Long-time averaging for integrable Hamiltonian dynamics

Eric Cancès (1, 2) , François Castella (3, 4) , Philippe Chartier (3) , Erwan Faou (4, 3) , Frédéric Legoll (2, 5, 6) , Claude Le Bris (1, 2) , Gabriel Turinici (2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MICMAC - Methods and engineering of multiscale computing from atom to continuum
3 IPSO - Invariant Preserving SOlvers
4 IRMAR - Institut de Recherche Mathématique de Rennes
5 navier umr 8205 - Laboratoire Navier
6 EDF R&D - EDF R&D

Given a Hamiltonian dynamics, we address the question of computing the space-average (referred as the ensemble average in the field of molecular simulation) of an observable through the limit of its time-average. For a completely integrable system, it is known that ergodicity can be characterized by a diophantine condition on its frequencies and that the two averages then coincide. In this paper, we show that we can improve the rate of convergence upon using a filter function in the time-averages. We then show that this convergence persists when a numerical symplectic scheme is applied to the system, up to the order of the integrator.

Domaines: Analyse numérique [math.NA]

Eric Cancès, François Castella, Philippe Chartier, Erwan Faou, Frédéric Legoll, et al.. Long-time averaging for integrable Hamiltonian dynamics. Numerische Mathematik, 2005, 100 (2), pp.211--232. ⟨10.1007/s00211-005-0599-0⟩. ⟨hal-00798337⟩