Stochastic Models of Higher Order Dielectric Responses

Abstract

The nonlinear response for systems exhibiting Markovian stochastic dynamics is calculated using time-dependent perturbation theory for the Green’s function, the conditional probability to find the system in a given configuration at a certain time given it was in another configuration at an earlier time. In general, the Green’s function obeys a so-called master-equation for the balance of the gain and loss of probability in the various configurations of the system. Using various models for the reorientational motion of molecules it is found that the scaled modulus of the third-order response, $X_3$, shows a hump-like behavior for random rotational motion in some cases and it exhibits “trivial” behavior, a monotonuos decay from a finite zero-frequency value to a vanishing high-frequency limit, if the model of isotropic rotational diffusion is considered. For the time-honored model of dipole reorientations in an asymmetric double-well potential, it is found that $X_3$ exhibits a peak in a certain temperature range around a characteristic temperature at which the zero-frequency limit vanishes. The fifth-order modulus $X_5$ shows hump-like behavior in two distinct temperature regimes located at temperatures, where $X_3$ behaves trivially. For a trap model with a Gaussian density of states, a model that exhibits some features of glassy relaxation, both nonlinear response functions can exhibit either trivial or hump-like behavior depending on the specific choice for some model parameters. The height of the peak shows various temperature dependencies from increasing with temperature, decreasing or a temperature-independent behavior.