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We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of a class of Glauber+Zero-range particle systems. The Zero-range part moves particles while preserving particle numbers, and the Glauber part governs the creation and annihilation of particles and is set to favor two levels of particle density. When the two parts are simultaneously seen in certain different time-scales, the Zero-range part being diffusively scaled while the Glauber part is speeded up at a lesser rate, a mean-curvature interface flow emerges, with a homogenized `surface tension-mobility parameter reflecting microscopic rates, between the two levels of particle density. We use relative entropy methods, along with a suitable `Boltzmann-Gibbs principle, to show that the random microscopic system may be approximated by a `discretized Allen-Cahn PDE with nonlinear diffusion. In turn, we show the behavior of this `discretized PDE is close to that of a continuum Allen-Cahn equation, whose generation and propagation interface properties we also derive.
In this report we discuss appropriate strategies for the tracking of charged particles in the limit of zero curvature. The suggested approach avoids special treatments and precision issues that frequently arise in that limit. We provide explicit expr
There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term. These RDEs have traveling wave solutions. When the speed of the wave is
In the nonlinear diffusion framework, stochastic processes of McKean-Vlasov type play an important role. In some cases they correspond to processes attracted by their own probability distribution: the so-called self-stabilizing processes. Such diffus
This is a contribution to the program of dynamical approach to mean curvature flow initiated by Colding and Minicozzi. In this paper, we prove two main theorems. The first one is local in nature and the second one is global. In this first result, we
Controlling large particle systems in collective dynamics by a few agents is a subject of high practical importance, e.g., in evacuation dynamics. In this paper we study an instantaneous control approach to steer an interacting particle system into a