Section: New Results
Atomistic to continuum methods
Participants : Claude Le Bris, Frédéric Legoll, Florian Thomines.
The project-team have continued their theoretical and numerical efforts on the general topic of "passage from the atomistic to the continuum". This concerns theoretical issues arising in this passage but also the development and the improvement of numerical simulations coupling the two scales. The lecture notes  review some of the numerical methods used in this context, along with some numerical analysis results. It also turns out that this topic shares many common features with the modelling of complex fluids (another domain in which the project-team has been strongly involved for many years), as explained in the review article  .
In collaboration with X. Blanc (Paris 6) and C. Patz (WIAS, Berlin), C. Le Bris and F. Legoll addressed questions related to finite temperature modeling of atomistic systems, and derivation of coarse-grained descriptions. The starting observation is that, for atomistic systems at constant temperature, relevant quantities are statistical averages of some functions (called observables in that context) with respect to the Gibbs measure. One particular case of interest is when the observable at hand does not depend on all the variables, but only on some of them (gathered in a region of interest, where some defects appear, for instance). In that case, a relevant quantity to compute is the free energy associated to these few degrees of freedom. In the one-dimensional setting, an efficient strategy, that bypasses the simulation of the whole system, has been proposed to compute this free energy, as well as averages of such observables. This strategy is based on a rigorous thermodynamical procedure. Encouraging results have been reported in  . Recent efforts in the project-team aimed at extending the strategy to more complex cases. Promising results have been obtained in the 2D scalar case  .
Another situation of major interest, beyond the static setting, is the dynamical case. Some preliminary work, on some simple models, has been conducted by C. Le Bris, in collaboration with X. Blanc (Paris 6) and P.-L. Lions (Collège de France).