Fractional and stochastic approaches to alternative conductivity in amorphous structures

We aim to derive a phenomenological approach to link the theories of anomalous transport governed by fractional calculus and stochastic theory with the conductivity behavior governed by the semi-empirical conductivity formalism involving Debye, Cole-Cole, Cole-Davidson, and Havriliak-Negami type conductivity equations. We want to determine the anomalous transport processes in complex or disordered media such as those occurring in the amorphous semiconductors and insulators via some familiar mathematical instruments and methods. Thus, we have started with the stochastic equation of motion called the Langevin equation, developed its equivalent master equation called Klein-Kramers or Fokker-Planck equation, and considered the space and time-fractional generalization of the master equation. Once we have derived the fractional master equation, then determined the observables or the mean value of the variables through some calculations and conditions. Finally, we have used the mean values in the conductivity formalism to obtain the average conductivity behavior.