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We study the transport properties of dilute electrolyte solutions on thebasis of the fluctuating hydrodynamic equation, which is a set of nonlinearLangevin equations for the ion densities and flow velocity. The nonlinearity ofthe Langevin equations generally leads to effective kinetic coefficients forthe deterministic dynamics of the average ion densities and flow velocity; theeffective coefficients generally differ from the counterparts in the Langevinequations and are frequency-dependent. Using the path-integral formalisminvolving auxiliary fields, we perform systematic perturbation calculations ofthe effective kinetic coefficients for ion diffusion, shear viscosity, andelectrical conductivity, which govern the dynamics on the large length scales.As novel contributions, we study the frequency dependence of the viscosity andconductivity in the one-loop approximation. Regarding the conductivity atfinite frequencies, we derive the so-called electrophoretic part in addition tothe relaxation part, where the latter has been obtained by Debye andFalkenhagen; it is predicted that the combination of these two parts gives riseto the frequency $\omega_{\rm max}$ proportional to the salt density, at whichthe real part of the conductivity exhibits a maximum. The zero-frequency limitsof the conductivity and shear viscosity coincide with the classical limitinglaws for dilute solutions, derived in different means by Debye, Falkenhagen,and Onsager. As for the effective kinetic coefficients for slow ion diffusionsin large length scales, our straightforward calculation yields the crosskinetic coefficient between cations and anions. Further, we discuss thepossibility of extending the present study to more concentrated solutions.

journal of statistical mechanics

Fluctuating hydrodynamics of dilute electrolyte solutions: systematic  perturbation calculation of effective transport coefficients governing  large-scale dynamics