Abstract
We consider a system of hard spheres in thermal equilibrium. Using Lanford's result about the convergence of the solutions of the BBGKY hierarchy to the solutions of the Boltzmann hierarchy, we show that in the low-density limit (Boltzmann-Grad limit): (i) the total time correlation function is governed by the linearized Boltzmann equation (proved to be valid for short times), (ii) the self time correlation function, equivalently the distribution of a tagged particle in an equilibrium fluid, is governed by the Rayleigh-Boltzmann equation (proved to be valid for all times). In the latter case the fluid (not including the tagged particle) is to zeroth order in thermal equilibrium and to first order its distribution is governed by a combination of the Rayleigh-Boltzmann equation and the linearized Boltzmann equation (proved to be valid for short times).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Grad, Principles of the Kinetic Theory of Gases, inHandbuch der Physik, S. Flügge, ed. (Springer, Berlin, 1958), Vol. 12.
O. E. Lanford, Time Evolution of Large Classical Systems, inDynamical Systems, Theory and Applications, J. Moser, ed. (Lecture Notes in Physics No. 38, Springer, Berlin, 1975), p. 1.
O. E. Lanford, On the Derivation of the Boltzmann Equation.Soc. Math. de France. Astérisque 40:117 (1976).
J. L. Lebowitz and J. Percus,Phys. Rev. 155:122 (1967).
J. L. Lebowitz, J. Percus, and J. Sykes,Phys. Rev. 188:487 (1969).
P. Resibois and J. L. Lebowitz,J. Stat. Phys. 12:483 (1975).
W. Braun and K. Hepp,Comm. Math. Phys. 56:101 (1977).
K. Hepp and E. H. Lieb,Helv. Phys. Acta 46, 573 (1973).
F. King, Ph.D. Thesis, Department of Mathematics, University of California at Berkeley (1975).
C. Cercignani,Transport Theory and Statistical Physics 2:211 (1972).
O. E. Lanford, Notes of the Lectures at the Troisième Cycle, Lausanne (1978), unpublished.
R. K. Alexander, Ph.D. Thesis, Department of Mathematics, University of California at Berkeley (1975).
C. Cercignani,The Boltzmann Equation (Elsevier, New York, 1976).
M. Klaus,Helv. Phys. Acta 48:99 (1975).
D. Ruelle,Statistical Mechanics: Rigorous Results (Benjamin, Reading, Mass., 1969).
R. L. Dobrushin and B. Tirozzi,Comm. Math. Phys. 54:173 (1977).
Author information
Authors and Affiliations
Additional information
Supported in part by NSF Grant PHY 78-22302.
Rights and permissions
About this article
Cite this article
van Beijeren, H., Lanford, O.E., Lebowitz, J.L. et al. Equilibrium time correlation functions in the low-density limit. J Stat Phys 22, 237–257 (1980). https://doi.org/10.1007/BF01008050
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01008050