  The fact that simulations of condensed matter problems are limited to sizes
far smaller than the macroscopic systems being emulated does have one redeeming
feature: The specific dependence on N, the number of molecules being simulated,
is sometimes itself a valuable piece of information. The isotropic phase of
liquid crystals provides a case in point. Light scattering and 2d-IR
experiments on isotropic-phase samples display increasingly large orientational
fluctuations as the samples approach their nematic phase. The growing length
scale of those locally ordered domains is readily seen in simulation as an ever
slower convergence of the distribution of orientational order parameters with
N. The rare-event character of the largest fluctuations make them difficult to
sample accurately. We show in this paper how taking a large deviation theory
perspective enables us to leverage simulation-derived information more
effectively. A key insight of large deviation theory is that extracting the
desired measurable, in our case, the extent of orientational order, is
completely equivalent to finding the size of the conjugate field required to
equilibrate that amount of order - and that knowing the relationship between
the two gives us the relative free energy. Indeed, a variety of thermodynamic
integration strategies used to evaluate free energy differences in simulation
are essentially founded on this idea. However, rather than applying an
artificially imposed external field, we use a priori statistical mechanical
insights into the small and large-field limits of the equation of state to
construct a simulation-guided, interpolated, equation of state. The free
energies that result turn out to need simulation information derived mostly
from the most probable configurations, rather than from the infrequent events,
and therefore use simulation information far more efficiently than conventional
methods.
