Dynamics of the Kac Ring Model with switching scatterers

TL;DR

The paper extends the classical Kac ring by introducing a Generalized Kac Ring (GKR) with two particle colors and scatterers that evolve via interactions, creating environment-mediated, nonlocal interactions. It establishes a deterministic cellular automaton framework with a rigidity parameter $r$ controlling the environment’s evolution and a selectivity mechanism that drives irreversibility. The authors prove that all trajectories are eventually periodic for any $r>0$ and analyze the resulting attractors, detailing frozen versus oscillating states, with explicit results in the one- and two-particle cases and insights into basins of attraction. This work advances the understanding of how particle–environment feedback can generate energy exchange and nontrivial long-time dynamics, offering a step toward incorporating thermodynamic concepts like equipartition and temperature in interacting, deterministic systems.

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.

Figures (4)

• Figure 1: The GKR model consists of a ring with $L$ sites, populated at time $t=0$ by $\hat{N}_b$ black particles and $\hat{N}_w$ white particles (depicted as solid and empty disks in the figure), with $N=\hat{N}_b+\hat{N}_w\le L$. Active and passive scatterers are represented by filled and empty triangles, respectively. At each integer time step, particles move clockwise to the nearest neighboring site on the ring. When a particle encounters an active scatterer, it instantaneously changes color: black becomes white, and white becomes black. In addition, a scatterer also switches its state (from active to passive or vice versa) after undergoing a fixed number of collisions with black particles. This threshold is defined by the parameter $r$, called rigidity.
• Figure 2: Specific interactions between particles and scatterers in the GKR model: (a) a black particle and an active scatterer, (b) a black particle and a passive scatterer, (c) a white particle and an active scatterer, (d) a white particle and a passive scatterer. An oriented arrow denotes the interaction exerted by the element at the tail on the element at the head, while the absence of an arrow indicates the lack of interaction.
• Figure 3: Behavior of $\chi(t)$ (filled disks), $\Phi(t)$ (filled triangles) and $\Sigma(t)$ (empty squares) for the GKR model with $L=4$ and $r=1$ (upper panel) and $r=2$ (lower panel). The initial configuration is the same considered in Table \ref{['tab:table1']}.
• Figure 4: Behavior of $\chi(t)$ for the GKR model with $L=5$ and $r=1$ for time-reversed (empty disks) and anticlockwise dynamics (filled squares). In both evolutions the initial configuration, which belongs to a periodic orbit, is $o_0(0) = 1$, $s_i(0) = -1$ for all $i \in \Lambda_L$.

Theorems & Definitions (11)

• Lemma 2.1
• proof
• Remark 1
• Corollary 1
• Corollary 2
• proof
• Lemma 3.2
• proof
• Lemma 4.3
• proof