  Multicanonical ensemble simulations for the simulation of first-order phase
transitions suffer from exponential slowing down. Monte Carlo autocorrelation
times diverge exponentially with free energy barriers $\Delta F$, which in
$L^d$ boxes grow as $L^{d-1}$. We exemplify the situation in a study of the 2D
Ising-model at temperature $T/T_c=0.63$ for two different lattice manifolds,
toroidal lattices and surfaces of cubes. For both geometries the effect is
caused by discontinuous droplet shape transitions between various classical
crystal shapes obeying geometrical constraints. We use classical droplet theory
and numerical simulations to calculate transition points and barrier heights.
On toroidal lattices we determine finite size corrections to the droplet free
energy, which are given by a linear combination of Gibbs-Thomson corrections,
capillary wave fluctuation corrections, constant terms and logarithmic terms in
the droplet volume. Tolman corrections are absent. In addition, we study the
finite size effects on the condensation phase transition, which occurs in
infinite systems at the Onsager value of the magnetization. We find that this
transition is of discontinuous order also. A combination of classical droplet
theory and Gibbs-Thomson corrections yields a fair description for the
transition point and for the droplet size discontinuity for large droplets. We
also estimate the nucleation barrier that has to be surmounted in the formation
of the stable droplet at coexistence.
