We have shown that the pair correlation function arising from stable configurations of interacting particles in a random pinscape can contain information about both the particle interactions and also the statistical properties of the underlying pinscape. Our analysis shows that this is the case in the regime where the pinning and pair interactions are of comparable strength. If the pinning completely dominates pair interactions, the pair correlation function will essentially reveal the spatial disorder of the active pinning sites. In the opposite limit, in which the pair interactions dominate, the pair correlation function will contain features corresponding to crystalline order. In both cases, little information can be extracted about the form of the interactions.
We have derived an expression for the pair correlation function in the limit of point-like disorder and low flux line density, Eq. (15). Our theory shows that the small r behavior of the pair correlation function depends both on the pair interaction and also on the availability of sufficiently strong pinning sites. We found that the distribution of the latter is not necessarily the same as the distribution of a priori available pinning sites. Rather, the dynamics favor strong pins. Inspection of the stable configurations revealed that the distribution of dynamically selected pins is consistent with a process of choosing the strongest out of n available pins. For the parameters used in our simulations, n = 2 described the dynamically selected pinning strength distributions very well. We expect however that the parameter n, while apparently independent of the width of the pinning strength distribution, should depend on both the concentration of pins and also the penetration depth of the flux line interaction. A more detailed investigation will be left for future work.
The theory presented in the previous sections is explicitly formulated for systems at low density. In this limit, structural correlations are dominated by otherwise isolated pairs of strongly pinned flux lines. Thus, the vectorial nature of their interaction forces need not be considered. At higher densities, each flux line interacts significantly with several neighbors and the vectorial nature of their forces must be taken into account. Constructing a more comprehensive theory valid at higher densities, while beyond the scope of the current paper, is currently under active investigation.
We now turn to a comparison of the qualitative features of the experimentally obtained pair correlation functions with those obtained from simulation. We find that the width and location of the first peak of g(r) is different: while the first peak of the pair correlation function as obtained from experiments (cf. Fig. 1) is broad and occurs around the mean flux line spacing, we see a much narrower peak in our simulations (cf. Fig. 2). Moreover the first peak occurs at a distance significantly smaller than the mean flux line spacing. Thus the numerical simulations have larger fluctuations in the local flux line density than are seen in experiment. A reason for this is that our particle-like treatment of flux lines ignores their magnetic properties. One expects that including boundary conditions for a uniform applied external magnetic field would favor a homogeneous flux line concentration and thus would penalize the large fluctuations in our simulations.
We thank Gene Mazenko and Stuart Rice for enlightening discussions. This work was supported in part by the National Science Foundation through the Science and Technology Center for Superconductivity under Award Number DMR-9120000 and in part by the MRSEC program of the National Science Foundation under Award Number DMR-9400379. C. H. S. would like to acknowledge the support of the National University of Singapore. M. M. acknowledges partial support from the Kerpisçi Foundation.