Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres

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December 2014, 7(4): 755-778. doi: 10.3934/krm.2014.7.755

Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres

Anton Trushechkin ^1,

1.	Steklov Mathematical Institute of the Russian Academy of Sciences, Gubkina str. 8, 119991 Moscow, Russian Federation

Received March 2014 Revised July 2014 Published November 2014

Abstract
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N. N. Bogolyubov discovered that the Boltzmann--Enskog kinetic equation has microscopic solutions. They have the form of sums of delta-functions and correspond to trajectories of individual hard spheres. But the rigorous sense of the product of the delta-functions in the collision integral was not discussed. Here we give a rigorous sense to these solutions by introduction of a special regularization of the delta-functions. The crucial observation is that the collision integral of the Boltzmann--Enskog equation coincides with that of the first equation of the BBGKY hierarchy for hard spheres if the special regularization to the delta-functions is applied. This allows to reduce the nonlinear Boltzmann--Enskog equation to the BBGKY hierarchy of linear equations in this particular case.
Also we show that similar functions are exact smooth solutions for the recently proposed generalized Enskog equation. They can be referred to as ``particle-like'' or ``soliton-like'' solutions and are analogues of multisoliton solutions of the Korteweg--de Vries equation.

Keywords: Boltzmann--Enskog equation, hard spheres, Enskog equation, generalized functions., Kinetic theory of gases.

Mathematics Subject Classification: Primary: 35Q20, 82C40; Secondary: 46F99, 35R09, 45K0.

Citation: Anton Trushechkin. Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres. Kinetic and Related Models, 2014, 7 (4) : 755-778. doi: 10.3934/krm.2014.7.755

References:

