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 ≤ 1, i.e., only the slow and the conventional diffusion can be described by this equation.