The Sixth-Moment Sum Rule for the Pair Correlations of the Two-Dimensional One-Component Plasma: Exact Result
The system under consideration is a two-dimensional one-component plasma in the fluid regime, at density n and arbitrary coupling Γ=βe2 (e=unit charge, β=inverse temperature). The Helmholtz free energy of the model, as the generating functional for the direct pair correlation c, is treated in terms of a convergent renormalized Mayer diagrammatic expansion in density. Using specific topological transformations within the bond-renormalized Mayer expansion, we prove that the nonzero contributions to the regular part of the Fourier component of c up to the k2-term originate exclusively from the ring diagrams (unable to undertake the bond-renormalization procedure) of the Helmholtz free energy. In particular, ĉ(k)=−Γ/k2+Γ/(8πn)−k2/[96(πn)2]+O(k4). This result fixes via the Ornstein–Zernike relation, besides the well-known zeroth-, second-, and fourth-moment sum rules, the new sixth-moment condition for the truncated pair correlation h, n(πΓn/2)3 ∫ r6h(r) dr=3(Γ−6)(8−3Γ)/4.