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
can exert a maximum force given by
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
shows that Pd(F) is very well approximated by n = 2, i.e.,
For exponentially distributed pinning
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 Fo 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.