  In this work, we provide a detailed theoretical analysis, supported by
numerical tests, of the reliability of the adaptive resolution simulation
(AdResS) technique in sampling the Grand Canonical ensemble. We demonstrate
that the correct density and radial distribution functions in the hybrid
region, where molecules change resolution, are two necessary conditions for
considering the atomistic and coarse-grained regions in AdResS equivalent to
subsystems of a full atomistic system with an accuracy up to the second order
with respect to the probability distribution of the system. Moreover, we show
that the work done by the thermostat and a thermodynamic force in the
transition region is formally equivalent to balance the chemical potential
difference between the different resolutions. From these results follows the
main conclusion that the atomistic region exchanges molecules with the
coarse-grained region in a Grand Canonical fashion with an accuracy up to (at
least) second order. Numerical tests, for the relevant case of liquid water at
ambient conditions, are carried out to strengthen the conclusions of the
theoretical analysis. Finally, in order to show the computational convenience
of AdResS as a Grand Canonical set up, we compare our method to the Insertion
Particle Method (IMP) in its most efficient computational implementation. This
fruitful combination of theoretical principles and numerical evidence
candidates the adaptive resolution technique as a natural, general and
efficient protocol for Grand Canonical Molecular Dynamics for the case of large
systems.
