A simple harmonic model as a caricature for mismatch and relaxation effects for ion hopping dynamics in solid electrolytes

We formulate a simple harmonic mean-field model with N+1 particles and analyse the relaxation processes following a jump of one of these particles. Either the particle can jump back (single-particle route) or the other N particles adjust themselves (multi-particle route). The dynamics of this model is solved analytically in the linear response regime. Furthermore we relate these results to a phenomenological approach by Funke and coworkers (concept of mismatch and relaxation: CMR) which has been successfully used to model conductivity spectra in the field of ion dynamics in solid electrolytes. Since the mean-field model contains the relevant ingredients of the CMR-approach, a comparison of the resulting rate equations with the CMR-equations becomes possible. Generalizations beyond the mean-field case are discussed.