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3 Discussion

3.1 (Non-)locality and (in-)finite propagation speed

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

[page 336, §1]    [336.2.1] Before discussing the (non-)locality of (-\Delta)^{{\alpha/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 \alpha=2 shows nonlocality in the sense that a localized initial condition such as u(x,0)=h(x)=\delta(x-x_{0}), vanishing everywhere except at x_{0} for t=0, spreads out instantaneously to all x such that u(x,t)\neq 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<\alpha\leq 2, it concerns the operator \partial/\partial t+(-\Delta)^{{\alpha/2}} and occurs only at t=0, the initial instant. [336.2.4] For \alpha=2 the operator \Delta is local and also \partial/\partial t+(-\Delta) is perfectly local for all t>0. [336.2.5] While an infinite propagation speed occurs also for 0<\alpha<2 another violation of locality occurs in this case. [336.2.6] This has more dramatic implications for experiment, as will now be discussed.

3.2 Probabilistic interpretation

[336.3.1] The fundamental difference between the cases \alpha=2 and 0<\alpha<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 (X_{t})_{{t\geq 0}} indexed by their starting point X_{0}=x\in\mathbb{B}(z,R) through the formula

u(x)=\langle u(X_{{T^{e}(\mathbb{R}^{d}\setminus\mathbb{B}(z,R)}})\rangle _{x}, (5)

where T^{e}(\mathbb{R}^{d}\setminus\mathbb{B}(z,R) denotes the first exit time of a path starting at X_{0}=x\in\mathbb{B}(z,R) and hitting the set \mathbb{R}^{d}\setminus\mathbb{B}(z,R) for the first time at t=T^{e}(\mathbb{R}^{d}\setminus\mathbb{B}(z,R). [336.3.3] The brackets \langle Y\rangle _{x} denote the expectation value of a random variable Y evaluated for the process (X_{t})_{{t\geq 0}} starting from x at t=0.

[336.4.1] For \alpha=2 the family of stochastic processes has almost surely continuous paths. [336.4.2] Because of this, a path starting from x\in\mathbb{B}(z,R) at t=0 will exit from \mathbb{B}(z,R) when hitting \partial\mathbb{B}(z,R)=\{ x\in\mathbb{R}^{d}:|x-z|=R\} for the first time.

[336.5.1] For 0<\alpha<2 on the other hand the families of stochastic processes have almost surely discontinuous paths that can jump over the boundary \partial\mathbb{B}(z,R). [336.5.2] As a result the first exit occurs not at the boundary but at some point X_{{T^{e}(\mathbb{R}^{d}\setminus\mathbb{B}(z,R))}} deep in the exterior region \mathbb{R}^{d}\setminus\mathbb{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 \alpha=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).

3.3 Stationary solutions

[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

u(x,t)=u(x). (6)

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

\displaystyle(-\Delta)^{{\alpha/2}}u(x)=0,\qquad x\in\mathbb{B} (7a)
\displaystyle u(x)=g(x),\qquad x\in\mathbb{R}^{d}\setminus\mathbb{B} (7b)

for suitable boundary data g(x) such that

\int\limits _{{\mathbb{R}^{d}\setminus\mathbb{B}}}\frac{|g(x)|}{1+|x|^{{d+\alpha}}}\;\mathrm{d}^{d}x<\infty (8)


[337.3.1] The solution of the fractional Riesz-Dirichlet problem for the case of a sphere \mathbb{B}=\mathbb{B}(z,R)=\{ x\in\mathbb{R}^{d}:|x-z|<R\} of radius R centered at z\in\mathbb{R}^{d} is the fractional Poisson integral [16]

u(x)=\frac{\Gamma\left(\frac{d}{2}\right)\sin\left(\frac{\pi\alpha}{2}\right)}{\pi^{{\frac{d}{2}+1}}}\int\limits _{{\mathbb{R}^{d}\setminus\mathbb{B}(z,R)}}\frac{\left|R^{2}-|x-z|^{2}\right|^{{\frac{\alpha}{2}}}}{\left|R^{2}-|y-z|^{2}\right|^{{\frac{\alpha}{2}}}\;|x-y|^{d}}g(y)\mathrm{d}^{d}y (9)

for x\in\mathbb{B}(z,R). [337.3.2] For \alpha\to 2 the solution reduces to the conventional Poisson integral

u(x)=\frac{\Gamma(d/2)}{2R\pi^{{d/2}}}\int\limits _{{\partial\mathbb{B}(z,R)}}\frac{R^{2}-|x-z|^{2}}{|x-y|^{d}}g(y)\mathrm{d}^{{d-1}}y (10)

for x\in\mathbb{B}(z,R) and u(x)=g(x) for x\in\partial\mathbb{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\in\mathbb{R}^{d}\setminus\mathbb{B}(z,R). [338.0.1] Physically this means that there are no atoms, molecules or tracer particles outside the spherical container \mathbb{B}(z,R). [338.0.2] Any particle that jumps out of \mathbb{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

u(x)=0 (11)

for all x\in\mathbb{R}^{d} and all 0<\alpha\leq 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 \mathbb{B}(z_{i},R_{i}) with i=1,...,n, \mathbb{B}(z_{i},R_{i})\cap\mathbb{B}(z_{j},R_{j})=\emptyset for all i\neq j and \mathbb{B}(z_{i},R_{i})\cap\mathbb{B}(z,R)=\emptyset for all i in which particles are kept (e.g. for replenishment). [338.1.3] This means that in these containers g(x)\neq 0 and particles jumping out of the sample region \mathbb{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 u_{i}\in\mathbb{R} denote the maximal concentration in each container. [338.1.6] Assume that

g(x)=\sum _{{i=1}}^{n}R_{i}^{{-d}}\phi _{i}\left(\frac{x-z_{i}}{R_{i}}\right) (12a)
\phi _{i}(x)=\begin{cases}u_{i}\exp\left(-\frac{1}{1-|x|^{2}}\right)&\mathrm{for~}x\in\mathbb{B}(0,1)\\
0&\mathrm{otherwise}\end{cases} (12b)

describes the concentration field in the region \mathbb{R}\setminus\mathbb{B}(z,R) outside the sample. [338.1.7] Other functions than \phi _{i}(x) with supp\phi _{i}\subset\mathbb{B}(z_{j},R_{j}) are possible. [338.1.8] Assume also that R_{i}\ll R for all i, so that in particular also suppg\cap\partial\mathbb{B}(z,R)=\emptyset holds.

[338.2.1] For \alpha=2 eq. (10) shows that the solution u(x)=0 remains unaffected by the containers \mathbb{B}(z_{i},R_{i}) and their content. [338.2.2] For 0<\alpha<2 on the other hand the solution changes and becomes nonzero. It is approximately

u(x)\approx\frac{\Gamma\left(\frac{d}{2}\right)\sin\left(\frac{\pi\alpha}{2}\right)}{\pi^{{\frac{d}{2}+1}}}\sum _{{i=1}}^{n}\frac{u_{i}\left|R^{2}-|x-z|^{2}\right|^{{\frac{\alpha}{2}}}}{\left|R^{2}-|z_{i}-z|^{2}\right|^{{\frac{\alpha}{2}}}\;|x-z_{i}|^{d}}\neq 0 (13)

for x\in\mathbb{B}(z,R). [338.2.3] This result implies that for 0<\alpha<2 the stationary solution inside the sample region \mathbb{B}(z,R) depends on the location and content of all other containers \mathbb{B}(z_{i},R_{i}) in the laboratory. [338.2.4] The sample in \mathbb{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.