Section: New Results

Computational Statistical Physics

Participants : Claude Le Bris, Frédéric Legoll, Tony Lelièvre, Francis Nier, Mathias Rousset, Gabriel Stoltz.

Free Energy calculations

For large molecular systems, the information of the whole configuration space may be summarized in a few coordinates of interest, called reaction coordinates. An important problem in chemistry or biology is to compute the effective energy felt by those reaction coordinates, called free energy.

In [39] , T. Lelièvre and G. Stoltz, in collaboration with physicists from CEA Saclay (especially, M. Athenes) studied a new adaptive technique of ABF type to compute on-the-fly the free energy of a system, without evaluating the second derivatives of the reaction coordinate. The method uses a Bayesian reinterpretation of an extended system where the reaction coordinate is considered as an additional variable.

In [44] , G. Fort (Telecom Paris), B. Jourdain (CERMICS), E. Kuhn (INRA), T. Lelièvre and G. Stoltz have studied the efficiency of the Wang-Landau algorithm, building on a previous study where they proved the convergence of this method. The aim was to obtain precise estimates of the exit times out of metastable states. This was done in two ways: a theoretical study in the simplest possible metastable situation, a system with three states; and a numerical study in a more realistic situation (a two-dimensional double well potential).

Sampling trajectories

There exist a lot of methods to sample efficiently Boltzmann-Gibbs distributions. The situation is much more intricated as far as the sampling of trajectories (and especially metastable trajectories) is concerned.

Recently, the quasi stationary distribution has been identified by the team as a good mathematical tool to analyze metastable trajectories, and to make a link between a continuous state space dynamics (Langevin dynamics) and a discrete state space dynamics (kinetic Monte Carlo models), see for examplelelievre-13. This perspective can also be used to analyze accelerated dynamics techniques which have been proposed by A. Voter in the late nineties, to simulate very efficiently the state-to-state dynamics associated with metastable trajectories. For example, in [33] , T. Lelièvre with D. Aristoff (University of Minnesota) propose a mathematical analysis of the Temperature Accelerated Dynamics. In [49] , T. Lelièvre and F. Nier have studied the quasi-stationary distribution for an overdamped Langevin process in a bounded domain. In the small temperature limit and by making the connection with boundary Witten Laplacians, they are able to accurately compute the spatial exit law along the boundary and non perturbative accurate formulas when the potential is changed inside the domain. This gives some insight into the foundations of the hyperdynamics method.

Finally, following a numerical observation in a previous work on the sampling of reactive trajectories by a multilevel splitting algorithm, F. Cérou (Inria Rennes), A. Guyader (Inria Rennes), T. Lelièvre and F. Malrieu (Université de Rennes) study theoretically in [19] the distribution of the lengths of these trajectories, using large deviation techniques.

Nonequilibrium systems

Let us also mention that the article [22] on a derivation of a Langevin-type dynamics for a heavy particle in a non-zero background flow, co-authored by M. Dobson, F. Legoll, T. Lelièvre, and G. Stoltz, has been published.

Sampling techniques

In [29] , T. Lelièvre studies with F. Nier and G. Pavliotis (Imperial College, London) the interest of using non-reversible dynamics (overdamped Langevin dynamics with a non-gradient drift term) to efficiently sample a given Boltzmann-Gibbs distribution.

Numerical analysis of simulation methods

Together with B. Leimkuhler and Ch. Matthews (Edinburgh University), G. Stoltz studied in [48] the discretization errors in the computation of average properties with Langevin dynamics integrated with splitting strategies. The average properties are either static (average of a given observable) or dynamic (transport coefficients). The main tool used in this analysis is the expansion of the transition operator in powers of the time step, with exact integral remainders; as well as fine estimates on the resolvent of the Langevin operators, especially in the so-called overdamped limit where the friction goes to infinity. Transport coefficients are studied either through errors in Green-Kubo formulae or errors in the linear response of nonequilibrium systems.

Coarse-graining of molecular systems

G. Stoltz, in collaboration with J.-B. Maillet and G. Faure, developed in [43] a potential energy function depending on the local density of the molecular fluid. The local density is evaluated with a three dimensional Voronoi tesselation, which proves more rigorous than the standard local averages typically found in the literature. The new potential allows to describe the compressibility of mesoparticles representing several molecules in a coarse-grained description of the atomic system. The quality of the potential has been assessed by reproducing equations of state and Hugoniot curves of model energetic materials.

Thermodynamic limit

The quasicontinuum method is an approach to couple an atomistic model with a coarse-grained approximation in order to compute the states of a crystalline lattice at a reduced computational cost compared to a full atomistic simulation. In that framework, the team has addressed questions related to the finite temperature modeling of atomistic systems and derivation of coarse-grained descriptions, such as canonical averages of observables depending only on a few variables. The work from F. Legoll and X. Blanc (Université Pierre et Marie Curie) is now published [12] .

When the temperature is small, a perturbation approach can be used to compute the canonical averages of these observables depending only on a few variables, at first order with respect to temperature. The work from F. Legoll in collaboration with E. Tadmor, W. K. Kim, L. Dupuy and R. Miller on the analysis of such an approach is now also published [32] .

Hamiltonian dynamics

Constant energy averages are often computed as long time limits of time averages along a typical trajectory of the Hamiltonian dynamics. One difficulty of such a computation is the presence of several time scales in the dynamics: the frequencies of some motions are very high (e.g. for the atomistic bond vibrations), while those of other motions are much smaller. This problem has been addressed in a two-fold manner.

Fast phenomena are often only relevant through their mean effect on the slow phenomena, and their precise description is not needed. The work from M. Dobson, C. Le Bris, and F. Legoll developing integrators for Hamiltonian systems with high frequencies (derived using homogenization techniques applied to the Hamilton-Jacobi PDE associated to the Hamiltonian ODE) is now published [22] .

Another track to simulate the system for longer times is to resort to parallel computations. An algorithm in that vein is the parareal in time algorithm. The work from C. Le Bris and F. Legoll, in collaboration with X. Dai and Y. Maday, studying several variants of the original plain parareal in time algorithm, is now also published [21] .

Effective dynamics

For a given molecular system, and a given reaction coordinate $\xi :{ℝ}^n\mapsto ℝ$, the free energy completely describes the statistics of $\xi \left(X\right)$ when $X\in {ℝ}^n$ is distributed according to the Gibbs measure. On the other hand, obtaining a correct description of the dynamics along $\xi$ is complicated. In this context, S. Lahbabi and F. Legoll have studied in [8] the case when the fine-scale, reference dynamics is a kinetic Monte Carlo model with small and fast time scales, and proved a path-wise convergence to a coarse kinetic Monte Carlo model only retaining slow degrees of freedom.

Another question is how to use a coarse-grained description (involving only the slow degrees of freedom) as a predictor for the dynamics of the actual reference system, involving all degrees of freedom. Together with G. Samaey (KU Leuven), F. Legoll and T. Lelièvre have addressed this question in the parareal framework, and shown in [28] that the precise coupling between both models should be done carefully in order for the algorithm to be efficient. In that case, the algorithm converges to the reference full dynamics.