Hi Miguel. In the anomalous diffusion case, the physical interpretation is clear. You have a diffusing particle which stays in a site for a random time interval, and, then, it instantaneously jumps elsewhere. The Caputo index alpha is nothing but the exponent of the distribution of these intervals. Even if the random intervals are independent and identically distributed random variables, the number of jumps up to a time instant t has infinite memory. In this framework, we can see that the Caputo derivative appears as a consequence of the generalized central limit theorem.

Now, let us forget the diffusive interpretation (IMHO the only case where the physical interpretation of fractional calculus is clear and nicely connected to probability theory). If you consider the definition of the Caputo derivative, you see that it is a convolution operator (see eq. 1.17 in this paper by Mainardi and Gorenflo: http://arxiv.org/pdf/0805.3823.pdf). The convolution kernel is a power-law with exponent -alpha. Therefore, if a system is well described by a dynamics driven by a Caputo derivative and something happens at alpha 0.78, and in the absence of other info, I am afraid one can just say that this is the exponent of the memory kernel. With more pieces of info, one could look at time subordination (or time transformation). Igor Podlubny has a paper on the geometrical and physical meaning of several fractional operators: http://arxiv.org/pdf/math/0110241.pdf pointing to several other pieces of literature. At this URL, you can find considerations made by Mainardi and Gorenflo on the physical meaning of fractional derivatives:

http://www.fracalmo.org/key-notes/index.htm

I hope this helps!

Dear Miguel,

to which of the many fractional derivatives are you referring?

If we consider anomalous diffusion and to the Caputo and Riesz-Feller derivatives, this paper of mine is a partial answer to your question:

http://pre.aps.org/abstract/PRE/v69/i1/e011107

http://arxiv.org/abs/cond-mat/0402657

In summary, the fractional index in the time derivative is related to memory whereas the fractional index in the space derivative is related to (extreme) non-locality in space.

Regards

Enrico

Hi Enrico,

I am mainly thinking in nonlinear oscillators, think for instance about the pendulum or the Duffing oscillator. The Caputo derivative is widely used, though as far as I know there are other definitions which are used for numerical purposes. From this point of view, when the exponent or parameter alpha=1, we have the ordinary situation with ordinary differential equations. In principle 0 < alpha < 1 for the fractional derivative.

I had heard previously about both interpretations that you give me: memory and non-locality. Nevertheless, they sound to me rather abstract or imprecise. Or at least I do not have a clear picture of its meaning. For instance: If I discover that certain new phenomenon arises at the transition of alpha at say 0.78. What can I conclude ? Yes, memory and non-locality, but what different memory and non-locality exist at say alpha=0.81 ?

Thanks !

Miguel

Hi Miguel. In the anomalous diffusion case, the physical interpretation is clear. You have a diffusing particle which stays in a site for a random time interval, and, then, it instantaneously jumps elsewhere. The Caputo index alpha is nothing but the exponent of the distribution of these intervals. Even if the random intervals are independent and identically distributed random variables, the number of jumps up to a time instant t has infinite memory. In this framework, we can see that the Caputo derivative appears as a consequence of the generalized central limit theorem.

Now, let us forget the diffusive interpretation (IMHO the only case where the physical interpretation of fractional calculus is clear and nicely connected to probability theory). If you consider the definition of the Caputo derivative, you see that it is a convolution operator (see eq. 1.17 in this paper by Mainardi and Gorenflo: http://arxiv.org/pdf/0805.3823.pdf). The convolution kernel is a power-law with exponent -alpha. Therefore, if a system is well described by a dynamics driven by a Caputo derivative and something happens at alpha 0.78, and in the absence of other info, I am afraid one can just say that this is the exponent of the memory kernel. With more pieces of info, one could look at time subordination (or time transformation). Igor Podlubny has a paper on the geometrical and physical meaning of several fractional operators: http://arxiv.org/pdf/math/0110241.pdf pointing to several other pieces of literature. At this URL, you can find considerations made by Mainardi and Gorenflo on the physical meaning of fractional derivatives:

http://www.fracalmo.org/key-notes/index.htm

I hope this helps!

Thank you for the references. They seem to be very useful for what I need.