Chemistry Reference
In-Depth Information
dephasing results of Carr-Purcell [29], Torrey [28], and Stejskal-Tanner [36] for
normal diffusion, which are based on the diffusion limit of the discrete time ran-
dom walk proposed by Einstein [13, 15]. Furthermore, it is easily generalized to
include the inertia of the nuclei, that is where the underlying statistics are governed
by the Ornstein-Uhlenbeck process [22] and to other more complicated situations
where the nuclei move in a field of force of potential
V
(
r
). Therefore, we have a
microscopic explanation of the dephasing process in free water; namely, it results
from the nucleus behaving as a random walker executing a jump of finite mean-
square length at uniform time intervals of finite mean duration so that the
only
variable is the orientation of the walker
.
IV.
FRACTIONAL DIFFUSION: POSSIBLE EXPLANATIONS OF THE
STRETCHED EXPONENTIAL BEHAVIOR USING THE
FRACTIONAL LANGEVIN EQUATION
As far as one possible explanation of the stretched exponential model proposed
by Bennett et al. [40] [Eq.(81)] based on the microscopic origins of anomalous
diffusion is concerned, we note that the finite jump-length variance and finite
average jump-time, in the theory of the normal Brownian motion, define a physical
length scale and a physical time scale [16]. Here the ordinary theorem applies. We
have mentioned that in anomalous diffusion, however, either the second moment
of the jump-length distribution or the first moment of the jump time distribution
diverges or both of them. We have seen that such motions are characterized by
heavy tailed probability distributions (i.e., power law tails) so that the central limit
theorem no longer applies [24, 116]. They are known by the generic title of CTRW
[20, 23, 117]. Examples are the Levy stable motion for which the mean-square
displacement diverges due to the occurrence of very long jumps [117]. Such a
nonlocal walk in space leads to enhanced diffusion and ultimately turbulence, as
the overall displacement is dominated by the largest jumps without any time cost,
(i.e., jumps of arbitrary length all take the same time if the jump length is a Levy
process). On the other hand, processes that are nonlocal in time consequently
exhibit memory effects, [i.e., the so-called long rests or fractal-time random-walk
model (fractus
latin
- broken)]. Here, in contrast to the continuum limit of the
discrete time random walk considered by Einstein in which the time intervals
between jumps are uniform, the walker may remain in a given configuration for an
arbitrarily long period before undertaking a jump. The fractal time random walk
invariably leads to subdiffusion as the random walker always risks being trapped
in some site for an arbitrarily long time before advancing a distance equal to the
finite jump length variance. As usual
α
is the fractal dimension of the set of waiting
times [15, 16], which is the scaling of the waiting time segments in the random
walk with magnification.
