In the host of numerical schemes devised to calculate free energy differences by way of geometric transformations, the adaptive biasing force algorithm has emerged as a promising route to map complex free-energy landscapes. It relies upon the simple concept that as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it yields a Hamiltonian devoid of an average force acting along the transition coordinate of interest. This means that sampling proceeds uniformly on a flat free-energy surface, thus providing reliable free-energy estimates. Much of the appeal of the algorithm to the practitioner is in its physically intuitive underlying ideas and the absence of any requirements for prior knowledge about free-energy landscapes. Since its inception in 2001, the adaptive biasing force scheme has been the subject of considerable attention, from in-depth mathematical analysis of convergence properties to novel developments and extensions. The method has also been successfully applied to many challenging problems in chemistry and biology. In this contribution, the method is presented in a comprehensive, self-contained fashion, discussing with a critical eye its properties, applicability, and inherent limitations, as well as introducing novel extensions. Through free-energy calculations of prototypical molecular systems, many methodological aspects are examined, from stratification strategies to overcoming the so-called hidden barriers in orthogonal space, relevant not only to the adaptive biasing force algorithm but also to other importance-sampling schemes. On the basis of the discussions in this paper, a number of good practices for improving the efficiency and reliability of the computed free-energy differences are proposed.

Mathematical methods for the magnetohydrodynamics of liquid metals Mathematical Methods for the Magnetohydrodynamics of Liquid Metals Numerical Mathematics and Scientific Computation Jean-Frédéric Gerbeau, Claude Le. Mathematical Methods for the Magnetohydrodynamics of Liquid Metals Towards a Design Tool for Self-heated Cells Producing Liquid Metal. Mathematical methods for the magnetohydrodynamics of liquid metals The compactness of weak solutions to the magnetohydrodynamic equations for the. Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Mathematical methods for the Magnetohydrodynamics of Liquid. Aug 31, 2006. This comprehensive text focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamic phenomena, with an Preprint-The University of Reading Towards a design tool for self-heated cells producing liquid metal by electrolysis. Sophie Poizeau the equations for liquid and gas flows simultaneously however, the methods used. Magnetohydrodynamics constitute the major cause of fluid flow in the.. 569-574. 22 Gerbeau, J.-F. 2006 Mathematical methods for. Mathematical Methods for the Magnetohydrodynamics of Liquid. Mathematical methods for the magnetohydrodynamics of liquid metals.

The objective of molecular dynamics computations is to infer macroscopic properties of matter from atomistic models via averages with respect to probability measures dictated by the principles of statistical physics. Obtaining accurate results requires efficient sampling of atomistic configurations, which are typically generated using very long trajectories of stochastic differential equations in high dimensions, such as Langevin dynamics and its overdamped limit. Depending on the quantities of interest at the macroscopic level, one may also be interested in dynamical properties computed from averages over paths of these dynamics.This review describes how techniques from the analysis of partial differential equations can be used to devise good algorithms and to quantify their efficiency and accuracy. In particular, a crucial role is played by the study of the long-time behaviour of the solution to the Fokker–Planck equation associated with the stochastic dynamics.

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.