Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process

• Main Author: Yuri Luchko
• Format: Article
• Language: English
• Published: MDPI AG 2016-02-01
• Series: Axioms
• Subjects: Caputo fractional derivative; Riesz fractional derivative; α-fractional diffusion equation; Mellin transform; fundamental solution; Mellin-Barnes integrals; entropy; entropy production rate
• Online Access: http://www.mdpi.com/2075-1680/5/1/6
• ISSN: 2075-1680

In this paper, the one-dimensional α-fractional diffusion equation is revisited. This equation is a particular case of the time- and space-fractional diffusion equation with the quotient of the orders of the time- and space-fractional derivatives equal to one-half. First, some integral representations of its fundamental solution including the Mellin-Barnes integral representation are derived. Then a series representation and asymptotics of the fundamental solution are discussed. The fundamental solution is interpreted as a probability density function and its entropy in the Shannon sense is calculated. The entropy production rate of the stochastic process governed by the α-fractional diffusion equation is shown to be equal to one of the conventional diffusion equation.