arXiv subjects

Gabriel Stoltz

Publications and source records attributed to Gabriel Stoltz.

100 records · Page 6Linked to original sources

arXiv2008

Adiabatic approximation, Gell-Mann and Low theorem and degeneracies: A pedagogical example

We study a simple system described by a 2x2 Hamiltonian and the evolution of the quantum states under the influence of a perturbation. More precisely, when the initial Hamiltonian is not degenerate,we check analytically the validity of the adiabatic approximation and verify that, even if the evolution operator has no limit for adiabatic switchings, the Gell-Mann and Low formula allows to follow the evolution of eigenstates. In the degenerate case, for generic initial eigenstates, the adiabatic approximation (obtained by two different limiting procedures) is either useless or wrong, and the Gell-Mann and Low formula does not hold. We show how to select initial states in order to avoid such failures.

cond-mat.str-el↗

arXiv2007

Local Exchange Potentials for Electronic Structure Calculations

The Hartree-Fock exchange operator is an integral operator arising in the Hartree-Fock method and replaced by a multiplicative operator (a local potential) in Kohn-Sham density functional theory. This article presents a detailed analysis of the mathematical properties of various local approximations to the nonlocal Hartree-Fock exchange operator, including the Slater potential, the optimized effective potential (OEP), the Krieger-Li-Iafrate (KLI) and common energy-denominator approximations (CEDA) to the OEP, and the effective local potential (ELP). In particular, we show that the Slater, KLI, CEDA potentials and the ELP can all be defined as solutions to certain variational problems. We also provide a rigorous derivation of the integral OEP equation and establish the existence of a solution to a system of coupled nonlinear partial differential equations defining the Slater approximation to the Hartree-Fock equations.

quant-ph↗

arXiv2007

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.

math.AP↗

arXiv2007

A reduced model for shock and detonation waves. II. The reactive case

We present a mesoscopic model for reactive shock waves, which extends a previous model proposed in [G. Stoltz, Europhys. Lett. 76 (2006), 849]. A complex molecule (or a group of molecules) is replaced by a single mesoparticle, evolving according to some Dissipative Particle Dynamics. Chemical reactions can be handled in a mean way by considering an additional variable per particle describing a rate of reaction. The evolution of this rate is governed by the kinetics of a reversible exothermic reaction. Numerical results give profiles in qualitative agreement with all-atom studies.

cond-mat.mtrl-sci↗

arXiv2007

Computation of free energy differences through nonequilibrium stochastic dynamics: the reaction coordinate case

The computation of free energy differences through an exponential weighting of out of equilibrium paths (known as the Jarzynski equality) is often used for transitions between states described by an external parameter $λ$ in the Hamiltonian. We present here an extension to transitions between states defined by different values of some reaction coordinate, using a projected Brownian dynamics. In contrast with other approaches, we use a projection rather than a constraining potential to let the constraints associated with the reaction coordinate evolve. We show how to use the Lagrange multipliers associated with these constraints to compute the work associated with a given trajectory. Appropriate discretizations are proposed. Some numerical results demonstrate the applicability of the method for the computation of free energy difference profiles.

cond-mat.stat-mech↗

arXiv2007

Computation of free energy profiles with parallel adaptive dynamics

We propose a formulation of adaptive computation of free energy differences, in the ABF or nonequilibrium metadynamics spirit, using conditional distributions of samples of configurations which evolve in time. This allows 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 show 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.

cond-mat.stat-mech↗

arXiv2006

Path sampling with Stochastic Dynamics: Some new Algorithms

We propose here some new sampling algorithms for Path Sampling in the case when stochastic dynamics are used. In particular, we present a new proposal function for equilibrium sampling of paths with a Monte-Carlo dynamics (the so-called ``brownian tube'' proposal). This proposal is based on the continuity of the dynamics with respect to the random forcing, and generalizes all previous approaches. The efficiency of this proposal is demonstrated using some measure of decorrelation in path space. We also discuss a switching strategy that allows to transform ensemble of paths at a finite rate while remaining at equilibrium, in contrast with the usual Jarzynski like switching. This switching is very interesting to sample constrained paths starting from unconstrained paths, or to perform simulated annealing in a rigorous way.

cond-mat.stat-mech↗

arXiv2006

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$_2$.

quant-ph↗

arXiv2006

An efficient sampling algorithm for Variational Monte Carlo

We propose a new algorithm for sampling the $N$-body density $|Ψ({\bf R})|^2/\int_{\mathbb{R}^{3N}} |Ψ|^2$ in the Variational Monte Carlo (VMC) framework. This algorithm is based upon a modified Ricci-Ciccotti discretization of the Langevin dynamics in the phase space $({\bf R},{\bf P})$ improved by a Metropolis acceptation/rejection step. We show through some representative numerical examples (Lithium, Fluorine and Copper atoms, and phenol molecule), that this algorithm is superior to the standard sampling algorithm based on the biased random walk (importance sampling).

cond-mat.other↗

arXiv2005

Equilibrium Sampling From Nonequilibrium Dynamics

We present some applications of an Interacting Particle System (IPS) methodology to the field of Molecular Dynamics. This IPS method allows several simulations of a switched random process to keep closer to equilibrium at each time, thanks to a selection mechanism based on the relative virtual work induced on the system. It is therefore an efficient improvement of usual non-equilibrium simulations, which can be used to compute canonical averages, free energy differences, and typical transitions paths.

cond-mat.stat-mech↗