Gabriel Stoltz

Free energy computations : a mathematical perspective

Sampling Methods Stochastic Differential Equations Meta-Stability Free Energy Perturbation Thermodynamic Integration Constrained Dynamics Non-Equilibrium Methods Fluctuation Identities Jarzynski Identity Adaptive Techniques Long Time Convergence Replica Selection Methods Selection Mechanisms Parallel Computation.

Computation of free energy profiles with parallel adaptive dynamics

We propose a formulation of an adaptive computation of free energy differences, in the adaptive biasing force or nonequilibrium metadynamics spirit, using conditional distributions of samples of configurations which evolve in time. This allows us to present a truly unifying framework for these methods, and to prove convergence results for certain classes of algorithms. From a numerical viewpoint, a parallel implementation of these methods is very natural, the replicas interacting through the reconstructed free energy. We demonstrate how to improve this parallel implementation by resorting to some selection mechanism on the replicas. This is illustrated by computations on a model system of conformational changes.

The electronic ground-state energy problem: a new reduced density matrix approach.

We present here a formulation of the electronic ground-state energy in terms of the second order reduced density matrix, using a duality argument. It is shown that the computation of the ground-state energy reduces to the search of the projection of some two-electron reduced Hamiltonian on the dual cone of N-representability conditions. Some numerical results validate the approach, both for equilibrium geometries and for the dissociation curve of N(2).

The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics

a la Talay-Tubaro on the invariant distribution for small stepsize, and compare the sampling bias obtained for various choices of splitting method. We further investigate the overdamped limit and apply the methods in the context of driven systems where the goal is sampling with respect to a non-equilibrium steady state. Our analyses are illustrated by numerical experiments.

Long-time convergence of an adaptive biasing force method

We propose a proof of convergence of an adaptive method used in molecular dynamics to compute free energy profiles. Mathematically, it amounts to studying the long-time behavior of a stochastic process which satisfies a non-linear stochastic differential equation, where the drift depends on conditional expectations of some functionals of the process. We use entropy techniques to prove exponential convergence to the stationary state.

Langevin dynamics with constraints and computation of free energy differences

In this paper, we consider Langevin processes with mechanical constraints. The latter are a fundamental tool in molecular dynamics simulation for sampling purposes and for the computation of free energy differences. The results of this paper can be divided into three parts. (i) We propose a simple discretization of the constrained Langevin process based on a standard splitting strategy. We show how to correct the scheme so that it samples {\em exactly} the canonical measure restricted on a submanifold, using a Metropolis rule in the spirit of the Generalized Hybrid Monte Carlo (GHMC) algorithm. Moreover, we obtain, in some limiting regime, a consistent discretization of the overdamped Langevin (Brownian) dynamics on a submanifold, also sampling exactly the correct canonical measure with constraints. The corresponding numerical methods can be used to sample (without any bias) a probability measure supported by a submanifold. (ii) For free energy computation using thermodynamic integration, we rigorously prove that the longtime average of the Lagrange multipliers of the constrained Langevin dynamics yields the gradient of a rigid version of the free energy associated with the constraints. A second order time discretization using the Lagrange multipliers is proposed. (iii) The Jarzynski-Crooks fluctuation relation is proved for Langevin processes with mechanical constraints evolving in time. An original numerical discretization without time-step error is proposed. Numerical illustrations are provided for (ii) and (iii).

The effective local potential method: Implementation for molecules and relation to approximate optimized effective potential techniques

We have recently formulated a new approach, named the effective local potential (ELP) method, for calculating local exchange-correlation potentials for orbital-dependent functionals based on minimizing the variance of the difference between a given nonlocal potential and its desired local counterpart [V. N. Staroverov et al., J. Chem. Phys. 125, 081104 (2006)]. Here we show that under a mildly simplifying assumption of frozen molecular orbitals, the equation defining the ELP has a unique analytic solution which is identical with the expression arising in the localized Hartree-Fock (LHF) and common energy denominator approximations (CEDA) to the optimized effective potential. The ELP procedure differs from the CEDA and LHF in that it yields the target potential as an expansion in auxiliary basis functions. We report extensive calculations of atomic and molecular properties using the frozen-orbital ELP method and its iterative generalization to prove that ELP results agree with the corresponding LHF and CEDA values, as they should. Finally, we make the case for extending the iterative frozen-orbital ELP method to full orbital relaxation.

Nonlinear Fluctuating Hydrodynamics in One Dimension: The Case of Two Conserved Fields

We study the BS model, which is a one-dimensional lattice field theory taking real values. Its dynamics is governed by coupled differential equations plus random nearest neighbor exchanges. The BS model has two locally conserved fields. The peak structure of their steady state space–time correlations is determined through numerical simulations and compared with nonlinear fluctuating hydrodynamics, which predicts a traveling peak with KPZ scaling function and a standing peak with a scaling function given by the maximally asymmetric Lévy distribution with parameter

Mesoscopic simulations of shock-to-detonation transition in reactive liquid high explosive

\alpha = 5/3