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 βnαt, 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.
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