Evolution equations for the probabilistic generalization of the Voigt profile function |
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Authors: | Gianni Pagnini Francesco Mainardi |
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Institution: | a ENEA, Centre “Ezio Clementel”, via Martiri di Monte Sole 4, I-40129 Bologna, Italy b Department of Physics, University of Bologna, and INFN, via Irnerio 46, I-40126 Bologna, Italy |
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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 Lévy stable distributions, a probabilistic generalization is proposed as the convolution of two arbitrary symmetric Lévy densities. We study the case when the widths of the distributions considered depend on a scale factor τ 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. |
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Keywords: | Voigt function Lé vy stable probability distributions Integro-differential equations Riesz space-fractional derivative |
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