Beyond Classical Diffusion: Fractional Derivatives in Transport and Stochastic Systems

TL;DR

This work shows how fractional reaction-diffusion equations emerge from continuous-time random walks with heavy-tailed waiting times and jumps, bridging microscopic CTRW dynamics with macroscopic PDEs via Montroll-Weiss analysis and scaling limits. It develops both density-based and stochastic formulations, highlighting Caputo time and Riesz space derivatives, and demonstrates through agent-based and macroscopic simulations how fractional diffusion yields heavier tails and nonlocal transport compared to classical diffusion. A new mass-conserving periodic-boundary scheme for the macroscopic Riesz derivative is proposed and validated against micro-scale simulations, illustrating the consistency between micro- and macro-level descriptions. The findings underscore the relevance of fractional calculus for modeling anomalous transport in heterogeneous biological media and point to future work on higher dimensions, reaction terms, and empirical validation.

Abstract

Integer-order differential operators were originally used to describe local and isotropic effects, in both space and time. However, in fields like biology, the modelling of complex phenomena with spatial heterogeneity necessitates more advanced approaches. The fractional calculus framework provides powerful tools for developing models that better capture the intricate dynamics of biological systems. This paper derives fractional reaction-diffusion equations from continuous-time random walks, highlighting the role of heavy-tailed distributions in the process. Both fractional partial differential equations, on the macroscopic level, as well as fractional stochastic differential equations, on the microscopic level, will be derived and simulated from, for simple Riesz-fractional diffusion models. A new numerical scheme that implements periodic boundary conditions is proposed to control the loss of mass density. We highlight the key differences between fractional and classical diffusion.

Figures (4)

• Figure 1: Comparison between agent-based (genuinely) fractional and (nearly) classical diffusion. The red dots represent agents dispersing according to the fractional order of $\alpha=1.5$ while the blue agents follow the (nearly) regular ($\alpha =1.99$) diffusion model. Snapshots of the simulation indicate how expansion of the colony evolves at different intermediate time points (a) $t = 10^{-6}$, (b) $t = 5\cdot 10^{-5}$, (c) $t = 2.5\cdot 10^{-5}$, and (d) $t = 10^{-4}$. The respective diffusion coefficients are chosen such that the bulk of the agents is spread over the same convex set in space, in both the fractional and (nearly) regular diffusion case. Nevertheless, the fractional diffusion model exhibits a heavier-tailed distribution that leads to a higher probability of long-range jumps.
• Figure 2: Statistical comparison of the agent-based distributions for the regular vs fractional diffusion of Figure \ref{['micro:d']}, at the final time $T$. (a) Boxplots showing the spread of position coordinates for all agents at the final time $T$, taking the outliers into consideration. (b) Same boxplots after removal of the outliers, for more emphasis on the central distribution. The blue and orange boxplots represent the distribution of the $x$-position coordinates of regularly ($\alpha = 1.99$), respectively fractionally ($\alpha = 1.5$) diffusing agents. Green and purple boxplots indicate how the final $y$-position coordinates are distributed for regularly, respectively (genuinely) fractionally diffusing agents.
• Figure 3: Macroscopic fractional vs regular diffusion, simulation results in one spatial dimension. The red curve represents the density spread of a fractionally ($\alpha =1.5$) diffusing population with diffusion coefficient $D_{\text{frac}} = 1$. The blue curve ($\alpha = 1.99$) indicates the density evolution of a regularly diffusing density, corresponding to a diffusion coefficient of $D_{\text{reg}}=0.02$. The diffusion coefficients were chosen such that the bulk of the densities remain co-localised over time. A pairwise comparison between regular and genuinely fractional diffusion at different time points (a) $t=0$, (b) $t=T/4$, (c) $t=T/2$, and (d) $t=T$ is presented in the above Figure \ref{['fig:macro']}.
• Figure 4: Macroscopic fractional vs regular diffusion simulation: zoomed-in picture at time $t=T$. To display the distinction between regularly ($\alpha=1.99$, blue curve) and genuinely fractional ($\alpha=1.5$, red curve) diffusion, we focus on the density (a) peaks and (b) tails, at the final time of the macroscopic density simulations, by zooming in on Figure \ref{['macro:d']}.

Theorems & Definitions (5)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 4.1: Grünwald-Letnikov operator
• Lemma 4.2