Statistical physics is important in understanding the physics of interacting many bodies. This has been historically developed by attempts to understand colliding gases and quantifying quantities like entropy, free energy, and other thermodynamic quantities. An important contribution in statistical physics was by Boltzmann in the form of the H-theorem, which considered collisions between particles and used the assumption of molecular chaos or Stosszahlansatz to understand macroscopic irreversibility. To elucidate these ideas, Mark Kac introduced a classical analog called Kac rings. In this work, we attempt to introduce quantum-ness in a Kac ring and study its entropy and recurrence, comparing and contrasting to corresponding trends in a classical Kac ring. We look at the trends of recurrence time for a system with a qubit as a pointer. We further study the time distribution of entropy for these systems.
We introduce quantum versions of the Kac Master Equation and the Kac Boltzmann Equation. We study the steady states of each of these equations, and prove a propagation of chaos theorem that relates them. The Quantum Kac Master Equation (QKME) describes a quantum Markov semigroup, while the Kac Boltzmann Equation describes a non-linear evolution of density matrices on the single particle state space. All of the steady states of the
N particle quantum system described by the QKME are separable, and thus the evolution described by the QKME is entanglement breaking. The results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory, and a quantitative study the rate of destruction of entanglement in a class of quantum Markov semigroups describing binary interactions.