Title: Dynamics of the Kac Ring Model with switching scatterers Show Chinese title

Title: Kac环模型中切换散射器的动力学

Abstract: 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 necessarily promote thermodynamic states, interactions are necessary for energy to be shared among the elementary constituents of matter, enabling the establishment of equipartition, which is a prerequisite for defining temperature. Therefore, our model is one step forward in this direction, elucidating the role of interactions and energy exchange. We prove 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. Show Chinese abstract

Abstract: 我们引入了一个广义的Kac环模型，其中粒子分为两种类型，黑色和白色。 黑色粒子通过所有粒子移动的环境进行修改，从而在它们之间引发间接且可能远距离的相互作用。 与Kac原始模型中的惰性散射体不同，我们设定中的散射体具有内部状态，在与黑色粒子相互作用时会发生变化，并可以被解释为环境的能量等级。 这使得模型自洽，因为它包含了由环境介导的粒子相互作用形式，这种相互作用会将系统引导至某种稳态。 尽管间接和远距离的相互作用不一定促进热力学状态，但相互作用对于能量在物质基本组成部分之间的共享是必要的，从而实现了均分，这是定义温度的前提条件。 因此，我们的模型在这个方向上前进了一步，阐明了相互作用和能量交换的作用。 我们证明了系统的任何初始状态都会收敛到一个时间周期状态（即相空间轨道），并描述了一些此类渐近周期状态的吸引盆地。