An extension of the Kac ring model

We introduce a unitary dynamics for quantum spins which is an extension of a model introduced by Mark Kac to clarify the phenomenon of relaxation to equilibrium. When the number of spins becomes very large, the magnetization satisfies an autonomous equation as a function of time with exponentially fast relaxation to the equilibrium magnetization as determined by the microcanonical ensemble. This is proved as a law of large numbers with respect to a class of initial data. The corresponding Gibbs-von Neumann entropy is also computed and its monotonicity in time discussed.