Our starting point is the

One of the most prominent and broadly-recognized applications of Fractional Calculus (FC) is for description of the anomalous transport processes [

In analogy to the case of the slow anomalous diffusion, some FC models for the supper-diffusion (fast diffusion) processes were introduced in form of the time-, space, or time-space-fractional PDEs. In particular, the time-fractional diffusion-wave PDE with the Caputo fractional derivative of order

However, the situation in the one-dimensional case (

In the literature, the entropy and the entropy production rates of the processes governed by some other particular cases of the time-space-fractional PDE with the Caputo time-fractional derivative of order

The rest of the paper is organized as follows. In

In this section, for the sake of the reader’s convenience we provide a sketch of derivation of the Mellin-Barnes integral representation of the fundamental solution to the time-space-fractional PDE with the Caputo time-fractional derivative and the fractional Laplacian (see Reference [

For a sufficiently well-behaved function

The normalization constant

It is worth mentioning that the fractional Laplacian

In what follows we consider an initial-value problem for the Equation (

If

Because the initial-value problem (

To derive a close form formula for the fundamental solution, we apply the Fourier transform (

In both cases, the unique solution to the initial-value problem for the fractional ODE (

Because of the asymptotic formula [

Because the function

In the case

Let us now consider the case

This representation will be used in the further discussions of the properties of

Now we employ the technique of the Mellin integral transform to deduce a Mellin-Barnes representation of the fundamental solution

If we denote by

As we can see, the integral at the right-hand side of the Equation (

By using the Mellin integral transform of the Mittag-Leffler function [

Then the Mellin convolution theorem (

Comparing the last formula with the expression (

It is worth mentioning that both the representation (

Another important point is the non-negativity property of the fundamental solution. 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 (pdf) evolving in time. As recently shown in [

For further properties and numerous particular cases of the fundamental solution

In this section, we treat the fundamental solution

For an

The entropy production rate is an important characteristic of a stochastic process that provides a measure of its irreversibility. For the conventional diffusion process (fundamental solution to the

The entropy production rate of the

The entropy production rate

Now let us proceed with calculation of the Shannon entropy of the fractional diffusion processes governed by the multi-dimensional time-space-fractional diffusion Equation (

Substituting (

In the last two integrals, we employ the variables substitution

The representation (

Surprisingly, the integral at the right-hand side of the Equation (

The second to last equality is a simple consequence of the formula for the residual of the Gamma-function at the point

Combining now the Equation (

Thus the entropy production rate of the

As in the case of the conventional diffusion, the entropy production rate of the

Another important finding follows from comparison of the entropy production rates of the

As one can see from the Equation (

In this paper, we derived and analyzed the fundamental solution to the

The entropy production rate is an increasing function in

Another important finding is that the entropy production rate of the

It would be important to study the problems discussed in this paper for other types of the fractional PDEs. In particular, the fractional diffusion equations with the Riesz–Feller fractional spatial derivative and/or the Riemann–Liouville time-fractional derivative would be worth investigating.

This research received no external funding.

The author declares no conflict of interest.