Effective dynamics using conditional expectations

Abstract

The question of coarse-graining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarse-grained variable ξ(x), where x describes the configuration of the system in a high-dimensional space R n , and ξ is a smooth function with value in R (typically a reaction coordinate). It is well known that, given a Boltzmann-Gibbs distribution on x ∈ R n , the equilibrium properties on ξ(x) are completely determined by the free energy. On the other hand, the question of the effective dynamics on ξ(x) is much more difficult to address. Starting from an overdamped Langevin equation on x ∈ R n , we propose an effective dynamics for ξ(x) ∈ R using conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones. We check numerically on some toy examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarse-grained dynamics.AMS classification scheme numbers: 35B40, 82C31, 60H10 ‡ In this article, we do not address the difficult question of how to find a good reaction coordinate. See for instance [24] for some discussion on that point.

Variational approach to coarse-graining of generalized gradient flows

“…The sequence ρ ε is tight in M([0, T ]×R 2 ) by estimate (49), which implies Part 1. The proof of part 2 is similar to Part 2 in Theorem 2.4, and the final estimate is a direct consequence of (50).…”

“…Existing quantitative methods such as [42,49] only work for gradient flows systems since they use crucially the gradient flow structures. The essential estimate that they need is the energy-dissipation inequality, which is similar to (4).…”

“…where we have used |∇ p H | 2 ≤ C(1 + H ) and the estimate (49) to arrive at the convergence statement. Note that (49) along with the definition of the rate functionals implies that I (ρ) < ∞ iff I α (ρ) < ∞.…”

“…Note that (49) along with the definition of the rate functionals implies that I (ρ) < ∞ iff I α (ρ) < ∞.…”

On two possible definitions of the free energy for collective variables

“…This possibility, which has been analysed in [27], yields a stochastic differential equation for z of the form,…”