next up previous
Next: Theoretical Description Up: Determining Pair Interactions from Previous: Numerical Implementation

Numerical Results

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 rp = 0.015. The inset to Fig. 2 shows the scaling behavior of $ \ln (-r^{\frac{1}{2}} \ln g(r)) $ with r. The solid line has slope $1/\lambda$. Note the good agreement with the liquid theory prediction, Eq. (8), for values of r in the initial lift-off of g(r). The deviations for larger r arise from excess correlations (pile-up) 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 liquid-like 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 rp. Fig. 3 shows the scaling of $ \ln (-r^{\frac{1}{2}} \ln g(r)) $ with r for different values of rp, ranging from $0.015~\mu$m to $0.05~\mu$m.

Figure: Dependence of the pair correlation function on the range of the pinning potential rp for exponentially distributed pinning strengths. Solid lines correspond to a liquid theory scaling, Eq.  (8), with $\lambda = 0.04~{\mu}$m. The values for rp are $0.05~\mu$m (a), $0.025~\mu$m (b), $0.017~\mu$m (c), and $0.015~\mu$m (d). Averages were calculated from 596, 600, 323 and 212 configurations, respectively.
\begin{figure}
\begin{center}
\includegraphics[width=3in]{fig3.eps} \end{center}\end{figure}

The asymptotic form, Eq. (8), is realized only in the limit of a short-ranged pinning potential ( $r_p \ll \lambda$).

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)], $ \nu = 2$, 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) half-gaussian, $ \nu = 2$, (c) exponential, $\nu = 1$, and (d) stretched exponential, $\nu = 1/2$. Solid lines indicate the scaling, Eq. (8), for $\lambda = 0.04$ (rp = 0.015). Averages were obtained from 148, 228, 212, and 152 configurations, respectively.
\begin{figure}
\begin{center}
\includegraphics[width=3in]{fig4b.eps} \end{center}\end{figure}

The pair correlation functions corresponding to the three broad distributions, $ \nu = 2$, 1 and 1/2 agree with the prediction of the liquid-like theory, Eq. (1), at least for small r. In contrast, Eq. (1) does not describe the case of identical pinning strengths; the rapid fall-off of $ \ln (-r^{\frac{1}{2}} \ln g(r)) $ indicates a steeper than expected lift-off of g(r).

Thus the results of our numerical simulations clearly show a liquid-like behavior of the form Eq. (8) as observed in experiments. Furthermore our simulations indicate that this scaling only arises in the limit of point-like 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 point-like 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 up previous
Next: Theoretical Description Up: Determining Pair Interactions from Previous: Numerical Implementation
David G. Grier
1998-11-19