[1]	L. Arkeryd and C. Cercignani, On the convergence of solutions of the Enskog equation to solutions of the Boltzmann equation, Comm. PDE, 14 (1989), 1071-1089. doi: 10.1080/03605308908820644.
[2]	L. Arkeryd and C. Cercignani, Global existence in $L_1$ for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation, J. Stat. Phys., 59 (1990), 845-867. doi: 10.1007/BF01025854.
[3]	N. Bellomo and M. Lachowicz, On the asymptotic theory of the Boltzmann and Enskog equations: A rigorous $H$-theorem for the Enskog equation, Springer Lecture Notes in Mathematics: Mathematical Aspects of Fluid and Plasma Dynamics, 1460 (1991), 15-30. doi: 10.1007/BFb0091358.
[4]	A. V. Bobylev, Tochnye resheniya uravneniya Boltsmana, (Russian) [Exact solutions of the Boltzmann equation], Dokl. Akad. Nauk SSSR, 225 (1975), 1296-1299.
[5]	L. Boltzmann, Vorlesungen Über Gastheorie, (German) [Lectures on gas theory], J. A. Barth, Leipzig, 1896/98.
[6]	N. V. Brilliantov and T. Pöschel, Kinetic theory of granular gases, Oxford University Press, 2004. doi: 10.1093/acprof:oso/9780198530381.001.0001.
[7]	N. N. Bogolyubov, Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls, Theor. Math. Phys., 24 (1975), 242-247.
[8]	N. N. Bogolubov and N. N. Bogolubov, Jr., Introduction to Quantum Statistical Mechanics, Gordon and Breach, New York, 2010. doi: 10.1142/7623.
[9]	M. S. Borovchenkova and V. I. Gerasimenko, On the non-Markovian Enskog equation for granular gases, J. Phys. A: Math. Theor., 47 (2014), 035001, 18pp; arXiv:1307.6676. doi: 10.1088/1751-8113/47/3/035001.
[10]	C. Cercignani, V. I. Gerasimenko and D. Ya. Petrina, Many-Particle Dynamics and Kinetic Equations, Kluwer Academic Publishing, Dordrecht, 1997. doi: 10.1007/978-94-011-5558-8.
[11]	C. Cercignani, On the Boltzmann equation for rigid spheres, Transport Theory and Statistical Physics, 2 (1972), 211-225. doi: 10.1080/00411457208232538.
[12]	C. Cercignani, Theory and Application of the Boltzmann Equation, Elsevier, New York, 1975.
[13]	V. I. Gerasimenko and I. V. Gapyak, Hard sphere dynamics and the Enskog equation, Kinet. Relat. Models, 5 (2012), 459-484; arXiv:1107.5572. doi: 10.3934/krm.2012.5.459.
[14]	S.-Y. Ha and S. E. Noh, Global weak solutions and uniform $L^p$-stability of the Boltzmann-Enskog equation, J. Differential Equations, 251 (2011), 1-25. doi: 10.1016/j.jde.2011.03.021.
[15]	R. Illner and M. Pulvirenti, A derivation of the BBGKY hierarchy for hard sphere particle systems, Trans. Theory Stat. Phys., 16 (1987), 997-1012. doi: 10.1080/00411458708204603.
[16]	V. V. Kozlov, Teplovoye Ravnovesie Po Gibbsu i Puankare, (Russian) [Thermal Equilibrium in the sense of Gibbs and Poincaré], Institute of Computer Research, Moscow-Izhevsk, 2002.
[17]	V. V. Kozlov, Thermal Equilibrium in the sense of Gibbs and Poincar'e, Dokl. Akad. Nauk, 382 (2002), 602-605.
[18]	V. V. Kozlov and D. V. Treshchev, Weak convergence of solutions of the Liouville equation for nonlinear Hamiltonian systems, Theor. Math. Phys., 134 (2003), 339-350. doi: 10.1023/A:1022697321418.
[19]	V. V. Kozlov and D. V. Treshchev, Evolution of measures in the phase space of nonlinear Hamiltonian systems, Theor. Math. Phys., 136 (2003), 1325-1335. doi: 10.1023/A:1025607517444.
[20]	M.-Y. Lee, T.-P. Liu and S.-H. Yu, Large-time behavior of solutions for the Boltzmann equation with hard potentials, Comm. Math. Phys., 269 (2007), 17-37. doi: 10.1007/s00220-006-0108-z.
[21]	T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608. doi: 10.1002/cpa.20011.
[22]	T.-P. Liu and S.-H. Yu, Green's function of Boltzmann equation, 3-D waves, Bulletin of Institute of Mathematics, Academia Sinica (New Series), 1 (2006), 1-78.
[23]	A. I. Mikhailov, Functional mechanics: Evolution of the moments of distribution function and the Poincaré recurrence theorem, $p$-Adic Numbers, Ultrametric Analysis and Applications, 3 (2011), 205-211; arXiv:1305.4057. doi: 10.1134/S2070046611030046.
[24]	D. Ya. Petrina and V. I. Gerasimenko, Mathematical problems of statistical mechanics of a system of elastic balls, Russian Mathematical Surveys, 45 (1990), 153-211. doi: 10.1070/RM1990v045n03ABEH002360.
[25]	D. Ya. Petrina and G. L. Garaffini, Analog of the Liouville equation and BBGKY hierarchy for a system of hard spheres with inelastic collisions, Ukrainian Mathematical Journal, 57 (2005), 967-990. doi: 10.1007/s11253-005-0242-3.
[26]	D. Ya. Petrina and G. L. Garaffini, Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions, Ukrainian Mathematical Journal, 58 (2006), 371-380. doi: 10.1007/s11253-006-0075-8.
[27]	E. V. Piskovskiy, On functional approach to classical mechanics, $p$-Adic Numbers, Ultrametric Analysis and Applications, 3 (2011), 243-247. doi: 10.1134/S2070046611030095.
[28]	E. V. Piskovskiy and I. V. Volovich, On the correspondence between Newtonian and functional mechanics, in Quantum Bio-Informatics IV. From Quantum Infarmation to Bio-Informatics, QP-PQ: Quantum Probab. White Noise Anal., vol. 28, World Scientific, Hackensack, NJ, (2011), 363-372. doi: 10.1142/9789814343763_0028.
[29]	P. Resibois, $H$-theorem for the (modified) nonlinear Enskog equation, Phys. Rev. Lett., 40 (1978), 1409-1411. doi: 10.1103/PhysRevLett.40.1409.
[30]	S. Simonella, Evolution of correlation functions in the hard sphere dynamics, J. Stat. Phys., 155 (2014), 1191-1221; arXiv:1205.2789. doi: 10.1007/s10955-013-0905-7.
[31]	H. Spohn, On the integrated form of the BBGKY hierarchy for hard spheres, preprint, arXiv:math-ph/0605068 (written in 1985).
[32]	A. S. Trushechkin, Microscopic solutions of the Boltzmann-Enskog equation and the irreversibility problem, Proc. Steklov Inst. Math., 285 (2014), 251-274.
[33]	A. S. Trushechkin, Irreversibility and the role of an instrument in the functional formulation of classical mechanics, Theoret. and Math. Phys., 164 (2010), 1198-1201. doi: 10.1007/s11232-010-0100-9.
[34]	A. S. Trushechkin and I. V. Volovich, Functional classical mechanics and rational numbers, p-Adic Numbers Ultrametric Anal. Appl., 1 (2009), 361-367; arXiv:0910.1502. doi: 10.1134/S2070046609040086.
[35]	A. A. Vlasov, Many-Particle Theory and Its Application to Plasma, Gordon and Breach, New York, 1961.
[36]	I. V. Volovich, The irreversibilty problem and functional formulation of classical mechanics, preprint,, , ().
[37]	I. V. Volovich, Randomness in classical mechanics and quantum mechanics, Found. Phys., 41 (2011), 516-528. doi: 10.1007/s10701-010-9450-2.
[38]	I. V. Volovich, Bogolyubov equations and functional mechanics, Theoret. and Math. Phys., 164 (2010), 1128-1135. doi: 10.1007/s11232-010-0090-7.

