[page 333, §1]

[333.1.1] Applications of fractional differentiation in physics [10] have
become widespread (see the reviews in [10, 12, 28, 29]
and references therein).
[333.1.2] Despite the wide dissemination of fractional models some basic questions
and fundamental problems persist [12].

[333.2.1] My objective in this note is to discuss the applicability of fractional Bochner-Levy diffusion and fractional Riesz potentials as a mathematical model for physical phenomena. [333.2.2] A lively scientific debate is presently concerned with boundary value problems for fractional Laplaceans and their relevance for experiment as witnessed by the special session devoted to nonlocal boundary value problems at the recent International Conference on Fractional Differentiation and Its Applications in Catania, June 23-25, 2014. [page 334, §0] [334.0.1] In view of this ongoing debate it seems timely to contribute to the discussion by exploring some thoughts of the present author [12] concerning implications of nonlocality for experiments, that were pointed out at the conference. [334.0.2] Originally, fractional derivatives and Riesz potentials were introduced as a convenient calculational tool (see e.g. [19, 22, 23]). [334.0.3] Recently, however, fractional differential equations have often been proposed as “generalizations” of more or less fundamental equations of physics and engineering. [334.0.4] Examples range from Schrödinger [17] and advection-dispersion equations [27, 3, 4] to viscoelasticity ([1]), suspension bridges ([24]) or loudspeaker coils ([25]). [334.0.5] Many generalizations remain formal in the sense that their rigorous relation to established theories is unknown and their limits of validity have not been worked out.

[334.1.1] Deep and fundamental principles of physics suggest that mathematical models of physical phenomena must always be formulated in terms of spatial derivatives of integer order (see [6]). [334.1.2] Electrodynamics, hydrodynamics, mechanics and quantum theory do not feature (spatial or temporal) fractional derivatives. [334.1.3] It is therefore not impossible that formal fractionalization of fundamental equations might produce interesting mathematical models that predict phenomena not observed in experiment.

[334.2.1] Given the large number of differential equations that have already been fractionalized this short note will restrict attention to a specific example. [334.2.2] Let me choose fractional Bochner-Levy diffusion as this specific example, because it has been a focus of attention both in mathematics and physics. [334.2.3] One can fractionalize the ordinary diffusion equation in several ways. [334.2.4] Replacing the Laplacean with a fractional derivative is one possibility. [334.2.5] It is called called Bochner-Levy or Riesz fractional diffusion [18, 22, 2]. [334.2.6] Another possibility, called Montroll-Weiss fractional diffusion, is to introduce a fractional time derivative [21, 26, 7, 13, 9, 11, 12]. [334.2.7] Mathematically it is equivalent to the the theory of continuous time random walks [20] as first observed in [8].