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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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