# American Institute of Mathematical Sciences

August  2018, 11(4): 911-931. doi: 10.3934/krm.2018036

## Microscopic solutions of the Boltzmann-Enskog equation in the series representation

 1 International Research Center M & MOCS, Università dell'Aquila, Palazzo Caetani, Cisterna di Latina, (LT) 04012, Italy 2 CNRS and UMPA (UMR CNRS 5669), École Normale Supérieure de Lyon, 46 allée dItalie, 69364 Lyon Cedex 07, France 3 Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina Street 8, Moscow 119991, Russia 4 National Research Nuclear University MEPhI, Kashirskoe Highway 31, Moscow 115409, Russia 5 National University of Science and Technology MISIS, Leninsky Avenue 2, Moscow 119049, Russia

Received  September 2017 Revised  January 2018 Published  April 2018

The Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equation is not well-defined. Here we give a rigorous mathematical meaning to the microscopic solutions to the Boltzmann-Enskog equation by means of a suitable series representation.

Citation: Mario Pulvirenti, Sergio Simonella, Anton Trushechkin. Microscopic solutions of the Boltzmann-Enskog equation in the series representation. Kinetic & Related Models, 2018, 11 (4) : 911-931. doi: 10.3934/krm.2018036
##### References:
 [1] R. K. Alexander, The Infinite Hard Sphere System, Ph. D thesis, Dep. of Mathematics, University of California at Berkeley, 1975. Google Scholar [2] L. Arkeryd and C. Cercignani, On the convergence of solutions of the Enskog equation to solutions of the Boltzmann equation, Comm. PDE, 14 (1989), 1071-1090. doi: 10.1080/03605308908820644. Google Scholar [3] 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. Google Scholar [4] N. Bellomo and M. Lachowicz, On the asymptotic equivalence between the Enskog and the Boltzmann equations, J. Stat. Phys., 51 (1988), 233-247. doi: 10.1007/BF01015329. Google Scholar [5] T. Bodineau, I. Gallagher and L. Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard-spheres, Inventiones, 203 (2016), 493-553. doi: 10.1007/s00222-015-0593-9. Google Scholar [6] T. Bodineau, I. Gallagher and L. Saint-Raymond, From hard sphere dynamics to the Stokes-Fourier equations: an $L^2$ analysis of the Boltzmann-Grad limit, Annals PDE, 3 (2017), Art.2,118 pp. doi: 10.1007/s40818-016-0018-0. Google Scholar [7] T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella, One-sided convergence in the Boltzmann-Grad limit, Ann. Fac. Sci. Toulouse Math. (to appear).Google Scholar [8] N. N. Bogoliubov, Problems of dynamic theory in statistical physics, Studies in Statistical Mechanics, North-Holland, Amsterdam; Interscience, New York, 1 (1962), 1–118. Google Scholar [9] N. N. Bogolyubov, Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls, Teoret. Mat. Fiz., 24 (1975), 242-247. Google Scholar [10] N. N. Bogolubov and N. N. (Jr.) Bogolubov, Introduction to Quantum Statistical Mechanics, Nauka, Moscow, 1984; World Scientific, Singapore, 2010. Google Scholar [11] C. Cercignani, V. I. Gerasimenko and D. Y. Petrina, Many-Particle Dynamics and Kinetic Equations, Kluwer Academic Publishing, Dordrecht, 1997. doi: 10.1007/978-94-011-5558-8. Google Scholar [12] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer–Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. Google Scholar [13] R. Denlinger, The propagation of chaos for a rarefied gas of hard spheres in the whole space, Thesis (Ph.D.)–New York University. 2016, arXiv: 1605.00589. Google Scholar [14] I. Gallagher, L. Saint Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2013. Google Scholar [15] V. I. Gerasimenko and I. V. Gapyak, Hard sphere dynamics and the Enskog equation, Kinet. Relat. Models, 5 (2012), 459-484. doi: 10.3934/krm.2012.5.459. Google Scholar [16] O. E. Lanford, Time evolution of large classical systems, Lect. Notes Phys., 38 (1975), 1-111. Google Scholar [17] M. Pulvirenti, On the Enskog hierarchy: Analiticity, uniqueness and derivability by particle systems, Rend. Circ. Mat. Palermo, 2 (1996), 529-542. Google Scholar [18] M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short-range potentials, Rev. Math. Phys., 26 (2014), 1-64. doi: 10.1142/S0129055X14500019. Google Scholar [19] M. Pulvirenti and S. Simonella, On the evolution of the empirical measure for the hard-sphere dynamics, Bull. Inst. Math. Academia Sinica, 10 (2015), 171-204. Google Scholar [20] M. Pulvirenti and S. Simonella, The Boltzmannn-Grad limit of a hard sphere system: Analysis of the correlation error, Inventiones, 207 (2017), 1135-1237. doi: 10.1007/s00222-016-0682-4. Google Scholar [21] S. Simonella, Evolution of correlation functions in the hard sphere dynamics, J. Stat. Phys., 155 (2014), 1191-1221. doi: 10.1007/s10955-013-0905-7. Google Scholar [22] H. Spohn, Large-Scale Dynamics of Interacting Particles, Springer, Berlin, 1991. doi: 10.1007/978-3-642-84371-6. Google Scholar [23] A. S. Trushechkin, Derivation of the particle dynamics from kinetic equations, p-Adic, Ultrametric Analysis and Applications, 4 (2012), 130-142. doi: 10.1134/S2070046612020057. Google Scholar [24] A. S. Trushechkin, Functional mechanics and kinetic equations, QP-PQ: Quantum probability and White Noise Analysis, 30 (2013), 339-350. doi: 10.1142/9789814460026_0029. Google Scholar [25] A. S. Trushechkin, Microscopic solutions of the Boltzmann-Enskog equation and the irreversibility problem, Proc. Steklov Inst. Math., 285 (2014), 251-274. doi: 10.1134/S008154381404018X. Google Scholar [26] A. S. Trushechkin, Microscopic and soliton-like solutions of the Boltzmann-Enskog and generalized Enskog equations for elastic and inelastic hard spheres, Kinetic and Relat. Models, 7 (2014), 755-778. doi: 10.3934/krm.2014.7.755. Google Scholar

