Long-range correlations of polarization and number densities in dilute electrolytes

In dilute electrolytes, we calculate the pair correlation functions among the solvent polarization p, the solvent density n₁, the cation density n₂, and the anion density n₃. We set up a simple Ginzburg-Landau free energy for these variables, so our results are valid at distances longer than the molecular size σ. In particular, we reproduce the Høye-Stell result for the polarization correlation ⟨p_α(r)p_β(0)⟩ (α, β = x, y, z) [J. S. Høye and G. Stell, J. Chem. Phys. 68, 4145 (1978)], which is proportional to the second derivative ∂²(e^-κr/r)/∂x_α∂x_β for r ≫ σ with κ being the Debye wave number. We also show that size asymmetry between the cations and the anions gives rise to similar long-range correlations in ⟨p_α(r)δn₁(0)⟩ and ⟨δn_i(r)δn₁(0)⟩ (i = 1, 2, 3). Moreover, we calculate the polarization time-correlation function. As a unique feature in dynamics, the longitudinal polarization fluctuations (∝∇ · p) consist of rapidly decaying and slowly decaying components, where the latter relax with the charge density ρ. As a result, the long-range part of the equal-time polarization correlation changes into a different long-ranged and long-lived form after the shorter polarization relaxation.