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).

• 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)

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