3 Discussion

[335.4.1] It is the primary objective of this note to contribute to the current debate by discussing fundamental differences between the cases α=2 and 0<α<2 in eq. (1) for applications to experiment. [335.4.2] The decisive difference between the cases α=2 and 0<α<2 is the locality of the Laplacean -Δ for α=2 in contrast with the nonlocality of the fractional Laplacean -Δα/2 for 0<α<2.

[page 336, §1]    [336.2.1] Before discussing the (non-)locality of -Δα/2 it seems important to distinguish it from another nonlocality appearing in eq. (1). [336.2.2] It is sometimes argued that also the case α=2 shows nonlocality in the sense that a localized initial condition such as u⁢x,0=h⁢x=δ⁢x-x0, vanishing everywhere except at x0 for t=0, spreads out instantaneously to all x such that u⁢x,t≠0 for all x for t>0. [336.2.3] This initially infinite “speed of propagation” violates relativistic locality. While this is true for all 0<α≤2, it concerns the operator ∂/∂⁡t+-Δα/2 and occurs only at t=0, the initial instant. [336.2.4] For α=2 the operator Δ is local and also ∂/∂⁡t+-Δ is perfectly local for all t>0. [336.2.5] While an infinite propagation speed occurs also for 0<α<2 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 α=2 and 0<α<2 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 u⁢x is given in terms of families of stochastic processes Xtt≥0 indexed by their starting point X0=x∈B⁢z,R through the formula

where Te(Rd∖B(z,R) denotes the first exit time of a path starting at X0=x∈B⁢z,R and hitting the set Rd∖B⁢z,R for the first time at t=Te(Rd∖B(z,R). [336.3.3] The brackets Yx denote the expectation value of a random variable Y evaluated for the process Xtt≥0 starting from x at t=0.

[336.4.1] For α=2 the family of stochastic processes has almost surely continuous paths. [336.4.2] Because of this, a path starting from x∈B⁢z,R at t=0 will exit from B⁢z,R when hitting ∂⁡B⁢z,R=x∈Rd:x-z=R for the first time.

[336.5.1] For 0<α<2 on the other hand the families of stochastic processes have almost surely discontinuous paths that can jump over the boundary ∂⁡B⁢z,R. [336.5.2] As a result the first exit occurs not at the boundary but at some point XTe⁢Rd∖B⁢z,R deep in the exterior region Rd∖B⁢z,R.

[336.6.1] In applications to particle diffusion the unknown u⁢x,t 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 α=2. [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 u⁢x,t 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.

[337.1.1] Whether or not a probabilistic interpretation applies, the discontinuity of the trajectories in the probabilistic interpretation leads to experimental difficulties. [337.1.2] This can be seen from considering the stationary states of (1),(2) and (4).

[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

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

for suitable boundary data g⁢x such that

holds.

[337.3.1] The solution of the fractional Riesz-Dirichlet problem for the case of a sphere B=B⁢z,R=x∈Rd:x-z<R of radius R centered at z∈Rd is the fractional Poisson integral [16]

for x∈B⁢z,R. [337.3.2] For α→2 the solution reduces to the conventional Poisson integral

for x∈B⁢z,R and u⁢x=g⁢x for x∈∂⁡B⁢z,R.

[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 g=0 for all x∈Rd∖B⁢z,R. [338.0.1] Physically this means that there are no atoms, molecules or tracer particles outside the spherical container B⁢z,R. [338.0.2] Any particle that jumps out of B⁢z,R 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

for all x∈Rd and all 0<α≤2.

[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 B⁢zi,Ri with i=1,…,n, B⁢zi,Ri∩B⁢zj,Rj=∅ for all i≠j and B⁢zi,Ri∩B⁢z,R=∅ for all i in which particles are kept (e.g. for replenishment). [338.1.3] This means that in these containers g⁢x≠0 and particles jumping out of the sample region B⁢z,R 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 ui∈R denote the maximal concentration in each container. [338.1.6] Assume that

describes the concentration field in the region R∖B⁢z,R outside the sample. [338.1.7] Other functions than ϕi⁢x with s⁢u⁢p⁢p⁢ϕi⊂B⁢zj,Rj are possible. [338.1.8] Assume also that Ri≪R for all i, so that in particular also suppg∩∂⁡B⁢z,R=∅ holds.

[338.2.1] For α=2 eq. (10) shows that the solution u⁢x=0 remains unaffected by the containers B⁢zi,Ri and their content. [338.2.2] For 0<α<2 on the other hand the solution changes and becomes nonzero. It is approximately

for x∈B⁢z,R. [338.2.3] This result implies that for 0<α<2 the stationary solution inside the sample region B⁢z,R depends on the location and content of all other containers B⁢zi,Ri in the laboratory. [338.2.4] The sample in B⁢z,R [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.