**Molecular dynamics.**

**Teams primarily involved:**

- CERMICS, ENPC – Molecular simulation
team (Eric
Cancès)

- Laboratoire
Jacques-Louis LIONS, Université Paris VI (Yves Achdou)

- Physics of Condensed Matter Unit, CEA-DAM, Ile de France
(Gilles Zérah)

- IPSO Project, Inria
Rennes (Philippe
Chartier)

Nowadays, the majority of people developing models, methods and
algorithms for molecular dynamics (MD) studies are physicists/chemists.
We believe that significant progress can be achieved by ideas
coming from the applied mathematics community. The main problem
arising in MD simulations is the length and time scales currently
reachable, and the characteristic time and length scales of the
system. Roughly speaking, the most performant codes can tackle
systems with a few million atoms during a microsecond –
whereas relevant times scales can be 9 orders of magnitude larger
as for biological systems!

Most computations are concerned with **long-time
simulations of systems**, in order to compute
thermodynamic integrals (under ergodicity conditions) or to learn
about systems’ evolutions. Mathematically, it can be seen as
the (long-)time integration of a Hamiltonian dynamic. Therefore,
the following issues arise:

**Long-time integration of Hamiltonian systems**, in particular for numerical discretizations of the original system, and development of performant algorithms (multiple time-steps methods, use of methods coming from ODE problems, etc)**Alternative strategies**for thermodynamical integrals (instead of one long trajectory, wouldn’t it be possible to use several short trajectories ? + paralelization)**Dimensionality reduction**in the problems through reduced effective description (often involving**stochastic**perturbations)

Some results have already been obtained in the setting of the ARC Prestissimo. They concern the improvement of convergence rates for thermodynamic integrals.

- E. Cancès, F. Castella, Ph. Chartier, E. Faou, C. Le Bris,
F. Legoll and G.
Turinici, High-order averaging schemes with error
bounds for thermodynamical properties calculations by molecular
dynamics simulations (rapport
de recherche
INRIA RR-4875), Journal of
Chemical Physics 121 (21), 10346-10355 (2004)

- E. Cancès, F. Castella, Ph. Chartier, E. Faou, F. Legoll,
C. Le
Bris, G. Turinici,
*Long-time averaging for integrable Hamiltonian dynamics*, Numerische Mathematik, 100 (2), 211-232 (2005).

Recent results focus on sampling issues in MD.

- E. Cancès, F. Legoll, and G. Stoltz,
*Theoretical and numerical comparison of some sampling methods*, sumitted to M2AN (2005).

- E. Cancès and G. Stoltz,
*A coupled NVE-NVT model for the calculation of dynamical properties in the canonical ensemble*, submitted to J. Chem. Phys. (2005)

- M. Rousset and G. Stoltz,
*An interacting particle system approach for molecular dynamics*, sumitted to M3AS (2005)