Effective dynamics using conditional expectations

The question of coarse-graining is ubiquitous in molecular dynamics. In this paper, we are interested in deriving effective properties for the dynamics of a coarse-grained variable xi(x), where x describes the configuration of the system in a high-dimensional space R-n, and xi is a given smooth function with value in R (typically a reaction coordinate). It is well known that, given a Boltzmann-Gibbs distribution on x is an element of R-n, the equilibrium properties on xi(x) are completely determined by the free energy. On the other hand, the question of the effective dynamics on xi(x) is much more difficult to address. Starting from an overdamped Langevin equation on x is an element of R-n, we propose an effective dynamics for xi(x). Rusing conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones, with precise error bounds. We check numerically on several examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarse-grained dynamics.