Are fractional derivatives an empirical fact of transport in complex media or is there a theoretical understanding behind it?
Why do fractional derivatives describe sub diffusive processes? Why do they describe flow in porous media?
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$\begingroup$ -1. What research have you done? Have you googled your title? Why do the results fail to answer your question? $\endgroup$– sammy gerbilCommented Sep 19, 2017 at 11:40
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$\begingroup$ "What research have you done?" I many references that point out that fractional derivatives describe flows in materials with fractal structures. I have googled (gscholar actually) my question so in fact I do know that FD are used for sub-D processes specifically when the assumption of diffusion described as a Markov Chain break down. But why? Is there anything non-empirical about these results? Is there a physical interpretation of FD that doesn't involved fractional dimensions? $\endgroup$– user16035Commented Sep 19, 2017 at 14:53
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Great question, this one made me learn as well.
Are fractional derivatives an empirical fact of transport in complex media or is there a theoretical understanding behind it?
There is a theoretical motivation behind using fractional calculus in the description of fluid dynamics. This goes back to Kolmogorov's initial realization in the early 1940s, you can read more about this here and another similar Physics SE is here. To briefly summarize, Kolmogorov's hypothesis about the energy spectra of turbulent fluids, also known as the Selection Rules, considered
$$E = Ck^\alpha\psi^\beta.$$
where $k$ is the wave number and $\psi$ is the energy per unit volume. Using known dimensions for each of these Kolmogorov found $\alpha=-5/3$ and $\beta=2/3$. From here, fluid dynamics started using fractional calculus after connecting the energy spectrum, given above, to length scale of a vortex.
So yes, there was a theoretical motivation behind it. In fact, empirically the hypothesis does not reproduce the behavior of fluids because Kolmogorov's arguments relied on using mean field theory while in fluid dynamics, mean field theory should not be used as it does not reproduce fluctuations of the energy dissipation rate. However, a mean field theory is still used today as it works well in capturing the small scale behaviors.
Why do fractional derivatives describe sub diffusive processes?
"Why" is a tough thing to answer without becoming philosophical, but I can say that quite simply, they are naturally equipped to handle these processes. Okay, so what does that mean? Well, integer derivatives require local information around a point as a function $f(x)$ is said to be differentiable at $x$ if it is continuous and $x$'s limit point $p$ is contained within an open neighborhood of the mapping of $x$, so $d(f(x),f(p))<\epsilon$ for some $\epsilon>0$ and $d(\cdot,\cdot)$ is some distance function on the metric space which $f(x)$ maps into.
Fractional derivatives, on the other hand, do not have the same machinery as integer derivatives as the notion of distance is harder to pin down. So you need some sort of boundary conditions in order to determine if a fractional derivative of a function exists.
Demanding these boundary conditions for fractional derivatives of functions to exist provides a natural way to tie in diffusive processes.
Why do they describe flow in porous media?
There might be a slight confusion here. The literature often refers to fractal derivatives as fractional derivatives, but they are different. Fractal derivatives are useful for porous media as the "porousness" is a type of fractal geometry. On these geometries, you can define a space-time and then a derivative based on the fractional dimension. Fluids are not fractal, but we've already discussed why motivation for fluid dynamicists use fractional derivatives there.
Hope this answered your question!
