Conjecture on kinetic equations for smooth decaying potentials

Let $\Phi$ be a smooth decaying potential and consider the scaling

$$\Phi_{\varepsilon}(x)=\varepsilon^\alpha\Phi(x/\varepsilon),\qquad N=\varepsilon^{-\beta},$$

with $\varepsilon\to0$, where $N$ is the average number of particles per macroscopic unit volume. Smooth decaying potentials conjecture. The normalized one-particle marginal $f^\varepsilon_1$ converges to a solution of a kinetic equation when $\alpha\in[0,1]$ and $\beta>0$ satisfy

$$\beta=2(1+\alpha).$$

The limiting equation is the Boltzmann equation for $\alpha=0$, the Landau equation for $\alpha\in(0,1)$, and the Balescu–Lenard equation for $\alpha=1$. This conjecture identifies the kinetic equation associated with each regime of the shrinking-interaction scaling, but the source does not specify which cases have been proved.