Dynamics of the Kac Ring Model with switching scatterers

We introduce a generalized version of the Kac ring model in which particles are of two types, black and white. Black particles modify the environment through which all particles move, thereby inducing indirect and potentially long-range interactions among them. Unlike the inert scatterers of Kac’s original model, the scatterers in our setting possess internal states that change upon interaction with black particles and can be interpreted as energy levels of the environment. This makes the model self-consistent, as it incorporates a form of particle interactions, mediated by the environment, that drives the system toward some kind of stationary state. Although indirect and long-range interactions do not typically promote thermodynamic states, interactions are necessary for energy to play a role and be shared among the elementary constituents of matter. Therefore, the inclusion of interactions in our model constitutes a step forward in the description of a macroscopic system in terms of its microscopic constituents. We find that any initial state of the system converges to a time periodic state (i.e. a phase space orbit) and describe basins of attraction for some of such asymptotic periodic states.