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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

 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

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.
Citation: Anton Trushechkin. Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres. Kinetic & 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.  doi: 10.1080/03605308908820644.  Google Scholar [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.  doi: 10.1007/BF01025854.  Google Scholar [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.  doi: 10.1007/BFb0091358.  Google Scholar [4] A. V. Bobylev, Tochnye resheniya uravneniya Boltsmana,, (Russian) [Exact solutions of the Boltzmann equation], 225 (1975), 1296.   Google Scholar [5] L. Boltzmann, Vorlesungen Über Gastheorie,, (German) [Lectures on gas theory], (1896).   Google Scholar [6] N. V. Brilliantov and T. Pöschel, Kinetic theory of granular gases,, Oxford University Press, (2004).  doi: 10.1093/acprof:oso/9780198530381.001.0001.  Google Scholar [7] N. N. Bogolyubov, Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls,, Theor. Math. Phys., 24 (1975), 242.   Google Scholar [8] N. N. Bogolubov and N. N. Bogolubov, Jr., Introduction to Quantum Statistical Mechanics,, Gordon and Breach, (2010).  doi: 10.1142/7623.  Google Scholar [9] M. S. Borovchenkova and V. I. Gerasimenko, On the non-Markovian Enskog equation for granular gases,, J. Phys. A: Math. Theor., 47 (2014).  doi: 10.1088/1751-8113/47/3/035001.  Google Scholar [10] C. Cercignani, V. I. Gerasimenko and D. Ya. Petrina, Many-Particle Dynamics and Kinetic Equations,, Kluwer Academic Publishing, (1997).  doi: 10.1007/978-94-011-5558-8.  Google Scholar [11] C. Cercignani, On the Boltzmann equation for rigid spheres,, Transport Theory and Statistical Physics, 2 (1972), 211.  doi: 10.1080/00411457208232538.  Google Scholar [12] C. Cercignani, Theory and Application of the Boltzmann Equation,, Elsevier, (1975).   Google Scholar [13] V. I. Gerasimenko and I. V. Gapyak, Hard sphere dynamics and the Enskog equation,, Kinet. Relat. Models, 5 (2012), 459.  doi: 10.3934/krm.2012.5.459.  Google Scholar [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.  doi: 10.1016/j.jde.2011.03.021.  Google Scholar [15] R. Illner and M. Pulvirenti, A derivation of the BBGKY hierarchy for hard sphere particle systems,, Trans. Theory Stat. Phys., 16 (1987), 997.  doi: 10.1080/00411458708204603.  Google Scholar [16] V. V. Kozlov, Teplovoye Ravnovesie Po Gibbsu i Puankare,, (Russian) [Thermal Equilibrium in the sense of Gibbs and Poincaré], (2002).   Google Scholar [17] V. V. Kozlov, Thermal Equilibrium in the sense of Gibbs and Poincar'e,, Dokl. Akad. Nauk, 382 (2002), 602.   Google Scholar [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.  doi: 10.1023/A:1022697321418.  Google Scholar [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.  doi: 10.1023/A:1025607517444.  Google Scholar [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.  doi: 10.1007/s00220-006-0108-z.  Google Scholar [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.  doi: 10.1002/cpa.20011.  Google Scholar [22] T.-P. Liu and S.-H. Yu, Green's function of Boltzmann equation, 3-D waves,, Bulletin of Institute of Mathematics, 1 (2006), 1.   Google Scholar [23] A. I. Mikhailov, Functional mechanics: Evolution of the moments of distribution function and the Poincaré recurrence theorem,, $p$-Adic Numbers, 3 (2011), 205.  doi: 10.1134/S2070046611030046.  Google Scholar [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.  doi: 10.1070/RM1990v045n03ABEH002360.  