Complete Decomposition of Anomalous Diffusion in Variable Speed Generalized Lévy Walks

TL;DR

The paper develops a complete decomposition of anomalous diffusion in Variable-Speed Generalized Lévy Walks (VGLWs) into the Joseph (J), Noah (L), and Moses (M) effects, revealing that the Noah exponent can be unbounded, unlike in previous Lévy-walk models. By deriving the velocity propagator $p(v,t)$ and velocity moments from the joint distribution $p(t,\tau, t')$, and connecting these to the TAMSD via a scaling Green–Kubo relation, the authors obtain explicit, region-dependent formulas for $J$, $L$, and $M$, and hence the MSD Hurst exponent $H$ through $H = J + L + M - 1$. The study maps a rich phase diagram with nine dynamical regimes (plus a non-scaling infinity region) across the parameter space, and confirms key predictions with large-scale simulations showing cases where $L>1$. The results generalize the GLW framework, illustrate how velocity-time coupling broadens anomalous-diffusion behavior, and have broad implications for transport in physical, biological, and financial contexts. Overall, the work provides a rigorous, quantitative framework for identifying and characterizing the dominant mechanisms driving anomalous diffusion in complex, time-inhomogeneous systems.

Abstract

Variable Speed Generalized Lévy Walks (VGLWs) are a class of spatio-temporally coupled stochastic processes that unify a broad range of previously studied models within a single parametrized framework. Their dynamics consist of discrete random steps, or flights, during which the walker's speed varies deterministically with both the elapsed time and the total duration of the flight. We investigate the anomalous diffusive behavior of VGLWs and analyze it through decomposition into the three fundamental constitutive effects that capture violations of the Central Limit Theorem (CLT): the Joseph effect, reflecting long-range increment correlations, the Noah effect, arising from heavy-tailed step-size distributions with infinite variance, and the Moses effect, associated with statistical aging and non-stationarity. Our results show that anomalous diffusion in VGLWs is typically generated by a nontrivial combination of all three effects, rather than being attributable to a single mechanism. Strikingly, we find that within the VGLW framework the Noah exponent $L$, which quantifies the strength of the Noah effect, is unbounded from above, revealing a richer and more extreme landscape of anomalous diffusion than in previously studied Lévy-walk-type models.

Figures (24)

• Figure 1: Phase diagram of the Mosses effect for variable speed generalized Lévy walk. The dotted line marks the onset for the infinite regime for $\eta=1$
• Figure 2: Phase diagram of the Noah effect for variable speed generalized Lévy walk .The dotted line marks the onset for infinite regime for $\eta=1$
• Figure 3: Phase diagram of the Joseph effect for variable speed generalized Lévy walk .The dotted line marks the onset for infite regime for $\eta=1$
• Figure 4: Phase diagram of the Hurst effect for variable speed generalized Lévy walk .The dotted line marks the onset for infinite regime for $\eta=1$
• Figure 5: Overall phase diagram of VGLWs based on the full set of scaling exponents $(H,J,L,M)$. The dotted line marks the onset of the non-scaling (infinite) regime for $\eta = 1$.