Generalized master equation for particle transport in binary random media with renewal statistics.

Abstract

Particle transport in binary stochastic mixtures is classically modeled assuming Markovian or exponential mixing statistics but in many applications material memory invalidates the Markov assumption. For non-Markovian mixing characterized by alternating renewal processes, a transport-theoretic framework is presented that provides an exact description of transport in nonscattering random binary media with general nonexponential statistics. Our approach is to Markovianize the problem by augmenting the (material type, particle flux) state space with the age or distance from the last interface. A Chapman-Kolmogorov equation is formulated for the joint probability density of the material type, particle flux, and age and subsequently reduced to a generalized master equation (GME) in differential form. This constitutes the primary result of this work. A state-updating Monte Carlo algorithm consistent with the GME is developed and benchmarked against analytical solutions for multiple chord-length laws. For purely absorbing renewal statistical media, the GME reproduces analytical benchmarks for the equilibrium age distribution, interior mean and variance of material-conditioned fluxes, and boundary transmittance. Simulations further demonstrate that a Markov (exponential) approximation of nonexponential statistics can introduce large errors in transmittance and interior flux profiles. Finally, the reintroduction of memory due to scattering is briefly addressed through heuristic considerations.