Coarse-grained variables

Metadynamics defines coarse-grained variables which are assumed to be slow coordinates of the system. Those coordinates are similar to the order parameters considered earlier in this chapter. The coarse variables are evolved independently following a steepest-descent equation. In the case of a single variable, Laio and Parrinello [34] use... 

When there are more macroscopic observables B,C,... the process can be continued. The end result is a collection of coarse-grained observables //, A, 5, C, ..., which all commute with one another. The Hilbert space H is decomposed in linear subspaces that are common eigenspaces of these observables. We shall call these subspaces phase cells and indicate them with a single label J. They correspond to definite values of the coarse-grained variables, which we shall now denote by Ej, AJy BJy Cj,. These phase cells are the macrostates. 

MTD is based on the assumption that the relevant processes can be described by a small set of coarse-grained variables, called collective variables (CVs), describing the essential degrees of freedom of the system. The CVs f ( r ) themselves are... 

The matrix L defining the coarse-grained Poisson bracket (7.7) is obtained by inserting the definitions (7.12)-(7.15) of the coarse-grained variables into Eq. (7.7). Details of the straightforward calculations are presented in Ref. [190]. 

The population densities are slowly varying functions of coarse-grained variables, and we may obtain hydrodynamic behavior by a Taylor expansion of (51) to second order in powers of g ... 

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