Section: New Results
The analysis of multi-scale phenomena and asymptotic problems aiming at identifying the influence of microscopic scales on the macroscopic observations is a hot topic in the team. Results have been obtained concerning the derivation of effective law describing the behavior of a particle interacting with a thermal bath or a set of oscillators. This work, which combines modeling efforts, analysis and large computations, is the object of a longstanding collaboration with P. Parris (Missouri-Rolla) see  and is the heart of the PhD thesis of B. Aguer. Some long time effective behavior of related models has been obtained in  .
At the same time, M. Rousset is working on the numerical simulation of stochastically perturbed Molecular Dynamics. The main goal is to handle in the same simulation the fastest time scales (the oscillations of molecular bindings), and the slowest time scales (the so-called reaction coordinates). Adaptive methods are a popular tool to accelerate the slow time scales by using the appropriate bias which is computed “on-line”. In  the long-time convergence analysis of the latter has been achieved. In  analysis of constrained dynamics is proposed, with associated numerical schemes. In  , a new method has been proposed which drastically slows down the fast frequencies with a penalty and accelerates simulations, while conserving the statistical behavior of molecular systems.
Recently, in  M. Rousset has initiated some new research on variance reduction in hybrid methods, where a ”fine-grained” model, typically a kinetic model, is simulated with particle/Monte-Carlo method, and the variance of the latter is reduced using the information of a ”coarse-grained” model, a PDE computed with a grid method.