Symmetric Fractional Diffusion and Entropy Production

Abstract

Abstract:The discovery of the entropy production paradox (Hoffmann et al., 1998) raised basic questions about the nature of irreversibility in the regime between diffusion and waves. First studied in the form of spatial movements of moments of H functions, pseudo propagation is the pre-limit propagation-like movements of skewed probability density function (PDFs) in the domain between the wave and diffusion equations that goes over to classical partial differential equation propagation of characteristics in the wave limit. Many of the strange properties that occur in this extraordinary regime were thought to be connected in some manner to this form of proto-movement. This paper eliminates pseudo propagation by employing a similar evolution equation that imposes spatial unimodal symmetry on evolving PDFs. Contrary to initial expectations, familiar peculiarities emerge despite the imposed symmetry, but they have a distinct character.

Between Waves and Diffusion: Paradoxical Entropy Production in an Exceptional Regime

“…Furthermore, we will also sketch out how this scaling symmetry plays out in a formally different extraordinary differential equation with space-fractional derivative, with completely different solutions and domains, leading to the same remarkable outcome [6,11]: the EPR increases in the reversible limit. We will also observe in passing that this paradox persists even for generalizations of entropy available in the literature such as Tsallis and Renyi entropies [4,6,[12][13][14]. In this context, we want to note that there might be implications of the entropy production paradox on applications known in finance [15,16], ecology [17], computational neuroscience [18], and physics [19][20][21].…”

“…as used in [6,[11][12][13], but other definitions are also possible [14]. While the space-fractional derivative is defined through transforms for historical reasons, this does not mean that the time-fractional derivative could not be defined alternatively through a suitable transform.…”

Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

“…Because the function E β −|κ| α t β is a radial function, we can rewrite the Equation (14) in the form [21]…”

“…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 β, 0 < β < 2 and the fractional spatial derivative (fractional Laplacian) of order α, 0 < α ≤ 2 have been considered. In References [13,14], the case of the one-dimensional space-fractional PDE with the first-order time derivative and fractional spatial derivative of order α, α ∈ (0, 2] was analyzed in detail. In Reference [15], a closed form formula for the Shannon entropy of the fundamental solution to the one-dimensional neutral-fractional PDE with the fractional derivatives of the same order α, 1 ≤ α ≤ 2 both in space and in time, was derived.…”

Thermodynamics of Superdiffusion Generated by Lévy–Wiener Fluctuating Forces

“…In the case of the free Lévy flights described by the stability index α, the above rate formula results in d dt S[p(x, t)] ∝ (αt) −1 , i.e., the entropy production rate is positive, decreases for increasing α, and attains a minimal value for α = 2, corresponding to generic Gaussian fluctuations, typical for states close to equilibrium [63,64]. Similarly, for Lévy flights in the quadratic potential, the evaluation of Equation (21) yields d dt S[p(x, t)] = 1 e αt −1 which coincides with the rate of a free Lévy process for short times (t 1).…”