A radiating spin chain as a model of irreversible dynamics

We construct a finite spin-1/2 chain model (quantum domino) interacting with a Fermi field, capable of emitting a scalar fermion from the last spin in the chain. The chain with dynamics gradually reversing the neighbouring spins emits eventually a fermion which escapes then to infinity, and the chain converges to a stationary state. We determine the rate of convergence of the system for large time to a different "macroscopic" state connected with the emission of a fermion. We prove that the probability of fermion emission as a function of time t approaches to unity "almost exponentially". We propose that this fast rate of convergence could serve as an approximate theoretical possibility for the "effective" description of the quantum measurement process in the sense proposed by K. Hepp. This all will be preceded by an outline of explicitly solvable dynamics of infinite version of the spin chain which exhibits transition from a locally perturbed unstable stationary state to a truly macroscopically different (i.e. disjoint) state, but with slow convergence for t approaching to infinity.