  In order to extract maximal information from neutron-star merger signals,
both gravitational and electromagnetic, we need to ensure that our theoretical
models/numerical simulations faithfully represent the extreme physics involved.
This involves a range of issues, with the finite temperature effects regulating
many of the relevant phenomena. As a step towards understanding these issues,
we explore the conditions for $\beta$-equilibrium in neutron star matter for
the densities and temperatures reached in a binary neutron star merger. Using
the results from our out-of-equilibrium merger simulation, we consider how
different notions of equilibrium may affect the merger dynamics, raising issues
that arise when attempting to account for these conditions in future
simulations. These issues are both computational and conceptual. We show that
the effects lead to, in our case, a softening of the equation of state in some
density regions, and to composition changes that affect processes that rely on
deviation from equilibrium, such as bulk viscosity, both in terms of the
magnitude and the equilibration timescales inherent to the relevant set of
reactions. We also demonstrate that it is difficult to determine exactly which
equilibrium conditions are relevant in which regions of the matter due to the
dependence on neutrino absorption, further complicating the calculation of the
reactions that work to restore the matter to equilibrium.
