  The question of coarse-graining is ubiquitous in molecular dynamics. In this
article, 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 smooth function
with value in $\R$ (typically a reaction coordinate). It is well known that,
given a Boltzmann-Gibbs distribution on $x \in \R^n$, 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 $x \in
\R^n$, we propose an effective dynamics for $\xi(x) \in \R$ using conditional
expectations. Using entropy methods, we give sufficient conditions for the time
marginals of the effective dynamics to be close to the original ones. We check
numerically on some toy 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.
