Effect of different waiting time processes with memory to anomalous diffusion dynamics in an external force fields

In this paper, we study the anomalous diffusion of a particle in an external force field whose motion is governed by nonrenewal continuous time random walks with memory. In our models, the waiting time involves Riemann–Liouville fractional derivative or Riemann–Liouville fractional integral. We obtain the systematic observation on the mean squared displacement, the Fokker–Planck-type dynamic equations and their stationary solutions. These processes obey a generalized Einstein–Stokes–Smoluchowski relation, and observe the second Einstein relation. The asymptotic behavior of waiting times and subordinations are of stretched Gaussian distributions. We also discuss the time averaged in the case of an external force field, and show that the process exhibits aging and ergodicity breaking.