A unifying approach to diffusive transport in heterogeneous media

Abstract

We introduce the concept of Randomly Modulated Gaussian Processes as a unifying framework for modeling, analyzing and classifying anomalous diffusion models in heterogeneous media. This formulation incorporates correlations in the displacements together with correlated fluctuations of their amplitudes. Most known models of anomalous diffusion (including Continuous-Time Random Walk and fractional Brownian motion) and random diffusivity can be described and generalized within this framework. Moreover, the unified view identifies the main statistical properties to be probed experimentally for a reliable classification of diffusive dynamics. The proposed matrix formulation facilitates the computation of the first four moments and allows for a systematic statistical characterization of the considered processes. The necessary and sufficient conditions are provided for the emergence of anomalous diffusion. General expressions for the non-Gaussian parameter, ergodicity breaking parameter and covariance of squared increments are derived. Biological applications of this framework for systematic analysis and biophysical interpretations of experimental single-particle trajectories are discussed.

Figures (2)

• Figure 1: Three-dimensional diagram of annealed diffusion models in heterogeneous media. The direction $C$ corresponds to first-order correlations of displacements going from uncorrelated (at the origin) to power-law correlated. The direction $J_i$ corresponds to the type of modulations, whether they are deterministic, random but constant, fluctuating with positive long-time limit, or fluctuating with freezing. Finally, the $\textrm{cov}(J_i,J_j)$ direction corresponds to correlations of modulations, ranging from uncorrelated (near origin) to power-law correlated. Stars denote processes with trajectory-wise stationary increments and disks denote those with non-stationary increments. The abbreviated models are described in the text.
• Figure 2: (A) Mean-squared displacement as a function of time. (B) Non-Gaussian parameter as a function of time. (C) Ergodicity breaking parameter as a function of trajectory duration. (D) Autocovariance of squared increments as a function of time. Lines show theoretical predictions, whereas symbols present simulations that are obtained for each model by generating $M=10^6$ trajectories. Five models are compared: fBm (with $H=0.35$ and $J_i=1$), DD-Exp (with $\langle D\rangle=1$ with $\tau=10$), SD-Exp (with two states $D_1=1$, $D_2=2$ and switching probabilities $p_1 = p_2 = 0.01$), sBm (with $\alpha=0.7$), and CTRW-Pow (with $\alpha=0.7$ and $D_0=1$). See Appendix E for more information on simulation procedures.