  Entropy has become increasingly central to characterize, understand and even
guide assembly, self-organization and phase transition processes. In this work,
we build on the analogous role of partition functions (or free energies) in
isothermal ensembles and that of entropy in adiabatic ensembles. In particular,
we show that the grand-isobaric adiabatic $(\mu,P,R)$ ensemble, or Ray
ensemble, provides a direct route to determine the entropy. This allows us to
follow the variations of entropy with the thermodynamic conditions and thus to
explore phase transitions. We test this approach by carrying out Monte Carlo
simulations on Argon and Copper in bulk phases and at phase boundaries and
assess the reliability and accuracy of the method through comparisons with the
results from flat-histogram simulations in isothermal ensembles and with the
experimental data. Advantages of the approach are multifold and include the
direct determination of the $\mu-P$ relation, without any evaluation of
pressure via the virial expression, the precise control of the system size and
of the number of atoms via the input value of $R$, and the straightforward
computation of enthalpy differences for isentropic processes, which are key
quantities to determine the efficiency of thermodynamic cycles. A new insight
brought by these simulations is the highly symmetric pattern exhibited by both
systems along the transition, as shown by scaled temperature-entropy and
pressure-entropy plots.
