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The Model

We consider a two-dimensional system of N particles in a quenched random pinscape consisting of discrete pinning sites positioned at random. The particles, which can be interpreted as stiff magnetic flux lines, have positions that are specified by (two-dimensional) vectors ${\bf r}_i$. As a convention, particle and pinning site locations are subscripted with Latin and Greek indices, respectively. The force on the ith flux line at position ${\bf r}_i$ is given by

{\bf F}_i = - \sum_{j \neq i}^{N} {\bf\nabla}_i
U({\bf r}...
...pha} {\bf\nabla}_i V_{\alpha}({\bf r}_i -
{\bf r}_{\alpha}).
\end{displaymath} (3)

Here the first and second sum are the contributions to the total force on a given particle due to its interactions with the other particles and with the pinning sites; U(r) is the interparticle interaction, whereas $V_\alpha(r)$ is the particle- pin interaction. For stiff magnetic flux lines, the interparticle potential U(r) (per unit length) is given in the low-density limit by the asymptotic form Eq. (2). We assume that the pinning potential due to a single pin is attractive and short ranged,

V_{\alpha}(r) = - V_{\alpha} e^{-r^2/r_p^2},
\end{displaymath} (4)

where $V_{\alpha}$ is the pin's strength, and rp its range.

The locations of the pinning sites ${\bf r}_\alpha$ are assumed to be distributed at random. We seek stable and static flux line configurations $\{ {\bf r}^{*}_i \}$ given by solutions to

{\bf F}_i({\bf r}^*_1,{\bf r}^*_2, \ldots,{\bf r}^*_N) = {\bf0},
\end{displaymath} (5)

with the additional stability constraint that the matrix $-\partial{{\bf F}_i}/\partial{{\bf r}^*_j}$ be positive-definite. If $\{ {\bf r}^*_1,{\bf r}^*_2, \ldots,{\bf r}^*_N \}$ is such a configuration, the pair correlation function g*(r) for this configuration is given by

g^*(r) = \frac{\langle \int \rho^* ({\bf x}-{\bf r})
...f x} \rangle_\theta}
{[\int \rho^* ({\bf x}) \, d{\bf x}]^2},
\end{displaymath} (6)


\rho^*({\bf x}) = \sum_{i=1}^N \delta({\bf x} - {\bf r}^*_i),
\end{displaymath} (7)

and $\langle \ldots \rangle_\theta $ denoting an average over angles. The pair correlation function g(r) is obtained by averaging Eq. (6) over independent configurations.

If the analogy to liquid structure theory adequately describes the emergence of structure in this quenched random system, then Eqs. (1) and (2) imply that (to lowest order)

\ln (-r^{\frac{1}{2}} \ln g(r)) \propto - \frac{r}{\lambda}.
\end{displaymath} (8)

Thus liquid theory predicts that a plot of the corresponding quantity versus r should yield a linear relation with slope $1/\lambda$, as observed experimentally in the inset to Fig. 1.

Our numerical work involves solving Eq. (5) for static configurations and thereby obtaining the configuration-averaged pair correlation function. Details of the implementation and presentation of our numerical results are given in the following two sections.

next up previous
Next: Numerical Implementation Up: Determining Pair Interactions from Previous: Introduction
David G. Grier