An analytical representation for the simplest of the diffusions

TL;DR

This paper presents an analytical model for the spatial and temporal behavior of Brownian diffusion in a homogeneous, radially symmetric slab, emphasizing its practical relevance for experimental analysis and absorption effects.

Contribution

It introduces a new analytical representation for diffusion dynamics that accounts for absorption and can be applied to various initial conditions in experimental contexts.

Findings

Provides a quantitative description of absorption in diffusion profiles.

Enables analysis of time-resolved radial profiles for point-like excitation.

Facilitates unambiguous interpretation of Anderson localization experiments.

Abstract

An analytical representation for the spatial and temporal dynamics of the simplest of the diffusions -- Bronwian diffusion in an homogeneous slab geometry, with radial symmetry -- is presented. This representation is useful since it describes the time-resolved (as well as stationary) radial profiles, for point-like external excitation, which are more important in practical experimental situations than the case of plane-wave external excitation. The analytical representation can be used, under linear system response conditions, to obtain the full dynamics for any spatial and temporal profiles of initial perturbation of the system. Its main value is the quantitative accounting of absorption in the spatial distributions. This can contribute to obtain unambiguous conclusions in reports of Anderson localization of classical waves in three dimensions.