Heterogeneous Memorized Continuous Time Random Walks 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 correlated memorized waiting times, which involves Reimann–Liouville fractional derivative or Reimann–Liouville fractional integral. We show that the mean squared displacement of the test particle \(X_{x}\) which is dependent on its location \(x\) of the form (El-Wakil and Zahran, Chaos Solitons Fractals, 12, 1929–1935, 2001) $$\begin{aligned} \langle \mathbb {X}_x^2\rangle (t)=\langle (\Delta X_x(t))^2\rangle _0\sim |x|^{-\theta }t^{\gamma }, \quad 0<\gamma <1, \quad \theta =d_w-2, \end{aligned}$$ (1) where \(d_w>2\) is the anomalous exponent, the diffusion exponent \(\gamma \) is dependent on the model parameters. We obtain the Fokker–Planck-type dynamic equations, and their stationary solutions are of the Boltzmann–Gibbs form. These processes obey a generalized Einstein–Stokes–Smoluchowski relation and the second Einstein relation. We observe that the asymptotic behavior of waiting times and subordinations are of stretched Gaussian distributions. We also discuss the time averaged in the case of an harmonic potential, and show that the process exhibits aging and ergodicity breaking.