Theoretical Description

The analogy drawn between pinned flux lines and thermally disordered fluids yields an empirically satisfactory description of flux line pair correlations. To this point, though, it has been justified by heuristic arguments. This section introduces a statistical treatment of structure in strongly pinned flux lines which provides a superior description of disordered flux line distributions yet yields the liquid structure result, Eq. (1), as a limiting case.

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 $r_p/\sqrt{2}$ 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, $f(r) = -\partial U(r)/\partial r$.

From Eq. (4) we see that a pin $\alpha$ of strength $V_{\alpha}$ can exert a maximum force given by

$$F_\alpha = \sqrt{\frac{2}{e}} \: \frac{V_\alpha}{r_p}.$$ (13)

Let $P_d(F_\alpha)$ be the probability that a flux line has come to rest on a pin of maximal pinning force $F_\alpha$. The cumulative probability,

$$Q_d(F) = \int_{F}^{\infty} P_d(F_\alpha) dF_\alpha,$$ (14)

is the likelihood that the occupied pin is at least as strong as F. If we assume that flux lines are equally likely to be pinned anywhere in the sample as long as the corresponding pinning forces are large enough to overcome the flux line interaction, it follows that the pair correlation function g(r) has the form

$$g(r) = {Q_d}^2(f(r)) + \ldots .$$ (15)

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, $P_d(F_\alpha)$, does not coincide with the corresponding distribution of a priori available strengths, $P(F_\alpha)$. The latter is readily obtained from the pinning energy distribution, ${\cal P}_\nu(V_\alpha)$, using Eq. (13). Fig. 5 compares the distributions of dynamically selected and available maximal pinning forces for exponentially distributed ($\nu = 1$) pinning strengths, Eq. (11).

Figure 5: Distribution of available and dynamically selected maximal pinning forces. Shown here is the case of exponentially distributed pinning strengths, $\nu = 1$. The dashed line corresponds to the distribution of available maximal pinning forces as obtained from Eqs. (13) and (11). Data points correspond to the distribution obtained by determining the pinning strength of the pins associated with every pinned flux line. The solid line corresponds to Eq. (21), as explained in the text.

We see clearly that the particles preferentially occupy strong pins. This is consistent with our expectation that an isolated flux line, or one that is effectively isolated, is more likely to be in the domain of attraction of a strong pin in its vicinity.

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

$$Q_d(F) = 1 - \left [1 - Q(F) \right ]^n, \qquad {\mathrm where}$$ (16)

$$Q(F) = \int_F^\infty \, P(F_\alpha) \, dF_\alpha,$$ (17)

and thus

$$P_d(F) = n P(F) \left [\int_{0}^{F} \, P(F')dF'\right ]^{n-1}.$$ (18)

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.,

$$P_d(F) = 2 P(F) \int_{0}^{F} P(F')dF',$$ (19)

implying that its cumulative distribution Q_d(F) obeys,

$$Q_d(F) = 1 - \left [1 - Q(F) \right ]^2.$$ (20)

We find that n=2 also satisfactorily describes dynamically selected pinning in our simulations with half-gaussian and stretched exponentially distributed pinning strengths, $\nu = 2$ and $\nu = 1/2$.

For exponentially distributed pinning strengths, $\nu = 1$, we obtain

$$P_d(F) = \frac{2}{F_o} e^{-F/F_o} \left (1 - e^{-F/F_o} \right ),$$ (21)

where $F_o = \sqrt{2/e} \: V_o / r_p$. Thus,

$$Q_d(F) = 2 e^{-F/F_o}\left (1 - \frac{1}{2} e^{-F/F_o} \right ),$$ (22)

and, using Eq. (15),

$$g(r) = 4 e^{-2f(r)/F_o} \left [1 - \frac{1}{2} e^{-f(r)/F_o}\right ]^2 + \ldots.$$ (23)

Analogous predictions can be obtained for the cases $\nu = 2$ and $\nu = 1/2$.

Fig. 6 shows comparisons of the theoretically predicted scaling of $\ln (-r^{\frac{1}{2}} \ln g(r))$ with the values obtained from our simulations for $\nu = 2$, $\nu = 1$ and $\nu = 1/2$. 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.

Figure 6: Comparison of the behavior of the pair correlation function near lift-off against simulation results, liquid theory type scaling (dashed lines) and theoretical prediction, Eqs. (15) and (20), for half-gaussian (circles), exponential (diamonds), and stretched exponential $\nu = 1/2$ (boxes) distribution of pinning strengths. Note that the theoretical predictions contain no free parameter once n has been fixed (Fig. 5), while the liquid theory predictions are one-parameter fits to the form Eq. (1). Curves have been vertically offset for clarity.

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 $f(r) \approx F_o$, 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, $U(r) \propto r^{-3}$ and r^-7, scaled such that at $r = 4\lambda$ 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.