A Generalized Linear Transport Model for Spatially Correlated Stochastic Media
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 τ(s), for fixed physical distance s, thus becomes (1 + τ(s)/a)− a, with standard exponential decay recovered when a → ∞. 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 = ∞. Differential diffusion equations are not presently known when a < ∞, 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 τt ≫ 1. Random walk theory correctly predicts T∝τ− min {1, a/2}t in the absence of absorption. Finally, single scattering theory in d = 3 highlights the model’s violation of angular reciprocity when a < ∞, 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 and the other in a transition from one position to another in the medium.