We begin by assuming that the dominant contribution to the force on a given
flux line is due to its nearest neighbor.
Because of the short-range nature
of the flux line interactions, this assumption is justified in the limit of
low flux line concentration, where the average flux line separation, as
well as fluctuations around it, are much larger
than the London penetration depth. Furthermore, our simulations
reveal that more than 99 % of the flux lines in a stable configuration
are *pinned*, *i.e.* within a distance
of
of a pinning site. Thus assuming that the flux lines
are pinned in pairs is reasonable.
In particular, this approximation is valid
for pairs of flux lines pinned at distances smaller than average.
These pairs
determine the behavior of the pair correlation function near lift-off,
*i.e. *the region where the function rises from zero.
They occur only if
sufficiently strong pins are available. In other words, a pair of
nearest-neighbor flux lines can be
pinned at a separation *r* only if both
pins are strong enough to overcome the flux line repulsion force,
.

From Eq. (4) we see that a pin
of strength
can exert a maximum force given by

Let be the probability that a flux line has come to rest on a pin of maximal pinning force . The cumulative probability,

(14) |

is the likelihood that the occupied pin is at least as strong as

This expression should be valid in the low-density limit, where only pairwise-pinned flux lines have to be considered and screening effects due to the presence of other flux line pairs can be neglected. One can readily show that the resulting flux line configurations are stable. The distribution of dynamically selected pinning strengths, , does not coincide with the corresponding distribution of

On the basis of this observation, we hypothesize that a given flux line
dynamically selects the strongest of *n* pins in its immediate
vicinity so that

(17) |

and thus

Equations (15) - (18) and their successful application to our simulations are central results of this article.

Fig. 5
shows that *P*_{d}(*F*) is very well approximated by *n* = 2, *i.e.*,

implying that its cumulative distribution

We find that

For exponentially distributed pinning
strengths, ,
we obtain

where . Thus,

and, using Eq. (15),

Analogous predictions can be obtained for the cases and .

Fig. 6 shows comparisons of the theoretically
predicted scaling of
with the values
obtained from our simulations for ,
and .
These comparisons involve no adjustable parameters once *n* has been fixed
(Fig. 5), since the force scale
*F*_{o} was specified for each simulation.
The results for all three pinning distributions
are in very good agreement with the
form predicted by Eq. (15) for *g*(*r*) near lift-off.

Comparing the liquid structure Ansatz to the present theory
emphasizes fundamental differences between systems
with thermal fluctuations and quenched disorder.
In thermal systems such as simple liquids, structural correlations
arise from the interplay of interaction energies and the
thermal energy scale.
Systems such as pinned flux lines are dominated by quenched disorder
and select configurations in which *forces* are balanced.
Despite the fundamental differences in the derivations of
Eqs. (1) and (15), the two approaches
yield remarkably similar predictions.
This resemblance is particularly close for
pinned flux lines because their interaction force, *f*(*r*), is
proportional to the potential, *U*(*r*), to logarithmic
accuracy in the regime of interest.
Thus, for
,
Eq. (15) reduces
to the same function form as Eq. (1).

This liquid-like form for *g*(*r*) is
obtained only if the distribution of dynamically
selected pins is exponential.
Eqs. (15) and
(16) show
that, in the general case,
prefactors and different forms of leading
order behavior can arise depending
on the tails of the (dynamical) pinning
energy distribution.
Our results in Fig. 6 suggest,
however, that this dependence is
weak and will become detectable only
for very small values of *r*.
This is consistent with
the success of Eq. (1) in describing
experimental data.

Even though the liquid structure Ansatz applies only
under limited circumstances, Eqs. (15) -
(18) should apply quite generally in the
low density limit.
For example, the liquid structure Ansatz fails
to describe flux line correlations on pinscapes
generated from pins of identical strength.
Fig. 4(a) reveals that this system
differs from the more broadly
distributed cases in that it gives rise to a distinctively
steep lift-off in the pair correlation function
not accounted for by Eq. (1).
Eq. (15), in contrast, predicts a step-like
increase in *g*(*r*) at the smallest
separation for which
a pair of flux lines can be pinned against their mutual
repulsion.
Analyzing the
configurations contributing
to the lift-off of *g*(*r*) shows that the small degree
of rounding apparent in Fig. 4(a) is due to
many-body effects, *i.e.* flux lines pinned by
interactions with
two or more neighboring flux lines.
Comparing this discrepancy to the good
agreement
with the theoretical prediction in the case of broadly distributed
pinning strengths leads us to conclude
that, in these cases, the disorder due to variation
in pinning strength dominates the contribution
arising from the spatial disorder of the flux line
configurations.

Equation (15) also should work equally
well for any inter-particle pair potential which is sufficiently
short ranged that many-body correlations are negligible.
To test this idea, we performed
simulations with a power-law
pair potentials,
and *r*^{-7}, scaled
such that at
they exerted the same magnitude of force
as that derived from Eq. (2).
As expected, Eq. (15) succeeded quantitatively
for the *r*^{-7} potential, but failed for the
longer-ranged *r*^{-3} potential.