
-
Previous Article
Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel
- KRM Home
- This Issue
-
Next Article
Traveling wave and aggregation in a flux-limited Keller-Segel model
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 |
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.
References:
[1] |
R. K. Alexander,
The Infinite Hard Sphere System, Ph. D thesis, Dep. of Mathematics, University of California at Berkeley, 1975. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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). |
[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. |
[9] |
N. N. Bogolyubov,
Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls, Teoret. Mat. Fiz., 24 (1975), 242-247.
|
[10] |
N. N. Bogolubov and N. N. (Jr.) Bogolubov,
Introduction to Quantum Statistical Mechanics, Nauka, Moscow, 1984; World Scientific, Singapore, 2010. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
O. E. Lanford,
Time evolution of large classical systems, Lect. Notes Phys., 38 (1975), 1-111.
|
[17] |
M. Pulvirenti,
On the Enskog hierarchy: Analiticity, uniqueness and derivability by particle systems, Rend. Circ. Mat. Palermo, 2 (1996), 529-542.
|
[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. |
[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.
|
[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. |
[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. |
[22] |
H. Spohn,
Large-Scale Dynamics of Interacting Particles, Springer, Berlin, 1991.
doi: 10.1007/978-3-642-84371-6. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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). |
[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. |
[9] |
N. N. Bogolyubov,
Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls, Teoret. Mat. Fiz., 24 (1975), 242-247.
|
[10] |
N. N. Bogolubov and N. N. (Jr.) Bogolubov,
Introduction to Quantum Statistical Mechanics, Nauka, Moscow, 1984; World Scientific, Singapore, 2010. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
O. E. Lanford,
Time evolution of large classical systems, Lect. Notes Phys., 38 (1975), 1-111.
|
[17] |
M. Pulvirenti,
On the Enskog hierarchy: Analiticity, uniqueness and derivability by particle systems, Rend. Circ. Mat. Palermo, 2 (1996), 529-542.
|
[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. |
[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.
|
[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. |
[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. |
[22] |
H. Spohn,
Large-Scale Dynamics of Interacting Particles, Springer, Berlin, 1991.
doi: 10.1007/978-3-642-84371-6. |
[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. |
[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. |
[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. |
[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. |



[1] |
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 |
[2] |
Viktor I. Gerasimenko, Igor V. Gapyak. Hard sphere dynamics and the Enskog equation. Kinetic and Related Models, 2012, 5 (3) : 459-484. doi: 10.3934/krm.2012.5.459 |
[3] |
Jacek Polewczak, Ana Jacinta Soares. On modified simple reacting spheres kinetic model for chemically reactive gases. Kinetic and Related Models, 2017, 10 (2) : 513-539. doi: 10.3934/krm.2017020 |
[4] |
A. V. Bobylev, E. Mossberg. On some properties of linear and linearized Boltzmann collision operators for hard spheres. Kinetic and Related Models, 2008, 1 (4) : 521-555. doi: 10.3934/krm.2008.1.521 |
[5] |
Lvqiao Liu, Hao Wang. Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3113-3136. doi: 10.3934/cpaa.2020135 |
[6] |
Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic and Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051 |
[7] |
Mario Pulvirenti, Sergio Simonella. On the cardinality of collisional clusters for hard spheres at low density. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3903-3914. doi: 10.3934/dcds.2021021 |
[8] |
Céline Baranger, Marzia Bisi, Stéphane Brull, Laurent Desvillettes. On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases. Kinetic and Related Models, 2018, 11 (4) : 821-858. doi: 10.3934/krm.2018033 |
[9] |
Gilberto M. Kremer, Wilson Marques Jr.. Fourteen moment theory for granular gases. Kinetic and Related Models, 2011, 4 (1) : 317-331. doi: 10.3934/krm.2011.4.317 |
[10] |
Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic and Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020 |
[11] |
Shaofei Wu, Mingqing Wang, Maozhu Jin, Yuntao Zou, Lijun Song. Uniform $L^1$ stability of the inelastic Boltzmann equation with large external force for hard potentials. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1005-1013. doi: 10.3934/dcdss.2019068 |
[12] |
José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 |
[13] |
Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361 |
[14] |
Marc Briant. Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinetic and Related Models, 2017, 10 (2) : 329-371. doi: 10.3934/krm.2017014 |
[15] |
Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic and Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673 |
[16] |
Seiji Ukai. Time-periodic solutions of the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 579-596. doi: 10.3934/dcds.2006.14.579 |
[17] |
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic and Related Models, 2011, 4 (1) : 17-40. doi: 10.3934/krm.2011.4.17 |
[18] |
Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801 |
[19] |
Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869 |
[20] |
Thomas Carty. Grossly determined solutions for a Boltzmann-like equation. Kinetic and Related Models, 2017, 10 (4) : 957-976. doi: 10.3934/krm.2017038 |
2020 Impact Factor: 1.432
Tools
Metrics
Other articles
by authors
[Back to Top]