Persistent random walk on a site-disordered one-dimensional lattice: Photon subdiffusion

Abstract

We study the persistent random walk of photons on a one-dimensional lattice of random transmittances. Transmittances at different sites are assumed independent, distributed according to a given probability density f (t). Depending on the behavior of f (t) near t = 0, diffusive and subdiffusive transports are predicted by the disorder expansion of the mean square-displacement and the effective medium approximation. Monte Carlo simulations confirm the anomalous diffusion of photons. To observe photon subdiffusion experimentally, we suggest a dielectric film stack for realization of a distribution f (t).

Relativistic Brownian motion

“…(209) represent a source of concern if one wishes to adopt the telegraph equation (206) as a model for particle transport in a random medium. While these singularities may be acceptable in the case of photon diffusion [371][372][373][374][375], they seem unrealistic for massive particles, because such fronts would imply that a finite fraction of particles carries a huge amount of kinetic energy (much larger than mc 2 ). In view of these shortcomings, it appears reasonable to explore other constructions of relativistic diffusion processes [23,390].…”

“…13 In contrast to standard non-directed random walk models, which lead to the classical diffusion equation (1) when performing an appropriate continuum limit [74], the random jumps of a persistent walk take into account the history of a path by assigning a larger probability to those jumps that point in the direction of the motion before the jump [366,367]. Persistent random walk models can be used to describe the transmission of light in multiple scattering media [372] such as foams [373][374][375] and thin slabs [371,376]. Similarly, the telegraph equation (2) has been applied in various areas of physics over the past decades, e.g., to model the propagation of electric signals and heat waves.…”

“…Apart from 75 Debbasch and Rivet [12] discuss the difficulties that arise when attempting a similar reduction for the relativistic Ornstein-Uhlenbeck process. the theoretical challenge of developing a consistent relativistic diffusion theory, there exist several practical applications including, for example, the analysis of data from high energy collision experiments [320][321][322] or the diffusion of light through turbid media [372,504,505] and foams [373][374][375]. In this context, a frequently considered alternative to the classical diffusion equation (194) is given by the telegraph equation [10,27,29,320,321,366,370,[377][378][379][380] …”

“…Hence, in contrast to the ordinary diffusion equation (194), the telegraph equation (206) relies on asymmetric transition probabilities, causing the non-vanishing probability concentration at the diffusion fronts. In more recent years, persistent diffusion models have been employed to describe the propagation of photons in thin slabs and foams [371][372][373][374][375].…”

Relativistic diffusion processes and random walk models

“…From the practical point of view, the concept of persistent diffusion can certainly be useful in some situations, e.g., if one aims at describing the propagation of photons in thin slabs or foams [11,36,37,38]. In general, however, it seems unlikely that the relativistic generalization of the simple nonrelativistic diffusion theory would require conceptual modifications as severe as the introduction of persistence.…”