Sie sind hier: ICP » R. Hilfer » Publikationen

3 Discussion

[3.1.1.1] In Figure 1 we display the function \psi(t;\omega,C) for C=1 and \omega=0.01,0.1,0.5,0.9,0.99 in a log-log plot. [3.1.1.2] The asymptotic behaviour (2.18) and (2.20) is clearly visible from the figure. [3.1.1.3] The fractional order \omega of the time derivative in (1.1) is restricted to 0<\omega\leq 1 as a result of the general theory [3]. [3.1.1.4] This and the behaviour of \psi(t) in figure 1 attributes special significance to the two limits \omega\rightarrow 1 and \omega\rightarrow 0.

[3.1.2.1] In the limit \omega\rightarrow 1 the fractional master equation (1.2) reduces to the ordinary master equation, and the waiting time density becomes exponential\psi(t;1,1)=\exp(-t). [3.2.0.1] In the limit \rightarrow 0 on the other hand equation (1.1) reduces to an eigenvalue or fixed point equation for the operator on the right hand side by virtue of \partial^{0}f/\partial t^{0}=f.

[3.2.1.1] While this is interesting in itself an even more interesting aspect is that the correspondingwaiting time density \psi(t) approaches the form\psi(t;\omega\rightarrow 0,1)\propto 1/t for which the normalization becomes logarithmically divergent. [3.2.1.2] This signals an onset of localization in this singular limit. [3.2.1.3] It is hoped that our results will stimulate further research into the fractal time concept.

Figure 1: Log-log plot of the waiting time density \psi(t;\omega,1) for \omega=0.01,0.1,0.5,0.9,0.99,1.0. The curves for \omega=1.0 and \omega=0.01 have been labeled in the figure, the other curves interpolate between them. For \omega=1 the waiting time density is exponential \psi(t)=\exp(-t) and for \omega\rightarrow 0 it approaches \psi(t)\rightarrow 1/t.