Fractional dynamics and nonlinear harmonic responses in dielectric relaxation of disordered liquids.

The problem of the nonlinear dielectric response due to the application of a strong electric field is reconsidered in the context of fractional kinetic equations. To accomplish that, we start from a fractional noninertial Fokker-Planck equation and restrict ourselves to the case of anomalous subdiffusive processes characterized by the critical exponent alpha ranging from 0 to 1, the limit of normal diffusion. In particular, we evaluate the first- and third-order nonlinear harmonic components of the electric polarization in the case of either a pure ac field or a strong dc bias field superimposed on a weak ac field. The stationary regime is therefore calculated from an infinite set of differential recurrence relations by using a perturbation method. The results so obtained are illustrated by three-dimensional dispersion and absorption plots in order to show the influence of alpha. Cole-Cole diagrams are also presented, allowing one to see that the arcs become more and more flattened as alpha-->0, and corresponding to a broadening of the absorption peaks as effectively observed in complex liquids. The theoretical model is supported by comparison with experimental data of the third-order nonlinear dielectric permittivity of a ferroelectric liquid crystal.