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

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

Mario Pulvirenti 1, , Sergio Simonella 2, and  Anton Trushechkin 3,4,5,

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.

Keywords: Kinetic theory of gases, hard spheres, Boltzmann-Enskog equation, empirical measure, microscopic solutions.
Mathematics Subject Classification: Primary: 82C05, 82C40; Secondary: 35Q20.
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

show all references

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

Figure 1.  $\mathbf {r_5} = \{1, 1, 2, 3, 2\}$
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Figure 5.  Initial configuration $\delta(\zeta_1-z_1)\delta(\zeta_2-z_2)\delta(\zeta_3-z_1)\delta(\zeta_4-z_1)\cdots$
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