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A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem
Hard sphere dynamics and the Enskog equation
1. | Institute of Mathematics of NAS of Ukraine, 3, Tereshchenkivs'ka Str., 01601 Kyiv-4, Ukraine |
2. | Taras Shevchenko National University of Kyiv, Department of Mechanics and Mathematics, 2, Academician Glushkov Av.03187, Kyiv, Ukraine |
References:
[1] |
C. Cercignani, V. I. Gerasimenko and D. Ya. Petrina, "Many-Particle Dynamics and Kinetic Equations," Kluwer Acad. Publ., 1997.
doi: 10.1007/978-94-011-5558-8. |
[2] |
C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Springer-Verlag, 1994. |
[3] |
H. Spohn, "Large Scale Dynamics of Interacting Particles," Springer-Verlag, 1991.
doi: 10.1007/978-3-642-84371-6. |
[4] |
D. Ya. Petrina, "Stochastic Dynamics and Boltzmann Hierarchy," Institute of Mathematics, Kyiv, 2008. |
[5] |
O. E. Lanford, Time evolution of large classical systems, in "Dynamical Systems, Theory and Application" (ed. J. Moser), Lecture Notes in Physics, 38, Springer, (1975), 1-111. |
[6] |
H. Grad, Principles of the kinetic theory of gases, in "Handbuch der Physik," 12, Springer, (1958), 205-294. |
[7] |
H. Spohn, Kinetic equations from Hamiltonian dynamics, Reviews of Modern Physics, 52 (1980), 569-615.
doi: 10.1103/RevModPhys.52.569. |
[8] |
D. Enskog, Kinetiche theorie der wärmeleitung, reibung und selbstdiffusion in gewissen werdichteten Gasen und flüβigkeiten, Kungl. Sv. Vetenskapsakademiens Handl., 63 (1922), 3-44. |
[9] |
N. Bellomo, M. Lachowicz, J. Polewczak and G. Toscani, "Mathematical Topics in Nonlinear Kinetic Theory II: The Enskog Equation," World Sci. Publ., 1991. |
[10] |
J. Polewczak, On some open problems in the revised Enskog equation for dense gases, in "Proc. ASCOM 99" (Eds. V. Ciancio, A. Donato, F. Oliveri and S. Rionero), World Sci. Publ., (2001), 382-396. |
[11] |
J. Polewczak, A review of the kinetic modelings for non-reactive and reactive dense fluids: Questions and problems, Rivista di Matematica della Universita' di Parma, 4 (2001), 23-55. |
[12] |
M. Pulvirenti, On the Enskog hierarchy: analyticity, uniqueness, and derivability by particle systems, Rediconti del Circolo Matematico di Palermo, Serie II, Suppl., 45 (1996), 529-542. |
[13] |
N. Bellomo and M. Lachowicz, Kinetic equations for dense gases: A review of mathematical and physical results, J. Modern Phys. B, 1 (1987), 1193-1206.
doi: 10.1142/S0217979287001687. |
[14] |
V. I. Gerasimenko and D. Ya. Petrina, Thermodynamical limit of nonequilibrium states of three dimensional hard spheres system, Theor. Math. Phys., 64 (1985), 130-149.
doi: 10.1007/BF01017041. |
[15] |
H. van Beijeren and M. H. Ernst, The modified Enskog equation, Physica, 68 (1973), 437-456.
doi: 10.1016/0031-8914(73)90372-8. |
[16] |
P. Re'sibois and M. De Leener, "Classical Kinetic Theory of Fluids," John Wiley and Sons, New York, 1977. |
[17] |
N. N. Bogolybov, "Problems of a Dynamical Theory in Statistical Physics," M.: Gostekhizdat, 1946. |
[18] |
V. I. Gerasimenko, T. V. Ryabukha and M. O. Stashenko, On the structure of expansions for the BBGKY hierarchy solutions, J. Phys. A: Math. Gen., 37 (2004), 9861-9872.
