Evolution equations of the probabilistic generalization of the Voigt profile function

TL;DR

This paper introduces evolution equations for a probabilistic generalization of the Voigt profile, modeled as a space-fractional diffusion process involving symmetric Lévy stable distributions, to account for inhomogeneity and non-stationarity.

Contribution

It proposes a novel probabilistic generalization of the Voigt profile using symmetric Lévy densities and derives corresponding space-fractional diffusion equations with two Riesz derivatives.

Findings

Derived evolution equations for the generalized Voigt profile.

Interpreted these equations as space-fractional diffusion equations.

Provided a framework for modeling inhomogeneous and non-stationary spectral profiles.

Abstract

The spectrum profile that emerges in molecular spectroscopy and atmospheric radiative transfer as the combined effect of Doppler and pressure broadenings is known as the Voigt profile function. Because of its convolution integral representation, the Voigt profile can be interpreted as the probability density function of the sum of two independent random variables with Gaussian density (due to the Doppler effect) and Lorentzian density (due to the pressure effect). Since these densities belong to the class of symmetric L\'evy stable distributions, a probabilistic generalization is proposed as the convolution of two arbitrary symmetric L\'evy densities. We study the case when the widths of the considered distributions depend on a scale-factor $\tau$ that is representative of spatial inhomogeneity or temporal non-stationarity. The evolution equations for this probabilistic generalization of the Voigt function are here introduced and interpreted as generalized diffusion equations containing two Riesz space-fractional derivatives, thus classified as space-fractional diffusion equations of double order.