  The configurational entropy is among the key observables to characterize
experimentally the formation of a glass. Physically, it quantifies the
multiplicity of metastable states in which an amorphous material can be found
at a given temperature, and its temperature dependence provides a major
thermodynamic signature of the glass transition, which is experimentally
accessible. Measurements of the configurational entropy require, however, some
approximations which have often led to ambiguities and contradictory results.
Here we implement a novel numerical scheme to measure the configurational
entropy Sigma(T) in supercooled liquids, using a direct determination of the
free energy cost to localize the system within a single metastable state at
temperature T. For two prototypical glass-forming liquids, we find that
Sigma(T) disappears discontinuously above a temperature T_c, which is slightly
lower than the usual estimate of the onset temperature for glassy dynamics.
This observation is in good agreement with theoretical expectations, but
contrasts sharply with alternative numerical methods. While the temperature
dependence of Sigma(T) correlates with the glass fragility, we show that the
validity of the Adam-Gibbs relation (relating configurational entropy to
structural relaxation time) established in earlier numerical studies is smaller
than previously thought, potentially resolving an important conflict between
experiments and simulations.