show all references

References:

[1]	L. Arkeryd and C. Cercignani, On the convergence of solutions of the Enskog equation to solutions of the Boltzmann equation, Comm. PDE, 14 (1989), 1071-1089. doi: 10.1080/03605308908820644.
[2]	L. Arkeryd and C. Cercignani, Global existence in $L_1$ for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation, J. Stat. Phys., 59 (1990), 845-867. doi: 10.1007/BF01025854.
[3]	N. Bellomo and M. Lachowicz, On the asymptotic theory of the Boltzmann and Enskog equations: A rigorous $H$-theorem for the Enskog equation, Springer Lecture Notes in Mathematics: Mathematical Aspects of Fluid and Plasma Dynamics, 1460 (1991), 15-30. doi: 10.1007/BFb0091358.
[4]	A. V. Bobylev, Tochnye resheniya uravneniya Boltsmana, (Russian) [Exact solutions of the Boltzmann equation], Dokl. Akad. Nauk SSSR, 225 (1975), 1296-1299.
[5]	L. Boltzmann, Vorlesungen Über Gastheorie, (German) [Lectures on gas theory], J. A. Barth, Leipzig, 1896/98.
[6]	N. V. Brilliantov and T. Pöschel, Kinetic theory of granular gases, Oxford University Press, 2004. doi: 10.1093/acprof:oso/9780198530381.001.0001.
[7]	N. N. Bogolyubov, Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls, Theor. Math. Phys., 24 (1975), 242-247.
[8]	N. N. Bogolubov and N. N. Bogolubov, Jr., Introduction to Quantum Statistical Mechanics, Gordon and Breach, New York, 2010. doi: 10.1142/7623.
[9]	M. S. Borovchenkova and V. I. Gerasimenko, On the non-Markovian Enskog equation for granular gases, J. Phys. A: Math. Theor., 47 (2014), 035001, 18pp; arXiv:1307.6676. doi: 10.1088/1751-8113/47/3/035001.
[10]	C. Cercignani, V. I. Gerasimenko and D. Ya. Petrina, Many-Particle Dynamics and Kinetic Equations, Kluwer Academic Publishing, Dordrecht, 1997. doi: 10.1007/978-94-011-5558-8.
[11]	C. Cercignani, On the Boltzmann equation for rigid spheres, Transport Theory and Statistical Physics, 2 (1972), 211-225. doi: 10.1080/00411457208232538.
[12]	C. Cercignani, Theory and Application of the Boltzmann Equation, Elsevier, New York, 1975.
[13]	V. I. Gerasimenko and I. V. Gapyak, Hard sphere dynamics and the Enskog equation, Kinet. Relat. Models, 5 (2012), 459-484; arXiv:1107.5572. doi: 10.3934/krm.2012.5.459.
[14]	S.-Y. Ha and S. E. Noh, Global weak solutions and uniform $L^p$-stability of the Boltzmann-Enskog equation, J. Differential Equations, 251 (2011), 1-25. doi: 10.1016/j.jde.2011.03.021.
[15]	R. Illner and M. Pulvirenti, A derivation of the BBGKY hierarchy for hard sphere particle systems, Trans. Theory Stat. Phys., 16 (1987), 997-1012. doi: 10.1080/00411458708204603.
[16]	V. V. Kozlov, Teplovoye Ravnovesie Po Gibbsu i Puankare, (Russian) [Thermal Equilibrium in the sense of Gibbs and Poincaré], Institute of Computer Research, Moscow-Izhevsk, 2002.
[17]	V. V. Kozlov, Thermal Equilibrium in the sense of Gibbs and Poincar'e, Dokl. Akad. Nauk, 382 (2002), 602-605.
[18]	V. V. Kozlov and D. V. Treshchev, Weak convergence of solutions of the Liouville equation for nonlinear Hamiltonian systems, Theor. Math. Phys., 134 (2003), 339-350. doi: 10.1023/A:1022697321418.
[19]	V. V. Kozlov and D. V. Treshchev, Evolution of measures in the phase space of nonlinear Hamiltonian systems, Theor. Math. Phys., 136 (2003), 1325-1335. doi: 10.1023/A:1025607517444.
