Brownian motion in a polarizable lattice: Application to superionic conductors

We discuss the Brownian motion of a set of interacting particles. The general problem is formulated in terms of generalized Langevin equations and the frequency-dependent conductivity $\sigma \left(\omega \right)$ is computed from the velocity-velocity correlation function. In particular we consider a case pertaining to superionic conductors. We apply the formalism to a system consisting of a polarizable periodic sublattice built up of one ion species and oppositely charged mobile ions. Approximate solutions for $\sigma \left(\omega \right)$ are derived by choosing the simplest analytical memory functions which fulfill all asymptotic conditions. We show that lattice polarizability leads to structure at the transition frequency from diffusion-controlled to oscillation-controlled properties. The formalism is applied to $\alpha -\mathrm{AgI}$. The different interactions in superionic conductors, i.e., effective potential seen by mobile ion, lattice polarizability, and correlation of jumps of mobile ions, can be split and studied separately in their respective characteristic frequency regimes.