A Nonlinear Mechanism for Transient Anomalous Diffusion

TL;DR

This work addresses the emergence of transient anomalous diffusion using a locally nonlinear diffusion equation with a concentration-dependent diffusivity $D(\psi)=D_0+D_1\psi$. Through a perturbative analysis in the nonlinearity parameter $\varepsilon$, the authors derive a hierarchy of linear diffusion problems, with the zeroth-order solution given by the heat kernel and higher orders computed via Green's functions. They show that the mean squared displacement satisfies $\langle u^2\rangle \approx 2\tau + \varepsilon\sqrt{\dfrac{2\tau}{\pi}}$, and that the effective diffusion coefficient behaves as $D_{\text{eff}}(\tau) \approx 1 + \dfrac{\varepsilon}{2\sqrt{2\pi\tau}}$, indicating a transient anomalous regime that crosses over to classical Fickian diffusion at long times. The results provide a physically transparent mechanism for transient anomalies, arising from local interactions rather than memory effects, with implications for crowded or interacting systems where diffusion is density-dependent.

Abstract

Diffusion is a fundamental physical phenomenon with critical applications in fields such as metallurgy, cell biology, and population dynamics. While standard diffusion is well-understood, anomalous diffusion often requires complex non-local models. This paper investigates a nonlinear diffusion equation where the diffusion coefficient is linearly dependent on concentration. We demonstrate through a perturbative analysis that this physically-grounded model exhibits transient anomalous diffusion. The system displays a clear crossover from an initial subdiffusive regime to standard Fickian behavior at long times. This result establishes an important mechanism for trasient anomalous diffusion that arises purely from local interactions, providing an intuitive alternative to models based on fractional calculus or non-local memory effects.

Figures (2)

• Figure 1: Evolution of diffusion profiles for different values of the nonlinearity parameter $\varepsilon$. The profiles show how the concentration-dependent diffusivity affects the shape of the spreading distribution, with notable deviations from the Gaussian profile of linear diffusion.
• Figure 2: Effective Diffusion Coefficient Demonstrating Transient Anomalous Behavior. This plot shows the effective diffusion coefficient $D_\text{eff}$ versus dimensionless time $(\tau)$for different values of the nonlinearity parameter, $\varepsilon$. The graph clearly illustrates the system's crossover from an initial anomalous diffusion regime to standard Fickian diffusion at long times.