Decomposition of Anomalous Diffusion in various Stochastic Processes


This dissertation presents three research projects on the decomposition of anomalous diffusion in various stochastic processes. Stochastic processes that scale anomalously with time, such that the Mean-Squared Displacement (MSD) of the expanding particle is x2(t)⟩∼t2H where H is the \emph{Hurst exponent} and H≠1/2. Anomalous diffusion is known to occur in such processes due to auto-correlations, the \emph{Joseph effect}, the infinite variance of individual events, the \emph{Noah effect}, or the non-stationarity of increments, the \emph{Moses effect}. Scaling exponents quantifying each of the three effects are known as the \emph{Joseph}, \emph{Latent} and \emph{Moses} exponents, respectively. These scaling exponents are related to the \emph{Hurst exponent} through the scaling relation H=J+L+M−1. The first project focuses on decomposing anomalous diffusion into its fundamental constitutive causes in an aging system. The model process is a sum of increments that are iterates of the Pomeau-Manneville map, a chaotic dynamical system. The increments can have long-time correlations, fat-tailed distributions, and be non-stationary. Each of these properties can cause anomalous diffusion. The model can have either sub- or super-diffusive behavior, which is due to a combination of the three effects. Scaling exponents quantifying each of the three constitutive effects are calculated analytically and confirmed numerically. They are then related to the scaling of the distribution of the process through a scaling relation. Finally, the importance of the Moses effect in the anomalous diffusion of experimental systems is discussed. The second project studies the origins of anomalous diffusion in an ensemble of time-series generated experimentally or numerically, without knowing the exact underlying dynamics of the process. The increment distribution converges at increasing times to a time-invariant asymptotic shape after appropriate rescaling based on the quantification of the three effects. This asymptotic limit can be an equilibrium state, an infinite-invariant density, or an infinite-covariant density for different processes. The three effects in a non-linearly coupled Lévy walk model are quantified using time-series analysis methods, and the results are compared to theoretical predictions. The generality of this method is also discussed. The third project considers diffusion processes with spatially varying diffusivity, which can result in anomalous diffusion. Heterogeneous diffusion processes are analyzed for the cases of exponential, power-law, and logarithmic dependencies of the diffusion coefficient on the particle position. The importance of the functional form of the space-dependent diffusion coefficient and the initial conditions of the diffusing particle in determining the statistical properties of the system is observed. The model exhibits sub- and super-diffusive behavior depending on the value of the space-dependent diffusion coefficient. The numerical methods of time-series analysis quantify the three effects in a heterogeneous diffusion model and compare the results to theoretical predictions. A heterogeneous diffusion model is an alternative approach to non-ergodic, anomalous diffusion that may be especially useful for diffusion in heterogeneous media.



Anomalous diffusion, PM map, Levy walks, Hurst exponent, Joseph exponent, Latent exponent, Moses exponent


Portions of this document appear in: Meyer, Philipp G., Vidushi Adlakha, Holger Kantz, and Kevin E. Bassler. "Anomalous diffusion and the Moses effect in an aging deterministic model." New Journal of Physics 20, no. 11 (2018): 113033.; Aghion, Erez, Philipp G. Meyer, Vidushi Adalkha, Holger Kantz, and Kevin E. Bassler. "Moses, Noah and Joseph Effects in Coupled L\'evy Processes." arXiv preprint arXiv:2009.08702 (2020).