Controllability for chains of dynamical scatterers

TL;DR

This paper demonstrates that a chain of chaotic mechanical cells with heat baths at both ends is controllable, allowing the system to transition between any states, which implies a unique invariant measure out of equilibrium.

Contribution

We prove controllability of a class of chaotic mechanical chains with heat baths, establishing the ability to drive the system between any states and confirming the uniqueness of the invariant measure.

Findings

System is controllable via heat baths.

Any state can be reached from any other in finite time.

Existence of a unique invariant measure out of equilibrium.

Abstract

In this paper, we consider a class of mechanical models which consists of a linear chain of identical chaotic cells, each of which has two small lateral holes and contains a rotating disk at its center. Particles are injected at characteristic temperatures and rates from stochastic heat baths located at both ends of the chain. Once in the system, the particles move freely within the cells and will experience elastic collisions with the outer boundary of the cells as well as with the disks. They do not interact with each other but can transfer energy from one to another through collisions with the disks. The state of the system is defined by the positions and velocities of the particles and by the angular positions and angular velocities of the disks. We show that each model in this class is controllable with respect to the baths, i.e. we prove that the action of the baths can drive the system from any state to any other state in a finite time. As a consequence, one obtains the existence of at most one regular invariant measure characterizing its states (out of equilibrium).