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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.   Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.   Google Scholar

[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.   Google Scholar

[25]

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

[26]

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

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.   Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.   Google Scholar

[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.   Google Scholar

[25]

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

[26]

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

[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.  Google Scholar

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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