Chemistry Reference
In-Depth Information
It is clear that both jump times and lengths may exhibit
chaotic
behavior. Hence,
it is impossible to attribute underlying physical scales to such processes. For exam-
ple, the origin of the chaotic set of jump times may be due to a random distribution
of impurities within a matrix. Such a distribution will give rise to an extremely
broad range of microscopic transition rates. Hence, a
spatial
disorder correspond-
ing to a chaotic energy landscape (energetic disorder) will give rise to a
temporal
disorder because of the very large ranges of chaotic barrier heights involved, result-
ing in anomalous diffusion [15, 16]. Another microscopic explanation (the one that
will be used in this chapter as it lends itself to analysis using the Langevin equa-
tion) is that the anomalous behavior simply arises from the inclusion of memory
effects [15] in the normal Brownian motion, destroying its Markovian character
as highlighted by Mandelbrot and van Ness [3] (see [4] for a recent review).
The use of anomalous diffusion to describe relaxation behavior is very well es-
tablished in many fields of physics, including biophysics and physics in medicine
[73]. In the particular case of subdiffusive transport, for example, we mention
[73, 116, 118, 119] such diverse phenomena as charge-carrier transport in amor-
phous semiconductors, diffusion in percolative or porous systems, transport in
fractal geometries, as well as protein conformational dynamics. We have also
mentioned that in the NMR context the anomalous diffusion approach was used in
phenomenological fashion by Magin et al.[38], where the Bloch-Torrey equation
was converted to fractional form.
However, in using random-walk models in the context of microscopic expla-
nations, note that if diffusion in tissue is
restricted
or hindered, (i.e., it does
not take place in an infinite reservoir), then it may lead to very different signal
attenuations [30]. For example, Robertson [120] described a motional narrowing
long-time regime for diffusion between parallel planes when the signal decays
exponentially in time, unlike
t
3
as it is for unrestricted diffusion. Stejskal and
Tanner [121] showed that the signal has oscillatory behavior for narrow gradient
pulses and the related diffusion-diffraction patterns were observed by Callaghan
et al. [122]. The localization regime predicted by Stoller et al. [123] exhibits a
stretched-exponential behavior. In these and many other cases [30, 124], diffusion
may be considered as normal. It is a geometrical restriction alone that may lead to
deviations from the classical unrestricted diffusion.
Virtually all the microscopic approaches above, ultimately rely on the proba-
bility density function of the phase, a notable exception being that of Chen and
Widom [125] who used a frequency domain analysis based on the spectral func-
tion, characterizing fractal Brownian motion. However, since the random variable
underlying the process is the position of a nucleus, we have seen that a much
more transparent treatment of the phase diffusion could be achieved by means of
the Langevin equation. We have also seen that for normal diffusion, this is sim-
ply the Newtonian equation of motion of the nucleus, augmented by a systematic
frictional force proportional to the velocity, on which a very rapidly fluctuating
