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.