Hierarchy of evolution equations for correlations of hard-sphere fluids





BBGKY hierarchy, Liouville hierarchy, correlation function


In the communication we discuss an approach to describing the correlations in a system of many hard spheres based on the hierarchy of evolution equations for correlation functions. It is established that the constructed dynamics of correlations underlies the description of the dynamics of both finitely and infinitely many hardspheres governed by the BBGKY hierarchies for reduced distribution functions or reduced correlation functions.


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Cercignani, C., Gerasimenko, V. I. & Petrina, D. Ya. (2012). Many-particle dynamics and kinetic equations. Amsterdam: Springer.

Petrina, D. Ya. & Gerasimenko, V. I. (1990). Mathematical problems of statistical mechanics of a system of elastic balls. Russ. Math. Surv., 5, No. 3, pp. 153-211. https://doi.org/10.1070/RM1990v045n03ABEH002360


Gallagher, I., Saint-Raymond, L. & Texier, B. (2014). From Newton to Boltzmann: hard spheres and shortrange potentials. Zürich Lectures in Advanced Mathematics (vol. 18). Zürich: EMS Publ. House.


Bodineau, T., Gallagher, I., Saint-Raymond, L. & Simonella, S. (2020). Fluctuation theory in the Boltzmann-Grad limit. J. Stat. Phys., 180, pp. 873-895. https://doi.org/10.1007/s10955-020-02549-5


Duerinckx, M. & Saint-Raymond, L. (2021). Lenard-Balescu correction to mean-field theory. Probab. Math. Phys., 2, No. 1, pp. 27-69. https://doi.org/10.2140/pmp.2021.2.27


Duerinckx, M. (2021). On the size of chaos via Glauber calculus in the classical mean-field dynamics. Commun. Math. Phys., 382, pp. 613-653. https://doi.org/10.1007/s00220-021-03978-3


Simonella, S. (2014). Evolution of correlation functions in the hard sphere dynamics. J. Stat. Phys., 155, No. 6, pp. 1191-1221. https://doi.org/10.1007/S10955-013-0905-7


Pulvirenti, M. & Simonella, S. (2016). Propagation of chaos and effective equations in kinetic theory: a brief survey. Math. Mech. Complex Syst., 4, No. 3-4, pp. 255-274. https://doi.org/10.2140/memocs.2016.4.255


Ivankiv, L. I., Prykarpatsky, Y. A., Samoilenko, V. H. & Prykarpatski, A. K. (2021). Quantum current algebra symmetry and description of Boltzmann type kinetic equations in statistical physics. Symmetry, 13, No. 8, 1452. https://doi.org/10.3390/sym13081452


Gerasimenko, V. I. & Gapyak, I. V. (2014). The non-Markovian Fokker-Planck kinetic equation for a system of hard spheres. Dopov. Nac. akad. nauk Ukr., No. 12, pp. 29-35 (in Ukrainian). https://doi.org/10.15407/dopovidi2014.12.029


Gerasimenko, V. I. & Gapyak, I. V. (2012). Hard sphere dynamics and the Enskog equation. Kinet. Relat. Models., 5, No. 3, pp. 459-484. https://doi.org/10.3934/krm.2012.5.459


Gerasimenko, V. I. & Gapyak, I. V. (2021). Boltzmann-Grad asymptotic behavior of collisional dynamics. Rev. Math. Phys., 33, 2130001. https://doi.org/10.1142/S0129055X21300016


Gerasimenko, V. & Gapyak, I. (2018). Low-density asymptotic behavior of observables of hard sphere fluids. Adv. Math. Phys., 2018, 6252919. https://doi.org/10.1155/2018/6252919


Prigogine, I. (1962). Non-equilibrium statistical mechanics. New York: John Wiley & Sons.

Gerasimenko, V. I., Ryabukha, T. V. & Stashenko, M. O. (2004). On the structure of expansions for the BBGKY hierarchy solutions. J. Phys. A: Math. Gen., 37, pp. 9861-9872. https://doi.org/10.1088/0305- 4470/37/42/002




How to Cite

Gapyak, I., & Gerasimenko, V. . (2022). Hierarchy of evolution equations for correlations of hard-sphere fluids. Reports of the National Academy of Sciences of Ukraine, (3), 3–12. https://doi.org/10.15407/dopovidi2022.03.003