Hard sphere dynamics and the Enskog equation

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

Hard sphere dynamics and the Enskog equation

Viktor I. Gerasimenko ^1, and Igor V. Gapyak ^2,

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

Abstract
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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.

Keywords: BBGKY hierarchy, hard spheres., Enskog equation, scattering operators, kinetic cluster expansion.

Mathematics Subject Classification: Primary: 35Q20; Secondary: 47J3.

Citation: Viktor I. Gerasimenko, Igor V. Gapyak. Hard sphere dynamics and the Enskog equation. Kinetic & 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. Google Scholar
[2]	C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,", Springer-Verlag, (1994). Google Scholar
[3]	H. Spohn, "Large Scale Dynamics of Interacting Particles,", Springer-Verlag, (1991). doi: 10.1007/978-3-642-84371-6. Google Scholar
[4]	D. Ya. Petrina, "Stochastic Dynamics and Boltzmann Hierarchy,", Institute of Mathematics, (2008). Google Scholar
[5]	O. E. Lanford, Time evolution of large classical systems,, in, 38 (1975), 1. Google Scholar
[6]	H. Grad, Principles of the kinetic theory of gases,, in, 12 (1958), 205. Google Scholar
[7]	H. Spohn, Kinetic equations from Hamiltonian dynamics,, Reviews of Modern Physics, 52 (1980), 569. doi: 10.1103/RevModPhys.52.569. Google Scholar
[8]	D. Enskog, Kinetiche theorie der wärmeleitung, reibung und selbstdiffusion in gewissen werdichteten Gasen und flüβigkeiten,, Kungl. Sv. Vetenskapsakademiens Handl., 63 (1922), 3. Google Scholar
[9]	N. Bellomo, M. Lachowicz, J. Polewczak and G. Toscani, "Mathematical Topics in Nonlinear Kinetic Theory II: The Enskog Equation,", World Sci. Publ., (1991). Google Scholar
[10]	J. Polewczak, On some open problems in the revised Enskog equation for dense gases,, in, (2001), 382. Google Scholar
[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. Google Scholar
[12]	M. Pulvirenti, On the Enskog hierarchy: analyticity, uniqueness, and derivability by particle systems,, Rediconti del Circolo Matematico di Palermo, 45 (1996), 529. Google Scholar
[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. doi: 10.1142/S0217979287001687. Google Scholar
[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. doi: 10.1007/BF01017041. Google Scholar
[15]	H. van Beijeren and M. H. Ernst, The modified Enskog equation,, Physica, 68 (1973), 437. doi: 10.1016/0031-8914(73)90372-8. Google Scholar
[16]	P. Re'sibois and M. De Leener, "Classical Kinetic Theory of Fluids,", John Wiley and Sons, (1977). Google Scholar
[17]	N. N. Bogolybov, "Problems of a Dynamical Theory in Statistical Physics,", M.: Gostekhizdat, (1946). Google Scholar
[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. doi: 10.1088/0305-4470/37/42/002. Google Scholar
[19]	J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications,", Springer, (2006). Google Scholar
[20]	V. I. Gerasimenko, On the solutions of the BBGKY hierarchy for a one-dimensional hard-sphere system,, Theor. Math. Phys., 91 (1992), 120. doi: 10.1007/BF01019833. Google Scholar
[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). Google Scholar
[22]	V. I. Gerasimenko, Approaches to derivation of quantum kinetic equations,, Ukrainian J. Phys., 54 (2009), 834. Google Scholar
[23]	V. I. Gerasimenko and D. Ya. Petrina, On the generalized kinetic equation,, Reports of NAS of Ukraine, 7 (1997), 7. Google Scholar
[24]	V. I. Gerasimenko and D. Ya. Petrina, The generalized kinetic equation generated by the BBGKY hierarchy,, Ukrainian J. Phys., 43 (1998), 697. Google Scholar
[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. Google Scholar
[26]	G. E. Uhlenbeck and G. W. Ford, "Lecture in Statistical Mechanics,", American Mathematical Society Providence, (1963). Google Scholar
[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. doi: 10.1103/PhysRev.132.1388. Google Scholar
[28]	M. S. Green, Boltzmann equation from the statistical mechanical point of view,, J. Chem. Phys., 25 (1956), 836. doi: 10.1063/1.1743132. Google Scholar
[29]	E. G. D. Cohen, Cluster expansions and the hierarchy. I. Nonequilibrium distribution functions,, Physica, 28 (1962), 1045. doi: 10.1016/0031-8914(62)90009-5. Google Scholar

