  Exact results on particle-densities as well as correlators in two models of
immobile particles, containing either a single species or else two distinct
species, are derived. The models evolve following a descent dynamics through
pair-annihilation where each particle interacts at most once throughout its
entire history. The resulting large number of stationary states leads to a
non-vanishing configurational entropy. Our results are established for
arbitrary initial conditions and are derived via a generating-function method.
The single-species model is the dual of the 1D zero-temperature kinetic Ising
model with Kimball-Deker-Haake dynamics. In this way, both infinite and
semi-infinite chains and also the Bethe lattice can be analysed. The
relationship with the random sequential adsorption of dimers and weakly tapped
granular materials is discussed.
