  Pairwise forces between particles in cosmological N-body simulations are
generally softened to avoid hard collisions. Physically, this softening
corresponds to treating the particles as diffuse clouds rather than point
masses. For particles of unequal mass (and hence unequal softening length),
computing the softened force involves a nontrivial double integral over the
volumes of the two particles. We show that Plummer force softening is
consistent with this interpretation of softening while spline softening is not.
We provide closed-form expressions and numerical implementation for pairwise
gravitational force laws for pairs of particles of general softening scales
$\epsilon_1$ and $\epsilon_2$ assuming the commonly used cloud profiles: NGP,
CIC, TSC, and PQS. Similarly, we generalize Plummer force law into pairs of
particles of general softenings. We relate our expressions to the gaussian,
Plummer and spline force softenings known from literature. Our expressions
allow possible inclusions of pointlike particles such as stars or supermassive
black holes.
