Adaptive molecular resolution via a continuous change of the phase space dimensionality

For the study of complex synthetic and biological molecular systems by computer simulations one is still restricted to simple model systems or by far too small time scales. To overcome this problem multiscale techniques are being developed. However, in almost all cases, the regions and molecules of different resolution are kept fixed and are not in equilibrium with each other. We here give a basic theoretical framework for an efficient and flexible coupling of the different regimes. The approach leads to a concept, which can be seen as a geometry-induced phase transition, and to a counterpart of the equipartition theorem for fractional degrees of freedom. This represents the initial step in developing a general theoretical framework for computer simulation methods applying simultaneously different levels of resolution.

Figure 1

The free energy $F$ as a function of $x$. $A$ and $B$ are the regions with high and low levels of detail, respectively, while $\Delta$ is a transition regime. The constant values of $F_A$ and $F_B$ are arbitrary and do not play any role in the following treatment.

Figure 2

The molecular resolution of a simple two-dimensional circular molecule in the high- and low-resolution regions $A$ and $B$ and the transition region $\Delta$, respectively.