Eliminating Poincaré Recurrence & Maximizing Entropy in the Kac Ring Model

In 1956, Mark Kac created the Kac Ring model to provide an intuition for how the second law of thermodynamics can arise from time-symmetric interactions between molecules (i.e. provide an answer to Loschmidt's paradox.) The model also serves as an excellent way to introduce the concept of coarse graining, given the relative simplicity of its calculations. Despite its simplicity, the Kac Ring (and various modified versions of it) has a variety of applications such as spin systems (similar to the Ising model), simplified lattice gas models, entropy-driven systems and modeling quantum many-body systems.

However, a limitation of the model is that it exhibits Poincaré Recurrence (i.e. the system returns to its initial configuration after a finite period of time, which is certainly not realistic) as the basic Kac Ring model describes an isolated and finite system akin to the microcanonical ensemble. The recurrence time is related to the number of rings in the ensemble, so one can increase the size of the ensemble to the point that no observer would ever live to witness the recurrence, but this does not actually eliminate recurrence from the system. It may be tempting to simply dismiss this as a limitation of a finite system, however at this time it is not definitely known if the universe itself is finite or infinite, which makes the approach of increasing system size all the more unsatisfactory.

This work proposes two alternative modifications to the model that definitively remove Poincaré recurrence irrespective to ensemble size. The first modification is the addition of a "kickback" probability (probability for an individual ring/microsystem to go back one step at a given time); the second alternative is a "pause" probability (an individual ring stopping for a moment.) These modifications are thoroughly explored and shown to result in the removal of recurrence and fufillment of entropy maximization. The results of these modifications are used to argue for an approach of desynchronizing component microsystems in the ensemble as a means of making more realistic models that can have finite and relatively small ensembles. Further implications are discussed.