Abstract
For systems with finite phase space volume, the density of states can be viewed as a multiple of the probability density of the energy, when the phase space variables are independent uniformly distributed random variables. We show that the distribution of a random variable proportional to the sum of pairwise interactions of independent identically distributed random variables converges to a limiting distribution as the number of variables goes to infinity, when the interaction satisfies certain homogeneity requirements. The moments of this distribution are simple combinations of cyclic integrals of the potential function. The existence of this limit gives information about the structure function of some systems in statistical mechanics having pair-summable interactions, even in the absence of a thermodynamic limit. The result is applied to several examples, including systems of two-dimensional point vortices.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird,Molecular Theory of Gases and Liquids (Wiley, New York, 1954), Chapter 3.
S. F. Edwards and J. B. Taylor, Negative temperature states of two-dimensional plasmas and vortex fluids,Proc. R. Soc. Lond. A 336:257–271 (1974).
J. Frohlich and D. Ruelle, Statistical mechanics of vortices in an inviscid two-dimensional fluid,Commun. Math. Phys. 87:1–36 (1982).
H. Rubin and R. A. Vitale, Asymptotic distribution of symmetric statistics,Ann. Stat. 8:165–170 (1980).
Robert J. Serfling,Approximation Theorems of Mathematical Statistics (Wiley, New York, 1980), Chapter 5.
W. Feller,An Introduction to Probability Theory and Its Applications, Vol. II, 3rd ed. (Wiley, New York, 1968), p. 228.
A. P. Hillman, G. L. Alexanderson, and R. M. Grassl,Discrete and Combinatorial Mathematics (Dellen, San Francisco, 1987).
K. Gehringer, Nonparametric probability density estimation using normalized B-splines, Masters thesis, Department of Mathematical and Computer Sciences, University of Tulsa (1990).
L. J. Campbell, Vortex lattices in theory and practice, inMathematical Aspects of Vortex Dynamics, R. Caflisch, ed. (SIAM, Philadelphia, 1989), pp. 195–204.
K. O'Neil, On the Hamiltonian dynamics of vortex lattices,J. Math. Phys. 30:1373–1379 (1989).
F. J. Dyson, Statistical theory of the energy levels of complex systems. I.,J. Math. Phys. 3(1):140–156 (1962).
L. J. Campbell and K. A. O'Neil, Statistics of 2D point vortices and high Reynolds number fluids, preprint.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
O'Neil, K.A., Redner, R.A. On the limiting distribution of pair-summable potential functions in many-particle systems. J Stat Phys 62, 399–410 (1991). https://doi.org/10.1007/BF01020875
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01020875