  In particle-based stochastic reaction-diffusion models, reaction rate and
placement kernels are used to decide the probability per time a reaction can
occur between reactant particles, and to decide where product particles should
be placed. When choosing kernels to use in reversible reactions, a key
constraint is to ensure that detailed balance of spatial reaction-fluxes holds
at all points at equilibrium. In this work we formulate a general
partial-integral differential equation model that encompasses several of the
commonly used contact reactivity (e.g. Smoluchowski-Collins-Kimball) and volume
reactivity (e.g. Doi) particle models. From these equations we derive a
detailed balance condition for the reversible $\textrm{A} + \textrm{B}
\leftrightarrows \textrm{C}$ reaction. In bounded domains with no-flux boundary
conditions, when choosing unbinding kernels consistent with several commonly
used binding kernels, we show that preserving detailed balance of spatial
reaction-fluxes at all points requires spatially varying unbinding rate
functions near the domain boundary. Brownian Dynamics simulation algorithms can
realize such varying rates through ignoring domain boundaries during unbinding
and rejecting unbinding events that result in product particles being placed
outside the domain.