[20]	M.-Y. Lee, T.-P. Liu and S.-H. Yu, Large-time behavior of solutions for the Boltzmann equation with hard potentials, Comm. Math. Phys., 269 (2007), 17-37. doi: 10.1007/s00220-006-0108-z.
[21]	T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608. doi: 10.1002/cpa.20011.
[22]	T.-P. Liu and S.-H. Yu, Green's function of Boltzmann equation, 3-D waves, Bulletin of Institute of Mathematics, Academia Sinica (New Series), 1 (2006), 1-78.
[23]	A. I. Mikhailov, Functional mechanics: Evolution of the moments of distribution function and the Poincaré recurrence theorem, $p$-Adic Numbers, Ultrametric Analysis and Applications, 3 (2011), 205-211; arXiv:1305.4057. doi: 10.1134/S2070046611030046.
[24]	D. Ya. Petrina and V. I. Gerasimenko, Mathematical problems of statistical mechanics of a system of elastic balls, Russian Mathematical Surveys, 45 (1990), 153-211. doi: 10.1070/RM1990v045n03ABEH002360.
[25]	D. Ya. Petrina and G. L. Garaffini, Analog of the Liouville equation and BBGKY hierarchy for a system of hard spheres with inelastic collisions, Ukrainian Mathematical Journal, 57 (2005), 967-990. doi: 10.1007/s11253-005-0242-3.
[26]	D. Ya. Petrina and G. L. Garaffini, Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions, Ukrainian Mathematical Journal, 58 (2006), 371-380. doi: 10.1007/s11253-006-0075-8.
[27]	E. V. Piskovskiy, On functional approach to classical mechanics, $p$-Adic Numbers, Ultrametric Analysis and Applications, 3 (2011), 243-247. doi: 10.1134/S2070046611030095.
[28]	E. V. Piskovskiy and I. V. Volovich, On the correspondence between Newtonian and functional mechanics, in Quantum Bio-Informatics IV. From Quantum Infarmation to Bio-Informatics, QP-PQ: Quantum Probab. White Noise Anal., vol. 28, World Scientific, Hackensack, NJ, (2011), 363-372. doi: 10.1142/9789814343763_0028.
[29]	P. Resibois, $H$-theorem for the (modified) nonlinear Enskog equation, Phys. Rev. Lett., 40 (1978), 1409-1411. doi: 10.1103/PhysRevLett.40.1409.
[30]	S. Simonella, Evolution of correlation functions in the hard sphere dynamics, J. Stat. Phys., 155 (2014), 1191-1221; arXiv:1205.2789. doi: 10.1007/s10955-013-0905-7.
[31]	H. Spohn, On the integrated form of the BBGKY hierarchy for hard spheres, preprint, arXiv:math-ph/0605068 (written in 1985).
[32]	A. S. Trushechkin, Microscopic solutions of the Boltzmann-Enskog equation and the irreversibility problem, Proc. Steklov Inst. Math., 285 (2014), 251-274.
[33]	A. S. Trushechkin, Irreversibility and the role of an instrument in the functional formulation of classical mechanics, Theoret. and Math. Phys., 164 (2010), 1198-1201. doi: 10.1007/s11232-010-0100-9.
[34]	A. S. Trushechkin and I. V. Volovich, Functional classical mechanics and rational numbers, p-Adic Numbers Ultrametric Anal. Appl., 1 (2009), 361-367; arXiv:0910.1502. doi: 10.1134/S2070046609040086.
[35]	A. A. Vlasov, Many-Particle Theory and Its Application to Plasma, Gordon and Breach, New York, 1961.
[36]	I. V. Volovich, The irreversibilty problem and functional formulation of classical mechanics, preprint,, , ().
[37]	I. V. Volovich, Randomness in classical mechanics and quantum mechanics, Found. Phys., 41 (2011), 516-528. doi: 10.1007/s10701-010-9450-2.
[38]	I. V. Volovich, Bogolyubov equations and functional mechanics, Theoret. and Math. Phys., 164 (2010), 1128-1135. doi: 10.1007/s11232-010-0090-7.

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