[335.4.1] It is the primary objective of this note to contribute to the current debate by discussing fundamental differences between the cases and in eq. (1) for applications to experiment. [335.4.2] The decisive difference between the cases and is the locality of the Laplacean for in contrast with the nonlocality of the fractional Laplacean for .

[page 336, §1] [336.2.1] Before discussing the (non-)locality of it seems important to distinguish it from another nonlocality appearing in eq. (1). [336.2.2] It is sometimes argued that also the case shows nonlocality in the sense that a localized initial condition such as , vanishing everywhere except at for , spreads out instantaneously to all such that for all for . [336.2.3] This initially infinite “speed of propagation” violates relativistic locality. While this is true for all , it concerns the operator and occurs only at , the initial instant. [336.2.4] For the operator is local and also is perfectly local for all . [336.2.5] While an infinite propagation speed occurs also for another violation of locality occurs in this case. [336.2.6] This has more dramatic implications for experiment, as will now be discussed.

[336.3.1] The fundamental difference between the cases and can be understood from the deep and well known relation between the diffusion equation (1) and the theory of stochastic processses. [336.3.2] The probabilistic interpretation of is given in terms of families of stochastic processes indexed by their starting point through the formula

(5) |

where denotes the first exit time of a path starting at and hitting the set for the first time at . [336.3.3] The brackets denote the expectation value of a random variable evaluated for the process starting from at .

[336.4.1] For the family of stochastic processes has almost surely continuous paths. [336.4.2] Because of this, a path starting from at will exit from when hitting for the first time.

[336.5.1] For on the other hand the families of stochastic processes have almost surely discontinuous paths that can jump over the boundary . [336.5.2] As a result the first exit occurs not at the boundary but at some point deep in the exterior region .

[336.6.1] In applications to particle diffusion the unknown is often the concentration of atomic, molecular or tracer particles and fractional generalizations of Ficks law have been postulated [4, 27, 3]. [336.6.2] Note, however, that the probabilistic interpretation is frequently not physical even for . [336.6.3] There are at least two possible reasons: [336.6.4] Firstly, the underlying physical dynamics may not be stochastic. [336.6.5] Secondly, fundamental laws of probability [page 337, §0] theory may be violated as for the case of heat diffusion where is the temperature field. [337.0.1] In such cases the random “paths” are fictitious as are the “particles” and their “trajectories” in the sense that they cannot be observed directly in an experiment.

[337.2.1] To explore the physical consequences of the initial and boundary value problem (1),(2) and (4) it is useful to start with stationary solutions, i.e. solutions of the form

(6) |

[337.2.2] The fractional diffusion equation then reduces to the fractional Riesz-Dirichlet problem

(7a) | |||

(7b) |

for suitable boundary data such that

(8) |

holds.

[337.3.1] The solution of the fractional Riesz-Dirichlet problem for the case of a sphere of radius centered at is the fractional Poisson integral [16]

(9) |

for . [337.3.2] For the solution reduces to the conventional Poisson integral

(10) |

for and for .

[337.4.1] Although the fractional Poisson formula eq. (9) has been known for nearly 70 years [22] its crucial difference to (10) seems to have escaped the attention of those scientists, who propose eq. (1) or its variants as a mathematical model for physical phenomena. [337.4.2] Perhaps this is due to [page 338, §0] the fact that many workers assume explicitly or implicitly “absorbing” or “killing” boundaries for all . [338.0.1] Physically this means that there are no atoms, molecules or tracer particles outside the spherical container . [338.0.2] Any particle that jumps out of is considered to be instantaneously removed from the experiment. [338.0.3] The environment surrounding the experimental apparatus has to be kept absolutely clean at all times for these boundary conditions to apply. [338.0.4] Under these experimental conditions both equations, eq. (9) as well as eq. (10), agree and both predict

(11) |

for all and all .

[338.1.1] Consider next the case when there exist regions where the atomic, molecular or tracer particles are not instantaneously removed. [338.1.2] For simplicity let there exist several small nonoverlapping spherical containers with , for all and for all in which particles are kept (e.g. for replenishment). [338.1.3] This means that in these containers and particles jumping out of the sample region may land in one of these containers. [338.1.4] They are not removed until the container is filled and a maximum concentration is reached. [338.1.5] Let denote the maximal concentration in each container. [338.1.6] Assume that

(12a) | ||

with | ||

(12b) |

describes the concentration field in the region outside the sample. [338.1.7] Other functions than with are possible. [338.1.8] Assume also that for all , so that in particular also supp holds.

[338.2.1] For eq. (10) shows that the solution remains unaffected by the containers and their content. [338.2.2] For on the other hand the solution changes and becomes nonzero. It is approximately

(13) |

for . [338.2.3] This result implies that for the stationary solution inside the sample region depends on the location and content of all other containers in the laboratory. [338.2.4] The sample in [page 339, §0] cannot be shielded or isolated from other samples in the laboratory. [339.0.1] It should be easy to verify or falsify this prediction in an experiment.