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##### References:
 [1] R. K. Alexander, The Infinite Hard Sphere System, Ph. D thesis, Dep. of Mathematics, University of California at Berkeley, 1975. Google Scholar [2] L. Arkeryd and C. Cercignani, On the convergence of solutions of the Enskog equation to solutions of the Boltzmann equation, Comm. PDE, 14 (1989), 1071-1090. doi: 10.1080/03605308908820644. Google Scholar [3] 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. Google Scholar [4] N. Bellomo and M. Lachowicz, On the asymptotic equivalence between the Enskog and the Boltzmann equations, J. Stat. Phys., 51 (1988), 233-247. doi: 10.1007/BF01015329. Google Scholar [5] T. Bodineau, I. Gallagher and L. Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard-spheres, Inventiones, 203 (2016), 493-553. doi: 10.1007/s00222-015-0593-9. Google Scholar [6] T. Bodineau, I. Gallagher and L. Saint-Raymond, From hard sphere dynamics to the Stokes-Fourier equations: an $L^2$ analysis of the Boltzmann-Grad limit, Annals PDE, 3 (2017), Art.2,118 pp. doi: 10.1007/s40818-016-0018-0. Google Scholar [7] T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella, One-sided convergence in the Boltzmann-Grad limit, Ann. Fac. Sci. Toulouse Math. (to appear).Google Scholar [8] N. N. Bogoliubov, Problems of dynamic theory in statistical physics, Studies in Statistical Mechanics, North-Holland, Amsterdam; Interscience, New York, 1 (1962), 1–118. Google Scholar [9] N. N. Bogolyubov, Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls, Teoret. Mat. Fiz., 24 (1975), 242-247. Google Scholar [10] N. N. Bogolubov and N. N. (Jr.) Bogolubov, Introduction to Quantum Statistical Mechanics, Nauka, Moscow, 1984; World Scientific, Singapore, 2010. Google Scholar [11] C. Cercignani, V. I. Gerasimenko and D. Y. Petrina, Many-Particle Dynamics and Kinetic Equations, Kluwer Academic Publishing, Dordrecht, 1997. doi: 10.1007/978-94-011-5558-8. Google Scholar [12] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer–Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. Google Scholar [13] R. Denlinger, The propagation of chaos for a rarefied gas of hard spheres in the whole space, Thesis (Ph.D.)–New York University. 2016, arXiv: 1605.00589. Google Scholar [14] I. Gallagher, L. Saint Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2013. Google Scholar [15] V. I. Gerasimenko and I. V. Gapyak, Hard sphere dynamics and the Enskog equation, Kinet. Relat. Models, 5 (2012), 459-484. doi: 10.3934/krm.2012.5.459. Google Scholar [16] O. E. Lanford, Time evolution of large classical systems, Lect. Notes Phys., 38 (1975), 1-111. Google Scholar [17] M. Pulvirenti, On the Enskog hierarchy: Analiticity, uniqueness and derivability by particle systems, Rend. Circ. Mat. Palermo, 2 (1996), 529-542. Google Scholar [18] M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short-range potentials, Rev. Math. Phys., 26 (2014), 1-64. doi: 10.1142/S0129055X14500019. Google Scholar [19] M. Pulvirenti and S. Simonella, On the evolution of the empirical measure for the hard-sphere dynamics, Bull. Inst. Math. Academia Sinica, 10 (2015), 171-204. Google Scholar [20] M. Pulvirenti and S. Simonella, The Boltzmannn-Grad limit of a hard sphere system: Analysis of the correlation error, Inventiones, 207 (2017), 1135-1237. doi: 10.1007/s00222-016-0682-4. Google Scholar [21] S. Simonella, Evolution of correlation functions in the hard sphere dynamics, J. Stat. Phys., 155 (2014), 1191-1221. doi: 10.1007/s10955-013-0905-7. Google Scholar [22] H. Spohn, Large-Scale Dynamics of Interacting Particles, Springer, Berlin, 1991. doi: 10.1007/978-3-642-84371-6. Google Scholar [23] A. S. Trushechkin, Derivation of the particle dynamics from kinetic equations, p-Adic, Ultrametric Analysis and Applications, 4 (2012), 130-142. doi: 10.1134/S2070046612020057. Google Scholar [24] A. S. Trushechkin, Functional mechanics and kinetic equations, QP-PQ: Quantum probability and White Noise Analysis, 30 (2013), 339-350. doi: 10.1142/9789814460026_0029. Google Scholar [25] A. S. Trushechkin, Microscopic solutions of the Boltzmann-Enskog equation and the irreversibility problem, Proc. Steklov Inst. Math., 285 (2014), 251-274. doi: 10.1134/S008154381404018X. Google Scholar [26] A. S. Trushechkin, Microscopic and soliton-like solutions of the Boltzmann-Enskog and generalized Enskog equations for elastic and inelastic hard spheres, Kinetic and Relat. Models, 7 (2014), 755-778. doi: 10.3934/krm.2014.7.755. Google Scholar
$\mathbf {r_5} = \{1, 1, 2, 3, 2\}$
Initial configuration $\delta(\zeta_1-z_1)\delta(\zeta_2-z_2)\delta(\zeta_3-z_1)\delta(\zeta_4-z_1)\cdots$
Sum of compositions of trees
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