Google Scholar [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.  doi: 10.1007/s11253-005-0242-3.  Google Scholar [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.  doi: 10.1007/s11253-006-0075-8.  Google Scholar [27] E. V. Piskovskiy, On functional approach to classical mechanics,, $p$-Adic Numbers, 3 (2011), 243.  doi: 10.1134/S2070046611030095.  Google Scholar [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, (2011), 363.  doi: 10.1142/9789814343763_0028.  Google Scholar [29] P. Resibois, $H$-theorem for the (modified) nonlinear Enskog equation,, Phys. Rev. Lett., 40 (1978), 1409.  doi: 10.1103/PhysRevLett.40.1409.  Google Scholar [30] S. Simonella, Evolution of correlation functions in the hard sphere dynamics,, J. Stat. Phys., 155 (2014), 1191.  doi: 10.1007/s10955-013-0905-7.  Google Scholar [31] H. Spohn, On the integrated form of the BBGKY hierarchy for hard spheres, preprint,, , (1985).   Google Scholar [32] A. S. Trushechkin, Microscopic solutions of the Boltzmann-Enskog equation and the irreversibility problem,, Proc. Steklov Inst. Math., 285 (2014), 251.   Google Scholar [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.  doi: 10.1007/s11232-010-0100-9.  Google Scholar [34] A. S. Trushechkin and I. V. Volovich, Functional classical mechanics and rational numbers,, p-Adic Numbers Ultrametric Anal. Appl., 1 (2009), 361.  doi: 10.1134/S2070046609040086.  Google Scholar [35] A. A. Vlasov, Many-Particle Theory and Its Application to Plasma,, Gordon and Breach, (1961).   Google Scholar [36] I. V. Volovich, The irreversibilty problem and functional formulation of classical mechanics, preprint,, , ().   Google Scholar [37] I. V. Volovich, Randomness in classical mechanics and quantum mechanics,, Found. Phys., 41 (2011), 516.  doi: 10.1007/s10701-010-9450-2.  Google Scholar [38] I. V. Volovich, Bogolyubov equations and functional mechanics,, Theoret. and Math. Phys., 164 (2010), 1128.  doi: 10.1007/s11232-010-0090-7.  Google Scholar

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.  doi: 10.1080/03605308908820644.  Google Scholar [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.  doi: 10.1007/BF01025854.  Google Scholar [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.  doi: 10.1007/BFb0091358.  Google Scholar [4] A. V. Bobylev, Tochnye resheniya uravneniya Boltsmana,, (Russian) [Exact solutions of the Boltzmann equation], 225 (1975), 1296.   Google Scholar [5] L. Boltzmann, Vorlesungen Über Gastheorie,, (German) [Lectures on gas theory], (1896).   Google Scholar [6] N. V. Brilliantov and T. Pöschel, Kinetic theory of granular gases,, Oxford University Press, (2004).  doi: 10.1093/acprof:oso/9780198530381.001.0001.  Google Scholar [7] N. N. Bogolyubov, Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls,, Theor. Math. Phys., 24 (1975), 242.   Google Scholar [8] N. N. Bogolubov and N. N. Bogolubov, Jr., Introduction to Quantum Statistical Mechanics,, Gordon and Breach, (2010).  doi: 10.1142/7623.  Google Scholar [9] M. S. Borovchenkova and V. I. Gerasimenko, On the non-Markovian Enskog equation for granular gases,, J. Phys. A: Math. Theor., 47 (2014).  doi: 10.1088/1751-8113/47/3/035001.  Google Scholar [10] C. Cercignani, V. I. Gerasimenko and D. Ya. Petrina, Many-Particle Dynamics and Kinetic Equations,, Kluwer Academic Publishing, (1997).  doi: 10.1007/978-94-011-5558-8.  Google Scholar [11] C. Cercignani, On the Boltzmann equation for rigid spheres,, Transport Theory and Statistical Physics, 2 (1972), 211.  doi: 10.1080/00411457208232538.  Google Scholar [12] C. Cercignani, Theory and Application of the Boltzmann Equation,, Elsevier, (1975).   