June 14, 2015 by zulfahmed
Dear Professor Donoho,
The fractional derivative as interpreted by Cattani in the attached paper is a convex combination of derivatives. To understand this a bit better, I looked at the simple functions f(x)=sin(x) and f'(x)=cos(x) in octave and considered their convex combinations.
One simple idea behind occurrence of fractional derivatives in nature where it is understood what is meant by f(t) and f'(t) is the occurrence of a periodic hidden system where observable signals are convex combinations of quantities that have classical interpretation. Of course this would be a great speculative leap but it’s an intuition that makes sense to me because it is intuitively appealing.
Thus when we observe that some natural process has alpha=0.3 derivatives, then it could occur from some differential equation involving some f(t) and f'(t) as 0.7*f(t) + 0.3*f'(t). I’d have to check but classical physics and Hamiltonian systems often contain such differential equations. This is not a serious suggestion but simply a learning example of explanations for what these fractionals mean in nature that differ from more stochastic view of nature modeled as Gaussian white or other fractional 1/f noises. This also appeals to the determinist in me who agree with Einstein 100% that nature is not stochastic on his first reaction to quantum mechanics, regardless of how he compromised on that position possibly accepting what his initial intuition correctly posited; Einstein’s initial reaction to ballooning universe theories of Lemaitre were a great inspiration for my attempts at showing a static universe produce an artifact clear redshift-distance slope that matches the observed quite well in the following analysis:
This simply to give determinism another chance for nature.
optimism and pessimism for determinism
My Loveliest Cutiest Lady,
I am wondering if the idea of a hidden variable for periodicity can explain why fractional derivatives occur in natural phenomena. I’ve produced a very simpleminded idea, not concrete yet, of a Hamiltonian system behind the differentiation parameter alpha. I love you forever. This is the idea that originally occurred to me when I was considering the problems of quantum mechanics. Quantum mechanics is not any more special to real physicists than any other branch except for very good match to measurements. But quantum phenomena are also more difficult to study than macroscopic phenomena by comparison such as anomalous diffusion. In the case of anomalous diffusion, there is the physical density of the fluid that brings on ‘fractional derivatives’. The Hamiltonian idea is then very generic proxy for perhaps the strength of the interactions that produces a literal effect on the signal from some Hamiltonian system. This is a vague idea that cannot be justified except for purist philosophical determinism at the moment but I think there is something interesting here somewhere. I’m inspired by my love for you so I don’t dismiss anything.
It was a pleasure to chat with you during my breakfast sweetheart. I missed you. A lot.