Anomalous Diffusion in Dielectric Relaxation of Polyelectrolytes

Anomalous diffusion is considered in the context of fractional dynamics applied to dielectric relaxation of polyelectrolyte solutions. The approach starts from a fractional Smoluchowski equation in configuration space of molecular orientations of macroions and displacements of couterions. By using a perturbation procedure, we derive analytic expressions for the buildup and reversing field processes of the electric polarization given by the expectation value of the product of the first Legendre polynomial by the first Hermite polynomial. The first harmonic component of the ac dielectric response is also calculated. All these results are illustrated by plots demonstrating the effect of both the coupling (rotation-translation) parameter a and the critical exponent α (subdiffusion).