The radiative transport equation with waiting time and its diffusion approximation with a time-fractional derivative

TL;DR

The paper addresses anomalous diffusion in porous-media transport by deriving a time-fractional advection-dispersion equation from a radiative transport equation with waiting-time traps in the diffusive regime. Using diffusion-approximation and Laplace-transform techniques with interior/initial-layer decomposition, it shows that the macroscopic field $u(\mathbf r,t)$ satisfies $\partial_t u + \eta\partial_t^{\alpha}u - D_0\Delta u + \mathbf c\cdot\nabla u + \sigma_a u = 0$, where $\eta=\gamma^{\alpha}\sigma_{\rm trap}$ and $D_0=|\mathbf v_0|^2/[3(1-g)\sigma_s]$, and that $\partial_t^{\alpha}$ is the Caputo derivative. The heavy-tailed waiting-time distribution $w(\tau) \sim \tau^{-(1+\alpha)}$ with $0<\alpha<1$ underpins the fractional term, and the framework recovers the classical advection-dispersion equation when trapping is negligible. Numerical tests in 1D compare the full RTE solution with the fractional-diffusion model and demonstrate diffusion-limit validity, highlighting how trap strength and tail exponent govern the approach to diffusion. The work provides a physically clearer derivation of time-fractional diffusion from mesoscopic transport, offering a robust link between radiative transport and fractional dynamics for porous-media applications.

Abstract

Albeit the past intensive research, the governing equation of anomalous diffusion which is observed for the transport of particles underground is still an open problem. In this paper, as a governing equation, the advection-diffusion equation with a time-fractional derivative term is derived from the radiative transport equation with an integral term for waiting time.

Figures (2)

• Figure 1: Comparison of the solution (blue line) to the transport equation (\ref{['1dRTE']}), the solution (red line) to the fractional diffusion equation (\ref{['1dfDE']}), and the solution (ocher line) to the normal diffusion (\ref{['normaldiff']}) at $t=10\,{\rm min}$. In panel (a), the parameters are set to $\sigma_{\rm trap}=0.1\,{\rm min}^{-1}$, $\gamma=0.1\,{\rm min}$, $\alpha=1/2$. The same parameters are used in other panels except that (b) $\sigma_{\rm trap}=0.01\,{\rm min}^{-1}$ and (c) $\gamma=1\,{\rm min}$, respectively.
• Figure 2: Same as Fig. \ref{['fig1']} but $t=100\,{\rm min}$.