The question of coarse-graining is ubiquitous in many applied sciences, including molecular dynamics. In this work, 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, and $\xi$ is a smooth function. Typically, $X$ is a high-dimensional variable (representing for instance the positions of all the particles in the system), whereas $\xi(X)$ is a low-dimensional, coarse-grained information (e.g. a particular angle between some atoms of the molecule).
We assume that the configuration $X_t$ of the complete system evolves according to a given stochastic differential equation. We propose an effective closed dynamics that approximates (under time-scale separation assumptions) the evolution of $\xi(X_t)$. Such an effective dynamics may be useful to compute more efficiently e.g. transition rates from one configuration of the system to another one. Its derivation may also help understand how uncertainties of the reference model may affect the effective dynamics of the system at a coarse scale.
We will first consider reversible systems, and next turn to some non-reversible examples.
Based on joint works with T. Lelievre, S. Olla and U. Sharma.