Constructing a counterexample: positive Fisher-information production in 2D for discrete-angle kernels

Construct a probability density f on R^2 and a collision kernel of the form B(v−v*,θ)=|v−v*|^γ b(cos θ) whose angular measure b is supported on a finite set of deviation angles (as in Theorem 18.1(i)), such that the instantaneous production of Fisher information along the spatially homogeneous Boltzmann flow is positive; equivalently, show that I′(f)·Q(f,f) > 0 for such a choice.

Background

Theorem 18.1(i) shows that for certain measures concentrated on finitely many deviation angles, a key integral-inequality criterion fails (its best constant is zero). This raises the possibility that the Fisher information might actually increase for some data under such collision laws.

The author explicitly flags the question of turning this negative functional-inequality result into a concrete dynamical counterexample, i.e., an f for which the Fisher information increases initially.