  We study the thermalization of a composite quantum system consisting of
several subsystems, where only a small one of the subsystem contacts with a
heat bath in equilibrium, while the rest of the composite system is contact
free. We show that the whole composite system still can be thermalized after a
relaxation time long enough, if the energy level structure of the composite
system is connected, which means any two energy levels of the composite system
can be connected by direct or indirect quantum transitions. With an example
where an multi-level system interacts with a set of harmonic oscillators via
non-demolition coupling, we find that the speed of relaxation to the global
thermal state is suppressed by the multi-Franck-Condon factor due to the
displacements of the Fock states when the degrees of freedom is large.
