Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

Yuri Luchko

doi:10.3390/e21100973

Abstract

Our starting point is the n-dimensional time-space-fractional partial differential equation (PDE) with the Caputo time-fractional derivative of order and the fractional spatial derivative (fractional Laplacian) of order . 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 if and additionally for 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 and of the time and spatial derivatives and on the space dimension n and is given by the expression , t being the time variable. Even if it is an increasing function in , 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 , i.e., only the slow and the conventional diffusion can be described by this equation.

Highlights

One of the most prominent and broadly-recognized applications of Fractional Calculus (FC)is for description of the anomalous transport processes [1,2,3,4]
In Reference [5], the fundamental solution to the time-fractional diffusion equation with the Caputo fractional derivative of order β ∈ (0, 1) and the spatial Laplace operator was shown to be non-negative and normalized. It can be interpreted as a spatial probability density function evolving in time that provides a strong justification for the time-fractional diffusion equation with the time derivative of order β ∈ (0, 1) to act as a model for a diffusion process
As one can see from the Equation (37), the entropy production rates and other properties β of the fractional diffusion processes depend on the quotient α of the orders of the time- and space-fractional derivatives

Summary

Introduction

One of the most prominent and broadly-recognized applications of Fractional Calculus (FC). The entropy and the entropy production rates of the processes governed by the one-dimensional time-fractional diffusion equation with the Caputo time-fractional derivative of order β ∈ [1, 2] and the second spatial derivative (Laplace operator in the one-dimensional case) were discussed in [10,11].

Fundamental Solution to the Time-Space-Fractional PDE

The Entropy Production Rates of the Fractional Diffusion Processes

Conclusions