Google Scholar [13] V. I. Gerasimenko and I. V. Gapyak, Hard sphere dynamics and the Enskog equation,, Kinet. Relat. Models, 5 (2012), 459.  doi: 10.3934/krm.2012.5.459.  Google Scholar [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.  doi: 10.1016/j.jde.2011.03.021.  Google Scholar [15] R. Illner and M. Pulvirenti, A derivation of the BBGKY hierarchy for hard sphere particle systems,, Trans. Theory Stat. Phys., 16 (1987), 997.  doi: 10.1080/00411458708204603.  Google Scholar [16] V. V. Kozlov, Teplovoye Ravnovesie Po Gibbsu i Puankare,, (Russian) [Thermal Equilibrium in the sense of Gibbs and Poincaré], (2002).   Google Scholar [17] V. V. Kozlov, Thermal Equilibrium in the sense of Gibbs and Poincar'e,, Dokl. Akad. Nauk, 382 (2002), 602.   Google Scholar [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.  doi: 10.1023/A:1022697321418.  Google Scholar [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.  doi: 10.1023/A:1025607517444.  Google Scholar [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.  doi: 10.1007/s00220-006-0108-z.  Google Scholar [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.  doi: 10.1002/cpa.20011.  Google Scholar [22] T.-P. Liu and S.-H. Yu, Green's function of Boltzmann equation, 3-D waves,, Bulletin of Institute of Mathematics, 1 (2006), 1.   Google Scholar [23] A. I. Mikhailov, Functional mechanics: Evolution of the moments of distribution function and the Poincaré recurrence theorem,, $p$-Adic Numbers, 3 (2011), 205.  doi: 10.1134/S2070046611030046.  Google Scholar [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.  doi: 10.1070/RM1990v045n03ABEH002360.  Google Scholar [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.  doi: 10.1007/s11253-005-0242-3.  Google Scholar [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.  doi: 10.1007/s11253-006-0075-8.  Google Scholar [27] E. V. Piskovskiy, On functional approach to classical mechanics,, $p$-Adic Numbers, 3 (2011), 243.  doi: 10.1134/S2070046611030095.  Google Scholar [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, (2011), 363.  doi: 10.1142/9789814343763_0028.  Google Scholar [29] P. Resibois, $H$-theorem for the (modified) nonlinear Enskog equation,, Phys. Rev. Lett., 40 (1978), 1409.  doi: 10.1103/PhysRevLett.40.1409.  Google Scholar [30] S. Simonella, Evolution of correlation functions in the hard sphere dynamics,, J. Stat. Phys., 155 (2014), 1191.  doi: 10.1007/s10955-013-0905-7.  Google Scholar [31] H. Spohn, On the integrated form of the BBGKY hierarchy for hard spheres, preprint,, , (1985).   Google Scholar [32] A. S. Trushechkin, Microscopic solutions of the Boltzmann-Enskog equation and the irreversibility problem,, Proc. Steklov Inst. Math., 285 (2014), 251.   Google Scholar [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.  doi: 10.1007/s11232-010-0100-9.  Google Scholar [34] A. S. Trushechkin and I. V. Volovich, Functional classical mechanics and rational numbers,, p-Adic Numbers Ultrametric Anal. Appl., 1 (2009), 361.  doi: 10.1134/S2070046609040086.  Google Scholar [35] A. A. Vlasov, Many-Particle Theory and Its Application to Plasma,, Gordon and Breach, (1961).   Google Scholar [36] I. V. Volovich, The irreversibilty problem and functional formulation of classical mechanics, preprint,, , ().   Google Scholar [37] I. V. Volovich, Randomness in classical mechanics and quantum mechanics,, Found. Phys., 41 (2011), 516.  doi: 10.1007/s10701-010-9450-2.  Google Scholar [38] I. V. Volovich, Bogolyubov equations and functional mechanics,, Theoret. and Math. Phys., 164 (2010), 1128.  doi: 10.1007/s11232-010-0090-7.  Google Scholar
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