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Open AccessFeature PaperArticle

Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany
Entropy 2019, 21(10), 973; https://doi.org/10.3390/e21100973
Received: 18 September 2019 / Revised: 2 October 2019 / Accepted: 3 October 2019 / Published: 5 October 2019
(This article belongs to the Special Issue The Fractional View of Complexity)
Our starting point is the n-dimensional time-space-fractional partial differential equation (PDE) with the Caputo time-fractional derivative of order $β , 0 < β < 2$ and the fractional spatial derivative (fractional Laplacian) of order $α , 0 < α ≤ 2$ . For this equation, we first derive some integral representations of the fundamental solution and then discuss its important properties including scaling invariants and non-negativity. The time-space-fractional PDE governs a fractional diffusion process if and only if its fundamental solution is non-negative and can be interpreted as a spatial probability density function evolving in time. These conditions are satisfied for an arbitrary dimension $n ∈ N$ if $0 < β ≤ 1 , 0 < α ≤ 2$ and additionally for $1 < β ≤ α ≤ 2$ in the one-dimensional case. In all these cases, we derive the explicit formulas for the Shannon entropy and for the entropy production rate of a fractional diffusion process governed by the corresponding time-space-fractional PDE. The entropy production rate depends on the orders $β$ and $α$ of the time and spatial derivatives and on the space dimension n and is given by the expression $β n α t$ , t being the time variable. Even if it is an increasing function in $β$ , one cannot speak about any entropy production paradoxes related to these processes (as stated in some publications) because the time-space-fractional PDE governs a fractional diffusion process in all dimensions only under the condition $0 < β ≤ 1$ , i.e., only the slow and the conventional diffusion can be described by this equation.
MDPI and ACS Style

Luchko, Y. Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes. Entropy 2019, 21, 973.

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