August  2021, 41(8): 3903-3914. doi: 10.3934/dcds.2021021

On the cardinality of collisional clusters for hard spheres at low density

1. 

Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro 5, 00185 Rome – Italy, and, International Research Center M & MOCS, Università dell'Aquila, Palazzo Caetani, 04012 Cisterna di Latina, Italy

2. 

UMPA UMR 5669 CNRS, ENS de Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France

Received  May 2020 Revised  December 2020 Published  August 2021 Early access  January 2021

We resume the investigation, started in [2], of the statistics of backward clusters in a gas of $ N $ hard spheres of small diameter $ \varepsilon $. A backward cluster is defined as the group of particles involved directly or indirectly in the backwards-in-time dynamics of a given tagged sphere. We obtain an estimate of the average cardinality of clusters with respect to the equilibrium measure, global in time, uniform in $ \varepsilon, N $ for $ \varepsilon^2 N = 1 $ (Boltzmann-Grad regime).

Citation: 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
References:
[1]

R. K. Alexander, The Infinite Hard Sphere System, Thesis (Ph.D.)–University of California, Berkeley. 1975.

[2]

K. AokiM. PulvirentiS. Simonella and T. Tsuji, Backward clusters, hierarchy and wild sums for a hard sphere system in a low-density regime, Math. Models Methods Appl. Sci., 25 (2015), 995-1010.  doi: 10.1142/S0218202515500256.

[3]

T. BodineauI. GallagherL. Saint-Raymond and S. Simonella, One-sided convergence in the Boltzmann-Grad limit, Ann. Fac. Sci. Toulouse Math. (6), 27 (2018), 985-1022.  doi: 10.5802/afst.1589.

[4]

T. BodineauI. GallagherL. Saint-Raymond and S. Simonella, Fluctuation theory in the Boltzmann-Grad limit, J. Stat. Phys., 180 (2020), 873-895.  doi: 10.1007/s10955-020-02549-5.

[5]

D. BuragoS. Ferleger and A. Kononenko, Uniform estimates on the number of collisions in semi-dispersing billiards, Ann. of Math. 2, 147 (1998), 695-708.  doi: 10.2307/120962.

[6]

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.

[7]

R. Denlinger, The propagation of chaos for a rarefied gas of hard spheres in the whole space, Arch. Rat. Mech. Anal., 229 (2018), 885-952.  doi: 10.1007/s00205-018-1229-1.

[8]

A. Gabrielov, V. Keilis-Borok, Ya. Sinai and I. Zaliapin, Statistical properties of the cluster dynamics of the systems of statistical mechanics, in Boltzmann's Legacy, ESI Lectures in Mathematics and Physics, EMS Publishing House, (2008), 203–215. doi: 10.4171/057-1/13.

[9]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zürich Adv. Lect. in Math. Ser., 18, EMS, 2013.

[10]

V. I. Gerasimenko and I. V. Gapyak, The Boltzmann-Grad asymptotic behavior of collisional dynamics: A brief survey, Rev. Math. Phys., 33 (2021).

[11]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure App. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[12]

H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik 3, Springer-Verlag, (1958), 205–294.

[13]

R. Illner and M. Pulvirenti, Global Validity of the Boltzmann equation for a two–and three–dimensional rare gas in vacuum: Erratum and improved result, Comm. Math. Phys., 121 (1989), 143-146. 

[14]

F. G. King, BBGKY Hierarchy for Positive Potentials, Thesis (Ph.D.)-University of California, Berkeley. 1975.

[15]

O. E. Lanford, Time evolution of large classical systems, Lect. Notes Phys., 38 (1975), 1-111. 

[16]

T. J. Murphy and E. G. D. Cohen, On the sequences of collisions among hard spheres in infinite space, in Hard Ball Systems and the Lorentz Gas, Szász D. (eds). Enc. of Math. Sci. (Math. Phys. II) Springer, Berlin, Heidelberg, 101 (2000), 29–49. doi: 10.1007/978-3-662-04062-1_3.

[17]

R. I. A. PattersonS. Simonella and W. Wagner, Kinetic theory of cluster dynamics, Phys D, 335 (2016), 26-32.  doi: 10.1016/j.physd.2016.06.007.

[18]

R. I. A. PattersonS. Simonella and W. Wagner, A kinetic equation for the distribution of interaction clusters in rarefied gases, J. Stat. Phys., 169 (2017), 126-167.  doi: 10.1007/s10955-017-1865-0.

[19]

M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short-range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64 pp. doi: 10.1142/S0129055X14500019.

[20]

M. Pulvirenti and S. Simonella, The Boltzmann-Grad limit of a hard sphere system: Analysis of the correlation error, Invent. Math., 207 (2017), 1135-1237.  doi: 10.1007/s00222-016-0682-4.

[21]

M. Pulvirenti and S. Simonella, A kinetic model for epidemic spread, Math. Mech. Complex Syst., 8 (2020), 249-260.  doi: 10.2140/memocs.2020.8.249.

[22]

D. Serre, Hard spheres dynamics: Weak vs strong collisions, preprint, arXiv: 2002.09157.

[23]

J. G. Sina${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Construction of dynamics in one-dimensional systems of statistical mechanics, Teoret. Mat. Fiz., 11 (1972), 248-258. 

[24]

J. G. Sina${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Construction of a cluster dynamic for the dynamical systems of statistical mechanics, Vestnik Moskov. Univ. Ser. I Mat. Meh., 29 (1974), 152-158. 

