Solving Lyapunov equations for electrically driven ternary electrolytes: Application to long-range van der Waals interactions

Stochastic density-functional theory (SDFT) has been widely used to study the out-of-equilibrium properties of electrolyte solutions. Examples include investigations of electrical conductivity-both within and beyond linear response-and modifications of thermal van der Waals interactions in driven electrolytes. Within the approximation scheme derived from linearizing SDFT for fluctuations around mean densities, the steady-state correlation functions between the N ionic species are governed by linear Lyapunov equations of degree N(N+1)/2. Consequently, the system's complexity increases significantly when transitioning from binary to ternary electrolytes, and few analytical results exist for the latter. In this paper, we demonstrate how-for the specific case of electrolytes-the Lyapunov equations can be reduced to a system of N linear equations. We apply this reduction to compute the long-range component of the van der Waals interaction between two slabs containing a ternary electrolyte under an applied electric field parallel to the slabs. Unlike the binary electrolyte case, we show that the resulting van der Waals interaction for a ternary electrolyte depends on the ionic species' diffusion coefficients, highlighting its inherently out-of-equilibrium nature.