Charge fluctuations in Coulomb systems

It was shown by Martin and Yalcin that the mean-square fluctuation $〈Q_{\Lambda }^2〉$ in the net electric charge $Q_{\Lambda }$ contained in a subregion $\Lambda$ of an infinitely extended equilibrium Coulomb system (plasma, electrolytes, etc.) grows only as the surface area $S_{\Lambda }$ (not the volume) of $\Lambda$ and that $\frac{Q_{\Lambda }}{\sqrt{S_{\Lambda }}}$ has a Gaussian distribution as $\Lambda \to \infty$. We extend these results to joint charge fluctuations in different spatial regions: Let space be divided into disjoint regions ${\Lambda }_i$, $i=1,2,\dots$, say, cubes of length $L$. We show that as $L\to \infty$, the covariance in $\frac{Q_{{\Lambda }_i}}{L}$ behaves as $L^{-2\mathrm{}}〈Q_{{\Lambda }_i}{Q}_{{\Lambda }_j})=-\frac{1}{6}{L}^{-2\mathrm{}}〈Q_{{\Lambda }_i}^2〉=-\frac{1}{6}K$ if ${\Lambda }_i$ and ${\Lambda }_j$ are adjacent, and is zero if they do not have a common face. Furthermore, the variables $\frac{Q_{\Lambda }}{L}$ approach, as $L\to \infty$, a jointly Gaussian distribution. These results can be proven rigorously whenever the correlations in the system decay faster than the fourth power of the distance, which is known to happen in many cases. This behavior of charge fluctuations is shown to be required for the consistency of the usual statistical-mechanical treatment of neutralmolecular systems.