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.

[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. BodineauI. 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. BodineauI. 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. PulvirentiC. 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. BodineauI. 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. BodineauI. 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. PulvirentiC. 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.

Figure 1.  $\mathbf {r_5} = \{1, 1, 2, 3, 2\}$
Figure 5.  Initial configuration $\delta(\zeta_1-z_1)\delta(\zeta_2-z_2)\delta(\zeta_3-z_1)\delta(\zeta_4-z_1)\cdots$
Figure 6.  Sum of compositions of trees
[1]

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

[2]

Viktor I. Gerasimenko, Igor V. Gapyak. Hard sphere dynamics and the Enskog equation. Kinetic & 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 & 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 & Related Models, 2008, 1 (4) : 521-555. doi: 10.3934/krm.2008.1.521

[5]

Céline Baranger, Marzia Bisi, Stéphane Brull, Laurent Desvillettes. On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases. Kinetic & Related Models, 2018, 11 (4) : 821-858. doi: 10.3934/krm.2018033

[6]

Gilberto M. Kremer, Wilson Marques Jr.. Fourteen moment theory for granular gases. Kinetic & Related Models, 2011, 4 (1) : 317-331. doi: 10.3934/krm.2011.4.317

[7]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

[8]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[9]

Marc Briant. Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinetic & Related Models, 2017, 10 (2) : 329-371. doi: 10.3934/krm.2017014

[10]

Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic & Related Models, 2011, 4 (1) : 17-40. doi: 10.3934/krm.2011.4.17

[11]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801

[12]

Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869

[13]

Hongjun Yu. Global classical solutions to the Boltzmann equation with external force. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1647-1668. doi: 10.3934/cpaa.2009.8.1647

[14]

Seiji Ukai. Time-periodic solutions of the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 579-596. doi: 10.3934/dcds.2006.14.579

[15]

Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic & Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673

[16]

Thomas Carty. Grossly determined solutions for a Boltzmann-like equation. Kinetic & Related Models, 2017, 10 (4) : 957-976. doi: 10.3934/krm.2017038

[17]

Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks & Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943

[18]

Yemin Chen. Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Kinetic & Related Models, 2010, 3 (4) : 645-667. doi: 10.3934/krm.2010.3.645

[19]

Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic & Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345

[20]

Gabriella Puppo, Matteo Semplice, Andrea Tosin, Giuseppe Visconti. Kinetic models for traffic flow resulting in a reduced space of microscopic velocities. Kinetic & Related Models, 2017, 10 (3) : 823-854. doi: 10.3934/krm.2017033

2017 Impact Factor: 1.219

Metrics

  • PDF downloads (37)
  • HTML views (140)
  • Cited by (0)

[Back to Top]