Entropy, Vol. 21, Pages 973: Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes (Entropy)

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.