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In this section we present the results of our simulations described above.
Fig. 2 shows the scaling of the pair correlation function
obtained from averaging over 212 stable flux line configurations on
a pinscape generated with exponentially distributed pinning strengths
according to
Eq. (11), with r_{p} = 0.015.
The inset to Fig. 2
shows the scaling behavior of
with
r. The solid line has slope .
Note the good agreement with
the liquid theory prediction, Eq. (8), for values of r in the
initial liftoff of g(r). The deviations for larger r arise from
excess correlations (pileup) near the peak of g(r) not
accounted for in the naïve scaling Ansatz. Such excess correlations
also are seen experimentally [3,8].
Having thus observed numerically a liquidlike relation of the form
Eq. (8),
we next ask how the behavior of the
pair correlation function depends on
the range of the pinning potential
r_{p}.
Fig. 3 shows the scaling
of
with
r for different values of r_{p}, ranging
from m to m.
Figure:
Dependence of the pair correlation
function on the range of the
pinning potential r_{p} for exponentially
distributed pinning strengths.
Solid lines correspond to a liquid theory
scaling, Eq. (8),
with
m. The values for r_{p}
are m (a), m (b), m (c),
and m (d). Averages were calculated
from 596, 600, 323 and 212 configurations,
respectively.

The asymptotic form, Eq. (8),
is realized only in the limit of a
shortranged pinning potential (
).
The dependence of the pair correlation
function on the breadth of the pinning
strength distribution is shown in Fig. 4. Here we compare the
results for configurations on a pinscape
of spatially narrow pins whose strengths
were drawn from increasingly broad
distributions: delta function
[Eq. (10)], ,
1 and 1/2 [Eq. (11)].
Figure 4:
Dependence of the pair correlation
function on the distribution
of pinning strengths: (a) identical pinning strengths,
(b) halfgaussian, ,
(c) exponential,
,
and (d)
stretched exponential, .
Solid
lines indicate the scaling, Eq. (8),
for
(r_{p} = 0.015). Averages
were obtained from 148, 228, 212, and 152
configurations, respectively.

The pair correlation functions corresponding to the three broad
distributions, ,
1 and 1/2 agree with the prediction of
the liquidlike theory, Eq. (1), at least for small r.
In contrast, Eq. (1) does not describe the case of identical
pinning
strengths; the rapid falloff of
indicates a steeper than expected liftoff of g(r).
Thus the results of our numerical simulations clearly show a
liquidlike behavior
of the form Eq. (8) as observed in experiments. Furthermore
our simulations indicate that this scaling only
arises in the limit of pointlike pins drawn
from a broad distribution of pinning strengths.
Its successful application
to experimental data [3,8]
suggests that the pinscape in these materials
also is characterized by pointlike pins with a broad distribution
of strengths. Moreover, these observations serve as a warning that
simulation results based on identical
pinning strengths may differ subtly from the behavior of physical
systems.
Next: Theoretical Description
Up: Determining Pair Interactions from
Previous: Numerical Implementation
David G. Grier
19981119