Non-exponential extinction of radiation by fractional calculus modelling

Abstract

Possible deviations from exponential attenuation of radiation in a random medium have been recently studied in several works. These deviations from the classical Beer–Lambert law were justified from a stochastic point of view by Kostinski (2001) [1]. In his model he introduced the spatial correlation among the random variables, i.e. a space memory. In this note we introduce a different approach, including a memory formalism in the classical Beer–Lambert law through fractional calculus modelling. We find a generalized Beer–Lambert law in which the exponential memoryless extinction is only a special case of non-exponential extinction solutions described by Mittag–Leffler functions. We also justify this result from a stochastic point of view, using the space fractional Poisson process. Moreover, we discuss some concrete advantages of this approach from an experimental point of view, giving an estimate of the deviation from exponential extinction law, varying the optical depth. This is also an interesting model to understand the meaning of fractional derivative as an instrument to transmit randomness of microscopic dynamics to the macroscopic scale.