Abstract
We study equilibrium states of systems of hard spheres in the Boltzmann-Enskog limit (d→0, 1/v→∞ (z→∞), and d 3 (1/v)=const (d 3 z=const)). For this purpose, we use the Kirkwood-Salsburg equations. We prove that, in the Boltzmann-Enskog limit, solutions of these equations exist and the limit distribution functions are constant. By using the cluster and compatibility conditions, we prove that all distribution functions are equal to the product of one-particle distribution functions, which can be represented as power series in z=d 3 z with certain coefficients.
Similar content being viewed by others
References
O. E. Lanford, “Time evolution of large classical systems,” Lect. Notes Phys., No. 38, 1–111 (1975).
V. I. Gerasimenko and D. Ya. Petrina, “Existence of Boltzmann-Grad limit for an infinite hard sphere system,” Teor. Mat. Fiz., 83, No. 1, 92–114 (1990).
D. Ruelle, Statistical Mechanics. Rigorous Results, Benjamin, New York-Amsterdam (1970).
D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev, Mathematical Foundations of Classical Statistical Mechanics, Gordon and Breach (1989).
Author information
Authors and Affiliations
Additional information
Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 1, pp. 112–121, January, 1997.
Rights and permissions
About this article
Cite this article
Petrina, D.Y., Petrina, E.D. Existence of equilibrium states of systems of hard spheres in the Boltzmann-Enskog limit within the frame work of the grand canonical ensemble. Ukr Math J 49, 124–134 (1997). https://doi.org/10.1007/BF02486621
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02486621