  We report on a study of the thermodynamics of the Ginzburg-Landau model for
two-dimensional type-II superconductors near the transition from the normal
state to the Abrikosov-lattice state. We couch our analysis in terms of the
participation-ratio entropy, $s(P)$, which expresses the volume in
order-parameter-space with a given participation-ratio for the local superfluid
density. $s(P)$ completely determines the thermodynamics of the system. We
report on results for $s(P)$ obtained analytically using perturbation expansion
methods and numerically using Monte Carlo simulation methods and discuss the
weak first-order phase transition which occurs in this system in terms of the
properties of $s(P)$.