doi: 10.1088/0305-4470/37/42/002. |
[19] |
J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications," Springer, 2006. |
[20] |
V. I. Gerasimenko, On the solutions of the BBGKY hierarchy for a one-dimensional hard-sphere system, Theor. Math. Phys., 91 (1992), 120-132.
doi: 10.1007/BF01019833. |
[21] |
V. I. Gerasimenko and Zh. A. Tsvir, A description of the evolution of quantum states by means of the kinetic equation, J. Phys. A: Math. Theor., 43 (2010), 485203, 19 pp. |
[22] |
V. I. Gerasimenko, Approaches to derivation of quantum kinetic equations, Ukrainian J. Phys., 54 (2009), 834-846. |
[23] |
V. I. Gerasimenko and D. Ya. Petrina, On the generalized kinetic equation, Reports of NAS of Ukraine, 7 (1997), 7-12. |
[24] |
V. I. Gerasimenko and D. Ya. Petrina, The generalized kinetic equation generated by the BBGKY hierarchy, Ukrainian J. Phys., 43 (1998), 697-702. |
[25] |
G. Borgioli, V. I. Gerasimenko and G. Lauro, Derivation of a discrete Enskog equation from the dynamics of particles, Rend. Sem. Mat. Univ. Pol. Torino, 56 (1998), 59-69. |
[26] |
G. E. Uhlenbeck and G. W. Ford, "Lecture in Statistical Mechanics," American Mathematical Society Providence, Rhode Island, 1963. |
[27] |
M. S. Green and R. A. Piccirelli, Basis of the functional assumption in the theory of the Boltzmann equation, Phys. Rev., 132 (1963), 1388-1410.
doi: 10.1103/PhysRev.132.1388. |
[28] |
M. S. Green, Boltzmann equation from the statistical mechanical point of view, J. Chem. Phys., 25 (1956), 836-855.
doi: 10.1063/1.1743132. |
[29] |
E. G. D. Cohen, Cluster expansions and the hierarchy. I. Nonequilibrium distribution functions, Physica, 28 (1962), 1045-1065.
doi: 10.1016/0031-8914(62)90009-5. |
show all references
References:
[1] |
C. Cercignani, V. I. Gerasimenko and D. Ya. Petrina, "Many-Particle Dynamics and Kinetic Equations," Kluwer Acad. Publ., 1997.
doi: 10.1007/978-94-011-5558-8. |
[2] |
C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Springer-Verlag, 1994. |
[3] |
H. Spohn, "Large Scale Dynamics of Interacting Particles," Springer-Verlag, 1991.
doi: 10.1007/978-3-642-84371-6. |
[4] |
D. Ya. Petrina, "Stochastic Dynamics and Boltzmann Hierarchy," Institute of Mathematics, Kyiv, 2008. |
[5] |
O. E. Lanford, Time evolution of large classical systems, in "Dynamical Systems, Theory and Application" (ed. J. Moser), Lecture Notes in Physics, 38, Springer, (1975), 1-111. |
