Exponential decay of mass for inertial coalescing particles with Hamiltonian noise
We study a system of $N$ inertial particles on a two-dimensional torus $\T^2$, evolving under a second-order stochastic dynamics with position-dependent friction $\lambda$ and noise amplitude $\sigma$, and undergoing coalescence at rate $R_0$ when their distance falls below a threshold $\delta$. In the joint small-mass / small-correlation limit $\mu(\eps)\to 0$, $\mu(\eps)/\eps\to\al\in(0,\infty)$, the empirical measure of the surviving particles converges to a stochastic continuity equation with inertial drift~$g_\al$. Assuming that $\sigma$ is tangent to the level sets of a Hamiltonian $H=h_1(x_1)\,h_2(x_2)$ satisfying mild non-degeneracy and convexity-type conditions, and that $\lambda$ and the amplitude $\rho$ of $\sigma$ along $\xi=\nabla^\perp H$ are aligned with $H$, we prove that the expected total mass decays exponentially in time, with an explicit rate depending on $\al$ and on the values of $\lambda$ and $\rho$ on the separatrix $\{H=0\}$. The proof rests on a cell-by-cell analysis of the sign of $\div\,g_\al$ on the level sets of $H$, showing that the inertial drift pushes trajectories toward the separatrix at a quantitative rate.