  In this paper, we investigate the equilibrium statistical properties of both
the force and potential interpolations of adaptive resolution simulation
(AdResS) under the theoretical framework of grand-canonical like AdResS
(GC-AdResS). The thermodynamic relations between the higher and lower
resolutions are derived by considering the absence of fundamental conservation
laws in mechanics for both branches of AdResS. In order to investigate the
applicability of AdResS method in studying the properties beyond the
equilibrium, we demonstrate the accuracy of AdResS in computing the dynamical
properties in two numerical examples: The velocity auto-correlation of pure
water and the conformational relaxation of alanine dipeptide dissolved in
water. Theoretical and technical open questions of the AdResS method are
discussed in the end of the paper.
