Local equilibrium of particle density in planar Lorentz processes

Particles are injected into a large planar domain through the boundary and perform a random or sufficiently chaotic deterministic motion inside the domain. Our main example is the Sinai billiard, which periodically extended to our large planar domain, is referred to as the Lorentz process. Assuming that the particles move independently from one another and the boundary is also absorbing, we prove the emergence of local equilibrium of the particle density in the diffusive scaling limit in two scenarios. One scenario is an arbitrary domain with piece-wise smooth boundary and a carefully chosen injection rule; the other scenario is a rectangular domain and a much more general injection mechanism. We study the latter scenario in an abstract framework that includes Lorentz processes and random walks and hopefully allows for more applications in the future.