Conjecture on the limiting process with a growing number of light particles

Let the heavy disk have diameter $\mathbf{r}=M^{-\alpha}$ and let the number of light particles be $N=M^\beta$, with $\alpha,\beta>0$. Set $\tau=t\sqrt{M}$. Suppose that the initial state of each light particle is chosen independently, with position and velocity direction uniformly distributed and speed having a smooth density $\rho_0(v)$. Define

$$a_j(\rho)=\int |v|^j\rho(v)\,dv.$$

The limiting-process conjecture. The limiting process $\mathbf{Q}(\tau)$ is (a) straight motion if $\beta=\alpha+1/2-\epsilon$; (b) the integral of an Ornstein–Uhlenbeck process,

$$d\mathbf{Q}=\mathbf{V}\,d\tau,\qquad d\mathbf{V}=-\nu\mathbf{V}\,d\tau+\sigma\,d\mathbf{w},$$

where

$$\nu=c_1a_1(\rho_0),\qquad \sigma^2=c_2a_3(\rho_0),$$

if $\beta=\alpha+1/2$; and (c) a Brownian motion if $\beta=\alpha+1/2+\epsilon$.

This conjecture describes the three scaling regimes for the heavy disk as the number of light particles changes. The intermediate regime is expected to produce an integrated Ornstein–Uhlenbeck limit, while the subcritical and supercritical regimes are expected to produce straight-line and Brownian limits, respectively.