[6] |
H. Grad, Principles of the kinetic theory of gases, in "Handbuch der Physik," 12, Springer, (1958), 205-294. |
[7] |
H. Spohn, Kinetic equations from Hamiltonian dynamics, Reviews of Modern Physics, 52 (1980), 569-615.
doi: 10.1103/RevModPhys.52.569. |
[8] |
D. Enskog, Kinetiche theorie der wärmeleitung, reibung und selbstdiffusion in gewissen werdichteten Gasen und flüβigkeiten, Kungl. Sv. Vetenskapsakademiens Handl., 63 (1922), 3-44. |
[9] |
N. Bellomo, M. Lachowicz, J. Polewczak and G. Toscani, "Mathematical Topics in Nonlinear Kinetic Theory II: The Enskog Equation," World Sci. Publ., 1991. |
[10] |
J. Polewczak, On some open problems in the revised Enskog equation for dense gases, in "Proc. ASCOM 99" (Eds. V. Ciancio, A. Donato, F. Oliveri and S. Rionero), World Sci. Publ., (2001), 382-396. |
[11] |
J. Polewczak, A review of the kinetic modelings for non-reactive and reactive dense fluids: Questions and problems, Rivista di Matematica della Universita' di Parma, 4 (2001), 23-55. |
[12] |
M. Pulvirenti, On the Enskog hierarchy: analyticity, uniqueness, and derivability by particle systems, Rediconti del Circolo Matematico di Palermo, Serie II, Suppl., 45 (1996), 529-542. |
[13] |
N. Bellomo and M. Lachowicz, Kinetic equations for dense gases: A review of mathematical and physical results, J. Modern Phys. B, 1 (1987), 1193-1206.
doi: 10.1142/S0217979287001687. |
[14] |
V. I. Gerasimenko and D. Ya. Petrina, Thermodynamical limit of nonequilibrium states of three dimensional hard spheres system, Theor. Math. Phys., 64 (1985), 130-149.
doi: 10.1007/BF01017041. |
[15] |
H. van Beijeren and M. H. Ernst, The modified Enskog equation, Physica, 68 (1973), 437-456.
doi: 10.1016/0031-8914(73)90372-8. |
[16] |
P. Re'sibois and M. De Leener, "Classical Kinetic Theory of Fluids," John Wiley and Sons, New York, 1977. |
[17] |
N. N. Bogolybov, "Problems of a Dynamical Theory in Statistical Physics," M.: Gostekhizdat, 1946. |
[18] |
V. I. Gerasimenko, T. V. Ryabukha and M. O. Stashenko, On the structure of expansions for the BBGKY hierarchy solutions, J. Phys. A: Math. Gen., 37 (2004), 9861-9872.
doi: 10.1088/0305-4470/37/42/002. |
[19] |
J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications," Springer, 2006. |
[20] |
V. I. Gerasimenko, On the solutions of the BBGKY hierarchy for a one-dimensional hard-sphere system, Theor. Math. Phys., 91 (1992), 120-132.
doi: 10.1007/BF01019833. |
[21] |
V. I. Gerasimenko and Zh. A. Tsvir, A description of the evolution of quantum states by means of the kinetic equation, J. Phys. A: Math. Theor., 43 (2010), 485203, 19 pp. |
[22] |
V. I. Gerasimenko, Approaches to derivation of quantum kinetic equations, Ukrainian J. Phys., 54 (2009), 834-846. |
[23] |
V. I. Gerasimenko and D. Ya. Petrina, On the generalized kinetic equation, Reports of NAS of Ukraine, 7 (1997), 7-12. |
[24] |
V. I. Gerasimenko and D. Ya. Petrina, The generalized kinetic equation generated by the BBGKY hierarchy, Ukrainian J. Phys., 43 (1998), 697-702. |
[25] |
G. Borgioli, V. I. Gerasimenko and G. Lauro, Derivation of a discrete Enskog equation from the dynamics of particles, Rend. Sem. Mat. Univ. Pol. Torino, 56 (1998), 59-69. |
[26] |
G. E. Uhlenbeck and G. W. Ford, "Lecture in Statistical Mechanics," American Mathematical Society Providence, Rhode Island, 1963. |
[27] |
M. S. Green and R. A. Piccirelli, Basis of the functional assumption in the theory of the Boltzmann equation, Phys. Rev., 132 (1963), 1388-1410.
doi: 10.1103/PhysRev.132.1388. |
[28] |
M. S. Green, Boltzmann equation from the statistical mechanical point of view, J. Chem. Phys., 25 (1956), 836-855.
doi: 10.1063/1.1743132. |
[29] |
E. G. D. Cohen, Cluster expansions and the hierarchy. I. Nonequilibrium distribution functions, Physica, 28 (1962), 1045-1065.
doi: 10.1016/0031-8914(62)90009-5. |
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