Fourth Moment Sum Rule for the Charge Correlations of a Two-Component Classical Plasma

Abstract

We consider an ionic fluid made with two species of mobile particles carrying either a positive or a negative charge. We derive a sum rule for the fourth moment of equilibrium charge correlations. Our method relies on the study of the system response to the potential created by a weak external charge distribution with slow spatial variations. The induced particle densities, and the resulting induced charge density, are then computed within density functional theory, where the free energy is expanded in powers of the density gradients. The comparison with the predictions of linear response theory provides a thermodynamical expression for the fourth moment of charge correlations, which involves the isothermal compressibility as well as suitably defined partial compressibilities. The familiar Stillinger-Lovett condition is also recovered as a by-product of our method, suggesting that the fourth moment sum rule should hold in any conducting phase. This is explicitly checked in the low density regime, within the Abe-Meeron diagrammatical expansions. Beyond its own interest, the fourth-moment sum rule should be useful for both analyzing and understanding recently observed behaviours near the ionic critical point.

Underscreening in ionic liquids: a first principles analysis

“…(16) Note that λ 2 S can be expressed as the ratio −I 4 /I 2 where I 2 and I 4 are the coefficients of k 2 and k 4 terms in the small k expansion of S ZZ (k); these coefficients have been derived from second and fourth moment sum rules within linear response theory [18] and lead back to the result (15).…”

Critical charge and density coupling in ionic spherical models

“…More recently, Monte Carlo simulations have unequivocally pointed towards the Ising behavior [10]. But one must still recognize that no generally accepted theoretical description is available espe-cially as regards the various charge and density correlation functions at and close to criticality [3][4][5][6][11][12][13][14][15]]. Yet some progress have been made as regards the crossover Ginzburg temperature [16][17][18][19][20][21].…”

Form invariance of the moment sum-rules for jellium with the addition of short-range terms in the pair-potential

“…This work is a step forward in the understanding of the failure of the second and fourth moment sum-rules recently observed in the restricted primitive model (RPM) at criticality [11,12]. In fact, it was only until recently that the previously unknown form of the fourth moment sum-rule for the RPM was established using a semi-heuristic argument [13,14] claiming form invariance respect to the pure coulombic case. Our present result gives a rigorous first principle proof of the form invariance at least in the weak coupling regime and for the one-component Jellium.…”

“…It is still an open problem the extension of our study to the more general case of a mixture. A semi-heuristic derivation has recently been carried out [13,14] showing that the addition of the short-range term should not play any role at the level of the first three (even) structure factor moments for a neutral TCP without the background. Strictly speaking, in these derivations we had to use results that are only rigorously valid in the Debye regime, like the local neutrality of the homogeneous system.…”