Numerical methods in molecular dynamics
Tony Lelièvre · 2011
Molecular dynamics is now a very widely used tool to study matter at the molecular level. It is used in various fields, such as biology, chemistry or materials science. The aim is in particular to understand the relationships between the macroscopic properties of a molecular system and its atomistic features. For example, one would like to compute the constitutive relations for materials from molecular models, or predict the most likely conformations of a protein in a solvent from its amino acid sequence. One of the difficulties to reach this aim is related to timescales: the typical timescale of a molecular dynamics simulation is much smaller than the typical timescale at which the crucial events, from a macroscopic viewpoint, occur. This is related to the metastability of a molecular dynamics trajectory: the system stays for a very long time in some regions of the configuration space (called a metastable state), before hopping to another one, and it is difficult to observe and simulate such rare events. An associated feature is the multimodality of the statistical ensemble (a probability measure) sampled by the molecular dynamics trajectories. Many methods have been proposed in the molecular dynamics community to deal with these difficulties, and we will focus on two prototypical ones for which a mathematical analysis gives useful insights. We will first present adaptive importance sampling techniques, which have been proposed to sample efficiently statistical ensembles. Then, we will propose a mathematical analysis of the parallel replica algorithm which has been introduced by A.F. Voter to generate efficiently metastable dynamics.