Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

Abstract

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.

The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging

“…Furthermore, the entropy production rate of the quasidiffusion propagator in the one-dimensional and two-dimensional cases is the same as for Gaussian diffusion process [48][49][50]. However, in the one-dimensional case, the second spatial moment of the probability density function, P 1 (x, t), does not exist and the mean squared displacement of the diffusing particles is not finite, indicating the diffusion process is anomalous [48][49][50]. Similarly, for the two-dimensional case, the second moment of P 2 (x, t), does not exist for 0.5 < α ≤ 1 and the diffusion is anomalous with long-tailed waiting time and jump lengths.…”

“…Several interesting mathematical results have been derived from the Green's function in the special case of the CTRW model for 2α/β = 1. It has been shown that the oneand two-dimensional quasi-diffusion propagators are probability density functions that evolve spatially in time [48][49][50]. Furthermore, the entropy production rate of the quasidiffusion propagator in the one-dimensional and two-dimensional cases is the same as for Gaussian diffusion process [48][49][50].…”

“…It has been shown that the oneand two-dimensional quasi-diffusion propagators are probability density functions that evolve spatially in time [48][49][50]. Furthermore, the entropy production rate of the quasidiffusion propagator in the one-dimensional and two-dimensional cases is the same as for Gaussian diffusion process [48][49][50]. However, in the one-dimensional case, the second spatial moment of the probability density function, P 1 (x, t), does not exist and the mean squared displacement of the diffusing particles is not finite, indicating the diffusion process is anomalous [48][49][50].…”

“…Here we consider the mathematics of quasi-diffusion and quasi-diffusion imaging. The fundamental solution of the space-time diffusion-wave equation within the CTRW diffusion model has been extensively studied [43][44][45][46][47], as have the properties of the quasidiffusion model [47][48][49][50]. In particular, quasi-diffusion is referred to as α-fractional diffusion by [48,49] who suggest this special case of the CTRW diffusion model represents a natural fractionalisation of the diffusion process [49].…”

The Fractional View of Complexity

“…The manuscript “Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes”, by Yuri Luchko, derives and analyzes the fundamental solution to the n -dimensional time-space-fractional partial differential equation with the Caputo time-fractional derivative of order b , , and the fractional spatial derivative (fractional Laplacian) of order a , . An explicit formula for the entropy production rate of the fractional diffusion processes is presented [ 10 ].…”