  In this article we investigate the long time behavior of solutions to a class
of infinitely many master equations defined from transition rates that are
suitable for the description of a quantum system approaching thermodynamical
equilibrium with a heat bath at fixed temperature and a reservoir consisting of
one species of particles characterized by a fixed chemical potential. We do so
by proving a result which pertains to the spectral resolution of the semigroup
generated by the equations, whose infinitesimal generator is realized as a
trace-class self-adjoint operator defined in a suitable weighted sequence
space. This allows us to prove the existence of global solutions which all
stabilize toward the grand canonical equilibrium probability distribution as
the time variable becomes large, some of them doing so exponentially rapidly.
When we set the chemical potential equal to zero, the stability statements
continue to hold in the sense that all solutions converge toward the Gibbs
probability distribution of the canonical ensemble which characterizes the
equilibrium of the given system with a heat bath at fixed temperature.
