Abstract
We use computer simulations to probe the thermodynamic and dynamic properties of a glass-former that undergoes an ideal glass-transition because ofthe presence of randomly pinned particles.We find that even deep in the equilibrium glass state the system relaxes to some extent because ofthe presence of localized excitations that allow the systemto access different inherent structures,giving thus rise to a non-trivial contribution to the entropy. By calculating with high accuracy thevibrational part of the entropy, we show that also in the equilibrium glass state thermodynamicsand dynamics give a coherent picture and that glasses shouldnot be seen as a disordered solid inwhich the particles undergo just vibrational motion but instead as a system with a highly nonlinearinternal dynamics.