Abstract
The structure function S(k; β) for the one-dimensional one-component log–gas is the Fourier transform of the charge–charge, or equivalently the density–density, correlation function. We show that for |k|<min(2πρ, 2πρβ), S(k; β) is simply related to an analytic function f(k; β) and this function satisfies the functional equation f(k; β)=f(−2k/β; 4/β). It is conjectured that the coefficient of kj in the power series expansion of f(k; β) about k=0 is of the form of a polynomial in β/2 of degree j divided by (β/2)j. The bulk of the paper is concerned with calculating these polynomials explicitly up to and including those of degree 9. It is remarked that the small k expansion of S(k; β) for the two-dimensional one-component plasma shares some properties in common with those of the one-dimensional one-component log–gas, but these break down at order k8.
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REFERENCES
K. Aomoto, Jacobi polynomials associated with Selberg's integral, SIAM J. Math. Analysis 18:545–549 (1987).
R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982).
P. J. Forrester, Exact integral formulas and asymptotics for the correlations in the 1/r 2 quantum many body system, Phys. Lett. A 179:127–130 (1993).
P. J. Forrester, Recurrence equations for the computation of correlations in the 1/r 2 quantum many body system, J. Stat. Phys. 72:3–50 (1993).
P. J. Forrester, Addendum to Selberg correlation integrals and the 1/r 2 quantum many body system, Nucl. Phys. B 416:377–385 (1994).
P. J. Forrester and B. Jancovici, Exact and asymptotic formulas for overdamped Brownian dynamics, Physica A 238:405–424 (1997).
P. J. Forrester, B. Jancovici, and E. R. Smith, The two-dimensional one-component plasma at Γ = 2: The semi-periodic strip, J. Stat. Phys. 31:129–140 (1983).
P. J. Forrester and J. A. Zuk, Applications of the Dotsenko-Fateev integral in random-matrix models, Nucl. Phys. B 473:616–630 (1996).
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 4th ed. (Academic Press, New York, 1980).
A. J. Guttmann, Indicators of solvability for lattice models, Discr. Math. 217:167–189 (2000).
Z. N.C. Ha, Fractional statistics in one dimension: View from an exactly solvable model, Nucl. Phys. B 435:604–636 (1995).
J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, 2nd ed. (Academic Press, London, 1990).
B. Jancovici, Exact results for the two-dimensional one-component plasma, Phys. Rev. Lett. 46:386–388 (1981).
B. Jancovici, Classical Coulomb systems near a plane wall. II, J. Stat. Phys. 29:263–280 (1982).
K. W. J. Kadell, The Selberg-Jack symmetric functions, Adv. Math. 130:33–102 (1997).
P. Kalinay, P. Markos, L. Samaj, and I. Travenec, The sixth-moment sum rule for the pair correlations of the two-dimensional one-component plasma: Exact results, J. Stat. Phys. 98:639–666 (2000).
J. Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math Anal. 24:1086–1110 (1993).
J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (CUP, Cambridge, 1958).
I. G. Macdonald, Hall Polynomials and Symmetric Functions, 2nd ed. (Oxford University Press, Oxford, 1995).
M. L. Mehta, Random Matrices, 2nd ed. (Academic Press, New York, 1991).
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Forrester, P.J., Jancovici, B. & McAnally, D.S. Analytic Properties of the Structure Function for the One-Dimensional One-Component Log–Gas. Journal of Statistical Physics 102, 737–780 (2001). https://doi.org/10.1023/A:1004846818738
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DOI: https://doi.org/10.1023/A:1004846818738