# American Institute of Mathematical Sciences

September  2012, 5(3): 459-484. doi: 10.3934/krm.2012.5.459

## 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

Received  August 2011 Revised  January 2012 Published  August 2012

We develop a rigorous formalism for the description of the kinetic evolution of infinitely many hard spheres. On the basis of the kinetic cluster expansions of cumulants of groups of operators of finitely many hard spheres which are the generating operators of a nonperturbative solution of the Cauchy problem of the BBGKY hierarchy the nonlinear kinetic Enskog equation is derived. It is established that for initial states which are specified in terms of one-particle distribution functions the description of the evolution by the Cauchy problem of the BBGKY hierarchy and by the Cauchy problem of the generalized Enskog kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation are an equivalent. For the initial-value problem of the generalized Enskog equation the existence theorem is proved in the space of integrable functions.
Citation: Viktor I. Gerasimenko, Igor V. Gapyak. Hard sphere dynamics and the Enskog equation. Kinetic and Related Models, 2012, 5 (3) : 459-484. doi: 10.3934/krm.2012.5.459
##### 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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##### 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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