  We derive an adjoint method for the Direct Simulation Monte Carlo (DSMC)
method for the spatially homogeneous Boltzmann equation with a general
collision law. This generalizes our previous results in [Caflisch, R.,
Silantyev, D. and Yang, Y., 2021. Journal of Computational Physics, 439,
p.110404], which was restricted to the case of Maxwell molecules, for which the
collision rate is constant. The main difficulty in generalizing the previous
results is that a rejection sampling step is required in the DSMC algorithm in
order to handle the variable collision rate. We find a new term corresponding
to the so-called score function in the adjoint equation and a new adjoint
Jacobian matrix capturing the dependence of the collision parameter on the
velocities. The new formula works for a much more general class of collision
models.
