Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

Our starting point is the n-dimensional time-space-fractional partial differential equation (PDE) with the Caputo time-fractional derivative of order $\beta ,0<\beta <2$ and the fractional spatial derivative (fractional Laplacian) of order $\alpha ,0<\alpha \le 2$ . 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 $n\in \mathbb{N}$ if $0<\beta \le 1,0<\alpha \le 2$ and additionally for $1<\beta \le \alpha \le 2$ 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 $\beta$ and $\alpha$ of the time and spatial derivatives and on the space dimension n and is given by the expression $\frac{\beta n}{\alpha t}$ , t being the time variable. Even if it is an increasing function in $\beta$ , 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 $0<\beta \le 1$ , i.e., only the slow and the conventional diffusion can be described by this equation.