[25]

H. Spohn, Large-Scale Dynamics of Interacting Particles, Springer, Berlin, 1991.

[26]

L. N. Vaserstein, On systems of particles with finite range and/or repulsive interactions, Comm. Math. Phys., 69 (1979), 31-56. 

[27]

E. Wild, On Boltzmann's equation in the kinetic theory of gases, Proc. Cambridge Philos. Soc., 47 (1951), 602-609.  doi: 10.1017/s0305004100026992.

show all references

References:
[1]

R. K. Alexander, The Infinite Hard Sphere System, Thesis (Ph.D.)–University of California, Berkeley. 1975.

[2]

K. AokiM. PulvirentiS. Simonella and T. Tsuji, Backward clusters, hierarchy and wild sums for a hard sphere system in a low-density regime, Math. Models Methods Appl. Sci., 25 (2015), 995-1010.  doi: 10.1142/S0218202515500256.

[3]

T. BodineauI. GallagherL. Saint-Raymond and S. Simonella, One-sided convergence in the Boltzmann-Grad limit, Ann. Fac. Sci. Toulouse Math. (6), 27 (2018), 985-1022.  doi: 10.5802/afst.1589.

[4]

T. BodineauI. GallagherL. Saint-Raymond and S. Simonella, Fluctuation theory in the Boltzmann-Grad limit, J. Stat. Phys., 180 (2020), 873-895.  doi: 10.1007/s10955-020-02549-5.

[5]

D. BuragoS. Ferleger and A. Kononenko, Uniform estimates on the number of collisions in semi-dispersing billiards, Ann. of Math. 2, 147 (1998), 695-708.  doi: 10.2307/120962.

[6]

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.

[7]

R. Denlinger, The propagation of chaos for a rarefied gas of hard spheres in the whole space, Arch. Rat. Mech. Anal., 229 (2018), 885-952.  doi: 10.1007/s00205-018-1229-1.

[8]

A. Gabrielov, V. Keilis-Borok, Ya. Sinai and I. Zaliapin, Statistical properties of the cluster dynamics of the systems of statistical mechanics, in Boltzmann's Legacy, ESI Lectures in Mathematics and Physics, EMS Publishing House, (2008), 203–215. doi: 10.4171/057-1/13.

[9]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zürich Adv. Lect. in Math. Ser., 18, EMS, 2013.

[10]

V. I. Gerasimenko and I. V. Gapyak, The Boltzmann-Grad asymptotic behavior of collisional dynamics: A brief survey, Rev. Math. Phys., 33 (2021).

[11]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure App. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[12]

H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik 3, Springer-Verlag, (1958), 205–294.

[13]

R. Illner and M. Pulvirenti, Global Validity of the Boltzmann equation for a two–and three–dimensional rare gas in vacuum: Erratum and improved result, Comm. Math. Phys., 121 (1989), 143-146. 

[14]

F. G. King, BBGKY Hierarchy for Positive Potentials, Thesis (Ph.D.)-University of California, Berkeley. 1975.

[15]

O. E. Lanford, Time evolution of large classical systems, Lect. Notes Phys., 38 (1975), 1-111. 

[16]

T. J. Murphy and E. G. D. Cohen, On the sequences of collisions among hard spheres in infinite space, in Hard Ball Systems and the Lorentz Gas, Szász D. (eds). Enc. of Math. Sci. (Math. Phys. II) Springer, Berlin, Heidelberg, 101 (2000), 29–49. doi: 10.1007/978-3-662-04062-1_3.

[17]

R. I. A. PattersonS. Simonella and W. Wagner, Kinetic theory of cluster dynamics, Phys D, 335 (2016), 26-32.  doi: 10.1016/j.physd.2016.06.007.

[18]

R. I. A. PattersonS. Simonella and W. Wagner, A kinetic equation for the distribution of interaction clusters in rarefied gases, J. Stat. Phys., 169 (2017), 126-167.  doi: 10.1007/s10955-017-1865-0.

[19]

M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short-range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64 pp. doi: 10.1142/S0129055X14500019.

[20]

M. Pulvirenti and S. Simonella, The Boltzmann-Grad limit of a hard sphere system: Analysis of the correlation error, Invent. Math., 207 (2017), 1135-1237.  doi: 10.1007/s00222-016-0682-4.

[21]

M. Pulvirenti and S. Simonella, A kinetic model for epidemic spread, Math. Mech. Complex Syst., 8 (2020), 249-260.  doi: 10.2140/memocs.2020.8.249.

[22]

D. Serre, Hard spheres dynamics: Weak vs strong collisions, preprint, arXiv: 2002.09157.

[23]

J. G. Sina${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Construction of dynamics in one-dimensional systems of statistical mechanics, Teoret. Mat. Fiz., 11 (1972), 248-258. 

[24]

J. G. Sina${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Construction of a cluster dynamic for the dynamical systems of statistical mechanics, Vestnik Moskov. Univ. Ser. I Mat. Meh., 29 (1974), 152-158. 

[25]

H. Spohn, Large-Scale Dynamics of Interacting Particles, Springer, Berlin, 1991.

[26]

L. N. Vaserstein, On systems of particles with finite range and/or repulsive interactions, Comm. Math. Phys., 69 (1979), 31-56. 

[27]

E. Wild, On Boltzmann's equation in the kinetic theory of gases, Proc. Cambridge Philos. Soc., 47 (1951), 602-609.  doi: 10.1017/s0305004100026992.

Figure 1.  The trajectory of a backward cluster $ BC(1) $ at time $ t $ (of cardinality $ 3 $) is represented on the left. Its tree structure $ {\Gamma}_3 = (1,1,2) $ is given by the graph on the right
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