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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. Google Scholar
[2]	C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,", Springer-Verlag, (1994). Google Scholar
[3]	H. Spohn, "Large Scale Dynamics of Interacting Particles,", Springer-Verlag, (1991). doi: 10.1007/978-3-642-84371-6. Google Scholar
[4]	D. Ya. Petrina, "Stochastic Dynamics and Boltzmann Hierarchy,", Institute of Mathematics, (2008). Google Scholar
[5]	O. E. Lanford, Time evolution of large classical systems,, in, 38 (1975), 1. Google Scholar
[6]	H. Grad, Principles of the kinetic theory of gases,, in, 12 (1958), 205. Google Scholar
[7]	H. Spohn, Kinetic equations from Hamiltonian dynamics,, Reviews of Modern Physics, 52 (1980), 569. doi: 10.1103/RevModPhys.52.569. Google Scholar
[8]	D. Enskog, Kinetiche theorie der wärmeleitung, reibung und selbstdiffusion in gewissen werdichteten Gasen und flüβigkeiten,, Kungl. Sv. Vetenskapsakademiens Handl., 63 (1922), 3. Google Scholar
[9]	N. Bellomo, M. Lachowicz, J. Polewczak and G. Toscani, "Mathematical Topics in Nonlinear Kinetic Theory II: The Enskog Equation,", World Sci. Publ., (1991). Google Scholar
[10]	J. Polewczak, On some open problems in the revised Enskog equation for dense gases,, in, (2001), 382. Google Scholar
[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. Google Scholar
[12]	M. Pulvirenti, On the Enskog hierarchy: analyticity, uniqueness, and derivability by particle systems,, Rediconti del Circolo Matematico di Palermo, 45 (1996), 529. Google Scholar
[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. doi: 10.1142/S0217979287001687. Google Scholar
[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. doi: 10.1007/BF01017041. Google Scholar
[15]	H. van Beijeren and M. H. Ernst, The modified Enskog equation,, Physica, 68 (1973), 437. doi: 10.1016/0031-8914(73)90372-8. Google Scholar
[16]	P. Re'sibois and M. De Leener, "Classical Kinetic Theory of Fluids,", John Wiley and Sons, (1977). Google Scholar
[17]	N. N. Bogolybov, "Problems of a Dynamical Theory in Statistical Physics,", M.: Gostekhizdat, (1946). Google Scholar
[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. doi: 10.1088/0305-4470/37/42/002. Google Scholar
[19]	J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications,", Springer, (2006). Google Scholar
[20]	V. I. Gerasimenko, On the solutions of the BBGKY hierarchy for a one-dimensional hard-sphere system,, Theor. Math. Phys., 91 (1992), 120. doi: 10.1007/BF01019833. Google Scholar
[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). Google Scholar
[22]	V. I. Gerasimenko, Approaches to derivation of quantum kinetic equations,, Ukrainian J. Phys., 54 (2009), 834. Google Scholar
[23]	V. I. Gerasimenko and D. Ya. Petrina, On the generalized kinetic equation,, Reports of NAS of Ukraine, 7 (1997), 7. Google Scholar
[24]	V. I. Gerasimenko and D. Ya. Petrina, The generalized kinetic equation generated by the BBGKY hierarchy,, Ukrainian J. Phys., 43 (1998), 697. Google Scholar
[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. Google Scholar
[26]	G. E. Uhlenbeck and G. W. Ford, "Lecture in Statistical Mechanics,", American Mathematical Society Providence, (1963). Google Scholar
[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. doi: 10.1103/PhysRev.132.1388. Google Scholar
[28]	M. S. Green, Boltzmann equation from the statistical mechanical point of view,, J. Chem. Phys., 25 (1956), 836. doi: 10.1063/1.1743132. Google Scholar
[29]	E. G. D. Cohen, Cluster expansions and the hierarchy. I. Nonequilibrium distribution functions,, Physica, 28 (1962), 1045. doi: 10.1016/0031-8914(62)90009-5. Google Scholar

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