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The role of entropy in wave propagation

In recent years there has been a growing interest in accurate models of electromagnetic (em) wave propagation in urban areas, mostly due to the growing diffusion of personal wireless communications in densely populated environments. The analytical solution of Maxwell equations with the complicated boundary conditions imposed by the complex geometry of a typical urban area is impossible, and only numerical solutions, tailored to a single specified scenario, are attainable. A different approach consists in developing random models of the area of interest, characterized by a few parameters, hopefully simply measurable. One of such models relies on principles of percolation theory and assumes the urban area to be a regular grid of randomly occupied cells [l]. The propagating field is modelled as an ensemble of optical rays undergoing specular reflections in the environment, see left-hand side Fig 1. A related model of propagation considers point-like obstacles that re-radiate uniformly [a]. In this case the propagating wave is modeled as an ensemble of photons following piecewise linear, random trajectories, as depicted in the right-hand side of Fig. 1. The step lengths are assumed independent, identically distributed random variables, and the direction chosen at each step is assumed to be a uniform random variable. The two models represent two different physical scenarios. In the percolation model the obstacles essentially represent buildings and are assumed large enough that the laws of specular reflection can be applied. The random walk model applies to a smaller scale and can model microcellular environments where the obstacles (trees, vegetation, street signs, pedestrians, cars, building patches etc.) are assumed small compared to the inter-obstacles-hitting distance, and thus re-radiate uniformly. It also applies to a very sparse density of randomly oriented buildings.