  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.
