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Figure 2:

Let

Then, convoluting with Gaussian,

and

Figure 3:

is shown for a variety of
values while
keeping
constant in Fig.2, and similarly *G*_{e}(*t*) in
Fig.3. As
,
becomes trivial, so all the distributions and the relaxation function tend
to Gaussian in that limit. As
for
finite nonzero
,
the distributions and the relaxation
function evolve in a Lorentzian-like direction, but even in the limit, they
do not quite get there. The depth of the minimum of the limit relaxation
function is significantly deeper than the minimum of the Lorentzian
Kubo-Toyabe. At the limit,
,
the "Lorentzian
pathology'' does develop: the second moments of the distributions diverge,
and the relaxation function develops a slope at *t*=0. Note that taking
for any fixed value of
makes
the distributions' norms diverge, and drives
.

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*Jess H. Brewer *

2002-09-24