A temperature control technique for nonequilibrium molecular simulation

We describe a dynamical approach to thermal regulation in molecular dynamics. Temperature is moderated by a control law and an additional variable, as in Nose dynamics, but whose influence on the system decreases as the system approaches equilibrium. This device enables approximation of microcanonical averages and autocorrelation functions consistent with a given target temperature. Moreover, we demonstrate that the suggested technique is effective for the control of heat dissipation in a nonequilibrium setting, first by showing that the temperature control correctly regulates heat introduced by a rapid change to the system, and then by studying the slow relaxation of vibrational degrees of freedom (e.g., due to bonded atoms) in a solvent bath.     

Approximation numérique d'une classe de problèmes en homogénéisation stochastique

In this Note, we study a numerical strategy for the computation of the homogenized matrix for a stochastic elliptic problem that is a small perturbation of a periodic problem. We adapt the analysis introduced in [X. Blanc, C. Le Bris, P.-L. Lions, Stochastic homogenization and random lattices, J. Math. Pures Appl. 88 (2007) 34–63] to the case when the corrector problems are numerically solved, and we computationally assess the interest and the accuracy of the approach.     

Symplectic schemes for highly oscillatory Hamiltonian systems with varying fast frequencies

We derive symplectic integrators for a class of highly oscillatory Hamiltonian systems. Our approach is based upon a two-scale expansion of the solution to the Hamilton–Jacobi equation associated to the original dynamics. This Note presents an extension of the approach previously introduced in Le Bris and Legoll (2007, 2010) [10], [11] to the case where the fast frequencies of the system, instead of being constant, explicitly depend on the slow degrees of freedom.     

Long-time averaging for integrable Hamiltonian dynamics

Summary. Given a Hamiltonian dynamical system, we address the question of computing the limit of the time-average of an observable. For a completely integrable system, it is known that ergodicity can be characterized by a diophantine condition on its frequencies and that this limit coincides with the space-average over an invariant manifold. 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 symplectic numerical scheme is applied to the system, up to the order of the integrator.     

Effective Dynamics for a Kinetic Monte–Carlo Model with Slow and Fast Time Scales

We consider several multiscale-in-time kinetic Monte Carlo models, in which some variables evolve on a fast time scale, while the others evolve on a slow time scale. In the first two models we consider, a particle evolves in a one-dimensional potential energy landscape which has some small and some large barriers, the latter dividing the state space into metastable regions. In the limit of infinitely large barriers, we identify the effective dynamics between these macro-states, and prove the convergence of the process towards a kinetic Monte Carlo model. We next consider a third model, which consists of a system of two particles. The state of each particle evolves on a fast time-scale while conserving their respective energy. In addition, the particles can exchange energy on a slow time scale. Considering the energy of the first particle, we identify its effective dynamics in the limit of asymptotically small ratio between the characteristic times of the fast and the slow dynamics. For all models, our results are illustrated by representative numerical simulations.     

Analysis of a Prototypical Multiscale Method Coupling Atomistic and Continuum Mechanics： the Convex Case

In order to describe a solid which deforms smoothly in some region,but non smoothly in someother region,many multiscale methods have been recently proposed that aim at coupling an atomistic model(discrete mechanics) with a macroscopic model (continuum mechanics).We provide here a theoretical basis forsuch a coupling in a one-dimensional setting,in the case of convex energy.     

High-order averaging schemes with error bounds for thermodynamical properties calculations by molecular dynamics simulations

We introduce high-order formulas for the computation of statistical averages based on the long-time simulation of molecular dynamics trajectories. In some cases, this allows us to significantly improve the convergence rate of time averages toward ensemble averages. We provide some numerical examples that show the efficiency of our scheme. When trajectories are approximated using symplectic integration schemes (such as velocity Verlet), we give some error bounds that allow one to fix the parameters of the computation in order to reach a given desired accuracy in the most efficient manner.     

Approximation grossière dʼun problème elliptique à coefficients hautement oscillants

Our wish is to approximate an elliptic problem with highly oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillating coefficients in the equation can be incomplete or entirely missing. We investigate the links between this particular question and the classical theory of homogenization. On some illustrating examples we show the potential practical interest of the approach.     

Thermal Conductivity of the Toda Lattice with Conservative Noise

We study the thermal conductivity of the one dimensional Toda lattice perturbed by a stochastic dynamics preserving energy and momentum. The strength of the stochastic noise is controlled by a parameter γ. We show that heat transport is anomalous, and that the thermal conductivity diverges with the length n of the chain according to κ(n)∼n ^α , with 0<α≤1/2. In particular, the ballistic heat conduction of the unperturbed Toda chain is destroyed. Besides, the exponent α of the divergence depends on γ.