Section: New Results

Computational Statistical Physics

Participants : Giacomo Di Gesù, Thomas Hudson, Dorian Le Peutrec, Frédéric Legoll, Tony Lelièvre, Antoine Levitt, Boris Nectoux, Julien Roussel, Mathias Rousset, Gabriel Stoltz, Pierre Terrier, Pierre-André Zitt.

The work of the project-team in this area is concentrated on two new directions: the sampling of reactive trajectories (where rare events dictate the dynamics of the system), and the computation of average properties of nonequilibrium systems (which complements the more traditional field of techniques to compute free energy differences).

Sampling of reactive trajectories

Finding trajectories for which the system undergoes a significant change is a challenging task since the transition events are typically very rare. Several methods have been proposed in the physics and chemistry literature, and members of the project-team have undertaken their study in the past years.

A first class of techniques are the accelerated dynamics introduced by A. Voter (Los Alamos National Lab) and his collaborators. A short review on the mathematical analysis of these dynamics was written by T. Lelièvre, see [48] . In [23] , T. Lelièvre and F. Nier (Paris 13) analyze the low temperature asymptotics for Quasi-Stationary Distributions in a bounded domain. The objective of this analysis is to justify mathematically the validity of hyperdynamics.

Another class of techniques to compute reactive trajectories is based on splitting techniques. After the first result obtained in [12] , C.E. Bréhier, T. Lelièvre and M. Rousset pursued their analysis of the Adaptive Multilevel Splitting algorithm, which is a rare event simulation method. In [31] , a generalization of the method is proposed, and it is shown how to make the estimator unbiased in a discrete-in-time setting (which is generically the setting encountered in practice). Numerical experiments illustrate the performance of the method.

Nonequilibrium systems and non-reversible dynamics

In [38] , T. Lelièvre has studied with A. Duncan and G.A. Pavliotis nonreversible diffusion processes to sample a probability measure. It is shown that nonreversible dynamics are always better in terms of the asymptotic variance (statistical error), but the efficiency of the whole algorithm sensitively depends on the time discretization algorithm, which may induce some bias (deterministic error).

T. Lelièvre together with R. Assaraf, B. Jourdain and R. Roux, have analyzed in [27] the validity of non equilibrium molecular dynamics techniques to compute the derivative of an observable with respect to a parameter-dependent probability measure. The probability measure is defined as the stationary state of a non-reversible stochastic dynamics (in particular no analytical formula for this measure is available). Such computations are at the basis of the numerical approximation of transport coefficients in molecular dynamics.

Numerical analysis of simulation techniques

In [44] , G. Stoltz, together with A.-A. Homman (École des Ponts) and J.-B. Maillet (CEA/DAM), present new parallelizable numerical schemes for the integration of Dissipative Particle Dynamics with Energy conservation. So far, no numerical scheme was able to correctly preserve the energy over long times and give rise to small errors on average properties for moderately small timesteps, while being straightforwardly parallelizable. Two new methods are proposed, both of them straightforwardly parallelizable, and allowing to correctly preserve the total energy of the system. The accuracy and performance of these new schemes are illustrated both on equilibrium and nonequilibrium parallel simulations.

The discretization of overdamped Langevin dynamics, through schemes such as the Euler-Maruyama method, may lead to numerical methods which are unstable when the forces are non-globally Lipschitz. One way to stabilize numerical schemes is to superimpose some acceptance/rejection rule, based on a Metropolis-Hastings criterion for instance. However, rejections perturb the dynamical consistency of the resulting numerical method with the reference dynamics. G. Stoltz and M. Fathi (Berkeley) present in [40] some modifications of the standard stabilization of discretizations of overdamped Langevin dynamics by a Metropolis-Hastings procedure, which allow to either improve the strong order of the numerical method, or to decrease the bias in the estimation of transport coefficients characterizing the effective dynamical behavior of the dynamics. The latter approach relies on modified numerical schemes together with a Barker rule for the stabilization.

A. Levitt, in collaboration with C. Ortner (University of Warwick), has worked on the numerical analysis of saddle point search, an important step in the computation of reaction rates. While the convergence theory of minimization algorithms, such as the gradient method, is well-understood and standard, no such theory exists for saddle point algorithms such as the dimer method. Their work reveals a major obstruction to convergence: for some systems, the dimer method can oscillate indefinitely. This shows that there is no Lyapunov function for the associated flow, and highlights the fundamental difference between minimization and saddle search. Further work focuses on improving the reliability and convergence speed of such methods.

Free energy computations

The topic of free energy computations is still a significant research area of the project-team. T. Lelièvre has co-authored a review article [14] on the adaptive biasing force (ABF) method.

In addition, two new results have been obtained on the ABF method by H. Al Rachid (École des Ponts) in collaboration with T. Lelièvre: a numerical result concerning a projected version of the ABF algorithm, which enables to reduce the variance, see [25] ; and a theoretical result on the existence of a solution to the non linear Fokker Planck equation associated to the ABF process, see [49] .

T. Lelièvre and G. Stoltz, together with G. Fort (Télécom Paris) and B. Jourdain (École des Ponts), have studied the Self-Healing Umbrella Sampling (SHUS) method in [16] . This method is an adaptive biasing method to compute free energies on the fly by appropriately penalizing already visited regions. The convergence of the method relies on a rewriting as a stochastic approximation method with random steps, and can therefore be seen as a variation of the Wang-Landau method.

Convergence of processes

D. Le Peutrec and G. Di Gesù have studied in [37] the rate of convergence to equilibrium at low temperature of a stochastic interacting large particle system which can be seen as a spatially discrete approximation of the stochastic Allen-Cahn equation on the one-dimensional torus. Upper and lower bounds for the leading term of the associated spectral gap in the small temperature regime are proven, uniformly in the system size. It is also shown that the upper bound is sharp under a suitable control of the growth of the system size by the temperature.

The article [17] by B. Jourdain (École des Ponts), T. Lelièvre and B. Miasojedow (Warsaw) on the mean-field limit for the transient phase of the random walk Metropolis algorithm in the infinite dimension limit has been published in Annals of Applied Probability. In this article, the authors prove that the Metropolis Hastings algorithm converges to a nonlinear stochastic differential equation in the infinite dimensional limit.

Force fields and modeling

In [41] , G. Stoltz, together with G. Ferré (École des Ponts) and J.B Maillet (CEA/DAM), has presented a distance between atomic configurations, which is invariant with respect to permutations of the atoms. This distance is defined through a functional representation of atomic positions. It allows to directly compare different atomic environments with an arbitrary number of particles without going through a space of reduced dimensionality (i.e. fingerprints) as an intermediate step. Moreover, this distance is naturally invariant through permutations of atoms and through global rotations. This distance provides an important building block for the construction of accurate force-fields using machine learning techniques.

E. Cancès has contributed to the development of more efficient algorithms for polarizable force field molecular dynamics, which have been implemented and successfully tested on massively parallel computers [18] .

During the post-doctoral position of I.G. Tejada, G. Stoltz, F. Legoll and E. Cancès studied in collaboration with L. Brochard (École des Ponts) the derivation of a concurrent coupling technique to model fractures at the atomistic level by combining a reactive potential with a harmonic approximation; see [50] .