A Generalized Linear Transport Model for Spatially-Correlated Stochastic Media

Abstract

We formulate a new model for transport in stochastic media with long-range spatial correlations where exponential attenuation (controlling the propagation part of the transport) becomes power law. Direct transmission over optical distance $\tau(s)$, for fixed physical distance $s$, thus becomes $(1+\tau(s)/a)^{-a}$, with standard exponential decay recovered when $a\to\infty$. Atmospheric turbulence phenomenology for fluctuating optical properties rationalizes this switch. Foundational equations for this generalized transport model are stated in integral form for $d=1,2,3$ spatial dimensions. A deterministic numerical solution is developed in $d=1$ using Markov Chain formalism, verified with Monte Carlo, and used to investigate internal radiation fields. Standard two-stream theory, where diffusion is exact, is recovered when $a=\infty$. Differential diffusion equations are not presently known when $a<\infty$, nor is the integro-differential form of the generalized transport equation. Monte Carlo simulations are performed in $d=2$, as a model for transport on random surfaces, to explore scaling behavior of transmittance $T$ when transport optical thickness $\tau_\text{t} \gg 1$. Random walk theory correctly predicts $T \propto \tau_\text{t}^{-\min\{1,a/2\}}$ in the absence of absorption. Finally, single scattering theory in $d=3$ highlights the model's violation of angular reciprocity when $a<\infty$, a desirable property at least in atmospheric applications. This violation is traced back to a key trait of generalized transport theory, namely, that we must distinguish more carefully between two kinds of propagation: one that ends in a virtual or actual detection, the other in a transition from one position to another in the medium.