polymer representation of atoms has far reaching consequences, because it translates the quantum-classical coupling into the coupling of two effective classical regions characterized by a different number of (as a matter of fact) ”classical” degrees of freedom; thus the whole machinery of classical adaptive methods would apply straightforwardly. In this work we show that indeed this is the case. Obviously the approach here is suited only for a class of problems and, as it is designed now, certainly not for those problems where the quantum me- chanics refers to electrons. It must be noted that practical methods [6, 16, 17] which couple the two regimes when electrons are considered suffer from the same conceptual limitations underlined in this work and base their validity on numerical criteria only. In this specific case the conceptual discontinuity is in the arbitrary cut off of the electron wavefunction and in the non conservation of the number of electrons as the system evolves in time. Although we do not consider systems with electrons, the concept of mapping a quantum problem onto a classical one and then have a smooth adaptivity is anyway of general valence. In this sense this work represents a possible starting point for a general adaptive coupling of quantum and classical description of atoms and molecules. The Adaptive scheme: Regarding the adaptive scheme for the classical case, the basic requirement is a controlled procedure of changing the number of degrees of freedom based on solid physical principles and consistent with the thermodynamic equilibrium of the overall system [18–20]. In this respect the AdResS method meets the requirements above in a (ex- tensively tested) satisfactory way [21–25]. For this reason in this work the coupling between the polymer rings and the classical particles will be done within the AdResS framework. Be- low we introduce the basic features of AdResS relevant for this work. In the classical AdResS method the atomistic (high resolution) and the coarse-grained (low resolution) regime are coupled via a force interpolation (see e.g. [1, 8]): F$_{αβ}$ = w ( X$_{α}$ ) w ( X$_{β}$ ) F atom αβ + [1 − w ( X$_{α}$ ) w ( X$_{β}$ )] F cg αβ (1) where α and β indicates two molecules, F atom is the force derived from the atomistic force field and F cg from the corresponding coarse-grained potential, X is the x coordinate of the center of mass of the molecule and w is an interpolating function which smoothly goes from 0 to 1 (or vice versa) in a transition region (Δ) where the lower resolution is then slowly transformed (according to w ) in the high resolution (or vice versa), as illustrated in Fig.1 (see also [26]). An additional locally acting thermostat is employed to assure the overall 3