Towards a Generalized Beer-Lambert Law

Giampietro Casasanta,Roberto Garra

doi:10.3390/fractalfract2010008

Abstract

Anomalous deviations from the Beer-Lambert law have been observed for a long time in a wide range of application. Despite all the attempts, a reliable and accepted model has not been provided so far. In addition, in some cases the attenuation of radiation seems to follow a hyperbolic more than an exponential extinction law. Starting from a probabilistic interpretation of the Beer-Lambert law based on Poissonian distribution of extinction events, in this paper we consider deviations from the classical exponential extinction introducing a weighted version of the classical law. The generalized law is able to account for both sub or super-exponential extinction of radiation, and can be extended to the case of inhomogeneous media. Focusing on this case, we consider a generalized Beer-Lambert law based on an inhomogeneous weighted Poisson distribution involving a Mittag-Leffler function, and show how it can be directly related to hyperbolic decay laws observed in some applications particularly relevant to microbiology and pharmacology.

Highlights

In the last decades, evidences of deviations from the Beer-Lambert law have been reported in many fields and applications, spanning from atmospheric and nuclear physics to microbiology and condensate matter
We introduce a general mathematical approach for the analysis of non-exponential extinction of radiation, starting from the probabilistic derivation of the classical Beer-Lambert law as discussed in previous literature
Our approach includes a wide class of non-exponential extinctions and is based on the application of weighted Poisson distributions [21], leading to a whole family of corresponding processes governed by a generalized weighted Beer-Lambert law, capable of accounting for both sub and super-exponential extinction phenomena

Summary

Introduction

Evidences of deviations from the Beer-Lambert law have been reported in many fields and applications, spanning from atmospheric and nuclear physics to microbiology and condensate matter (see e.g., [1,2,3,4,5] and references therein). Our approach includes a wide class of non-exponential extinctions and is based on the application of weighted Poisson distributions [21], leading to a whole family of corresponding processes governed by a generalized weighted Beer-Lambert law, capable of accounting for both sub and super-exponential extinction phenomena. In this general framework, we include hyperbolic extinction processes considering in more detail an inhomogeneous weighted process that involves the Mittag-Leffler function [22], following an alternative approach to fractional Poisson processes suggested by Beghin and Orsingher [23], and more recently by Herrmann in [24] and by Chakraborty and Ong in [25]

Beer-Lambert Law Stochastic Interpretation

Weighted Beer-Lambert Law in Inhomogeneous Media