveys 1993, 48:81-109.
105. Tabachnikov S. Poncelet's theorem and dual billiards. L'Enseign. Math. 1993, 39:189-194.
106. Tabachnikov S. On the dual billiard problem. Adv. Math. 1995, 115:221-249.
107. Tabachnikov S. Fagnano orbits of polygonal dual billiards. Geom. Dedicata 1999, 77:279-286.
108. Lorentz H.A. The motion of the electrons in metal bodies. Proc. Amst. Acad. 1905, 7:438-453.
109. Vance W.N. Unstable periodic orbits and transport properties of nonequilibrium steady states. Phys. Rev. Lett. 1992, 69:1356.
110. Cvitanović P., Gaspard P., Schreiber T. Investigation of the Lorentz gas in terms of periodic orbits. Chaos 1992, 2:85.
111. Gallavotti G., Ornstein D. Billiards and Bernoulli schemes. Comm. Math. Phys. 1974, 38:83-101.
112. Bunimovich L.A., Sinai Ya.G., Chernov N.I. Statistical properties of two-dimensional hyperbolic billiards. Russiean Math. Surveys 1991, 46:47-106.
113. Bleher P. Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon. J. Statist. Phys. 1992, 66:315-373.
114. Friedman B., Martin R.F. Decay of the velocity autocorrelation function for the periodic Lorentz gas. Phys. Lett. A 1984, 105:23-26.
115. Friedman B., Martin R.F. Behavior of the velocity autocorrelation function for the periodic Lorentz gas. Physica D 1988, 30:219-227.
116. Friedman B., Oono Y., Kubo I. Universal behavior of Sinai billiard systems in the small-scatterer limit. Phys. Rev. Lett. 1988, 52:87-97. (1984) pp. 709-712.
117. Bouchaud J.P., Le Doussal P. Numerical study of a d-dimensional periodic Lorentz gas with universal properties. J. Stat. Phys. 1985, 41:225-248.
118. A. Latz, H. Van Beijeren, J.R. Dorman, Lyapunov spectrum and the conjugate pairing rule for a thermostatted random Lorentz gas: Kinetic theory, manuscript, 1996.
119. Dellago Ch., Posch H.A. Lyapunov spectrum and the conjugate pairing rule for a thermostatted random Lorentz gas: numerical simulations. Phys. Rev. Lett. 1997, 78:211-214.
120. Bowen R., Ruelle D. The ergodic theory of Axiom A flows. Invent. Math. 1975, 29:181-202.
121. Ruelle D. Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 1999, 95:393-468.
122. Ruelle D. A measure associated with Axiom-A attractors. Amer. J. Math. 1976, 98:619-654.
123. Sinai Ya.G. Gibbs measures in ergodic theory. Russian Math. Surveys 1972, 27:21-69.
124. Gallavotti G. New methods in nonequilibrium gases and fluids. Open Sys. Inf. Dynam. 1999, 6:101-136.
125. Gallavotti G., Ruelle D. SRB states and nonequilibrium statistical mechanics close to equilibrium. Comm. Math. Phys. 1997, 190:279-281.
126. Hu H., Young L.-S. Nonexistence of SBR measures for some diffeomorphisms that are almost Anosov. Ergodic Theory Dynam. Systems 1995, 15:67-76.
127. Young L.-S. What are SRB measures, and which dynamical systems have them?. J. Stat. Phys. 2002, 108:733-754.
128. Chernov N.I., Eyink G.L., Lebowitz J.L., Sinai Ya.G. Steady-states electrical condution in the periodic Lorentz gas. Comm. Math. Phys. 1993, 154:569-601.
129. Moran B., Hoover W. Diffusion in a periodic Lorentz gas. J. Stat. Phys. 1987, 48:709-726.
130. Evans D.J., Morriss G.P. Statistical Mechanics of non Equilibrium Liquids 1990, Academic Press, San Diego, CA.
131. Bonetto F., Daems D., Lebowitz J. Property of stationary nonequilibrium states in the thermostatted periodic Lorentz gas I: the one particle system. J. Stat. Phys. 2000, 101:35-60.
132. Wojtkowski M.P. Magnetic flows and gaussian thermostats on manifolds of negative curvature. Ergodic Fund. Math. 2000, 163:177-191.
133. Wojtkowski M.P. W-Flows on Weyl manifolds and gaussian thermostats. J. Math. Pures Appl. 2000, 79:953-974.
134. Bianca C., Rondoni L. The Nonequilibrium Ehrenfest Gas: a Chaotic Model with Flat Obstacles?. Chaos 2009, 19:013121.