| Popis: |
We prove exponential concentration estimates and a strong law of large numbers for a particle system that is the simplest representative of a general class of models for 2D grain boundary coarsening. The system consists of $n$ particles in $(0,\infty)$ that move at unit speed to the left. Each time a particle hits the boundary point $0$, it is removed from the system along with a second particle chosen uniformly from the particles in $(0,\infty)$. Under the assumption that the initial empirical measure of the particle system converges weakly to a measure with density $f_0(x) \in L^1_+(0,\infty)$, the empirical measure of the particle system at time $t$ is shown to converge to the measure with density $f(x,t)$, where $f$ is the unique solution to the kinetic equation with nonlinear boundary coupling $$\partial_t f (x,t) - \partial_x f(x,t) = -\frac{f(0,t)}{\int_0^\infty f(y,t)\, dy} f(x,t), \quad 0Comment: 23 pages |
