  We use computer simulations to probe the thermodynamic and dynamic properties
of a glass-former that undergoes an ideal glass-transition because of the
presence of randomly pinned particles. We find that even deep in the
equilibrium glass state the system relaxes to some extent because of the
presence of localized excitations that allow the system to access different
inherent structures, giving thus rise to a non-trivial contribution to the
entropy. By calculating with high accuracy the vibrational part of the entropy,
we show that also in the equilibrium glass state thermodynamics and dynamics
give a coherent picture and that glasses should not be seen as a disordered
solid in which the particles undergo just vibrational motion but instead as a
system with a highly nonlinear internal dynamics.
