  A family of collective variables is proposed to perform exact dynamical
coarse-graining even in systems without time scale separation. More precisely,
it is shown that these variables are not slow in general but they satisfy an
overdamped Langevin equation that statistically preserves the sequence in which
any regions in collective variable space are visited and permits to calculate
exactly the mean first passage times from any such region to another. The role
of the free energy and diffusion coefficient in this overdamped Langevin
equation is discussed, along with the way they transform under any change of
variable in collective variable space. These results apply both for systems
with and without inertia, and they can be generalized to using several
collective variables simultaneously. The view they offer on what makes
collective variables and reaction coordinates optimal breaks from the standard
notion that good collective variable must be slow variable, and it suggests new
ways to interpret data from molecular dynamic simulations and experiments.
