  In this paper we develop quantum dynamical methods capable of treating the
dynamics of chemically reacting systems in an optical cavity in the
vibrationally strong-coupling (VSC) limit at finite temperatures and in the
presence of a dissipative solvent in both the few and many molecule limits. In
the context of two simple models we demonstrate how reactivity in the {\em
collective} VSC regime does not exhibit altered rate behavior in equilibrium,
but may exhibit resonant cavity modification of reactivity when the system is
explicitly out of equilibrium. Our results suggest experimental protocols that
may be used to modify reactivity in the collective regime and point to features
not included in the models studied which demand further scrutiny.
