Disentangling Sources of Anomalous Diffusion

We show that some important properties of subdiffusion of unknown origin (including ones of mixed origins) can be easily assessed when finding the “fundamental moment” of the corresponding random process, i.e., the one which is additive in time. In subordinated processes, the index of the fundamental moment is inherited from the parent process and its time dependence from the leading one. In models of a particle’s motion in disordered potentials, the index is governed by the structural part of the disorder while the time dependence is given by its energetic part.

Figure 1

Simulation results for CTRW with Pareto distributed waiting times (triangles) and for the trap model (boxes). Every point is calculated from 2048 trajectories of length ${10}^6\tau$. Error bars are given for three points and are smaller than the symbols. Both models are (quasi-) subordinated to a process with uncorrelated increments, and their fundamental exponent is ${\gamma }_F=2$.

Figure 2

Simulation of barrier (upper panel) and mixed (lower panel) models. The fundamental exponent is plotted against the mean barrier height. Every point was calculated from 2048 trajectories of length ${10}^6\tau$. Error bars are given for the barrier model for three points and have at most the size of the symbols. Each dotted line in the lower plot corresponds to one fixed value of mean trap depth varying from zero to $3k_{B}T$.

Figure 3

The fundamental moments for the trap, barrier, and mixed models as functions of time. Here, we took $E_T=3$ and $E_B=3$, so that both in the barrier and in the trap model $⟨X^2(t)⟩\propto \sqrt{t}$. The upper and the lower dashed lines have slopes 1 and 0.5, respectively.