Section: New Results
Computational Statistical Physics
Participants : Thomas Hudson, Frédéric Legoll, Tony Lelièvre, Mathias Rousset, Gabriel Stoltz.
The work of the 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 completes the more traditional field of expertise associated with 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 team have undertaken their study in the past years.
A prominent example is the parallel replica method where several replicas of the system evolve on different processors, until one of them undergoes a transition. Several extensions and refinements to the original method were proposed by T. Lelièvre:

together with D. Aristoff and G. Simpson, he proposed in [7] an adaptation of the Parallel Replica method for Markov chains;

together with A. Binder and G. Simpson, he introduced in [17] a generalized parallel replica dynamics. The idea is to extend the applicability of the original algorithm by computing on the fly the socalled decorrelation time.
Another class of techniques to compute reactive trajectories is based on splitting techniques. C.E. Bréhier, T. Lelièvre and M. Rousset have performed in [21] an analysis of the Adaptive Multilevel Splitting algorithm, which is a rare event simulation method where several replicas are evolved concurrently, and selected to favor exploration in a given direction. The computational cost of the algorithm is studied in details in the limit of a large number of replicas.
Nonequilibrium systems
G. Stoltz, together with G. Pavliotis (Imperial College) and Rémi Joubaud, studied in [27] the response of equilibrium systems evolving according to a Langevin dynamics, to external, spacetime dependent forcings. In particular, they found out that, even if the external forcing is periodic in time and space with a vanishing spacetime average, the systems in general evolves with a nonzero average velocity. It may even be the case that the average velocity is in the direction opposite to the average forcing (when the latter is nonzero), which can be seen as an example of negative mobility. The behavior of the system over diffusive time scales (in the reference frame obtained by removing the average velocity) is also studied, for arbitrary forcing strengths. This work was initiated when G. Pavliotis was a visiting member of the team MATHERIALS.
A numerical analysis of the error arising in the computation of transport coefficients, with an emphasis on mobility and selfdiffusion, was provided by M. Fathi, A.A Homman and G. Stoltz in [25] in the case when MetropolisHastings algorithms are used to stabilize straightforward discretizations of overdamped Langevin dynamics.
Together with Herbert Spohn (TU München), G. Stoltz has verified the relevance of modecoupling predictions for the scaling of space time correlations of invariants for one dimensional systems subjected to a nonreversible deterministic dynamics perturbed by an exchange noise [32] . In particular, it has been confirmed that the equilibrium relaxation of the invariants involves two modes, a traveling sound mode and a standing heat mode (related to the energy current and height autocorrelation functions). Both modes exhibit a superdiffusive scaling, of Lévy type for the heat mode, and of KPZ type for the sound mode.
Free energy computations
The topic of free energy computations is still a significant research area of the team. T. Lelièvre and G. Stoltz, together with G. Fort and B. Jourdain, studied the SelfHealing Umbrella Sampling (SHUS) method in [26] . 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 WangLandau method. The efficiency of the SHUS algorithm was assessed for a model twodimensional system in terms of exit times out of a metastable set.
Concerning practical applications, G. Stoltz, together with A.A. Homman, E. Bourasseau, P. Malfreyt, L. Strafella and A. Ghoufi have worked on the computation of surface tension in droplets [10] , using alchemical transformations where the droplet volume is artificially varied.
Finally, T. Lelièvre, together with J. Comer, J.C. Gumbart, J. Hénin, A. Pohorille and C. Chipot, wrote a review article on the adaptive biasing force method [9] .
Thermodynamic limit
Another work in progress is related to the understanding of the origin of hysteresis in rubbermade materials. When submitted to cyclic deformations, the strainstress curve of these materials indeed shows a hysteresis behavior, which seems to be independent of the speed of loading.
Some years ago, members of the team have suggested a model, at a mesoscale, to explain this behavior. This model was written in terms of a system made of a finite number of particles. One of the aim of the postdoc of Thomas Hudson, who joined the team in Sept. 2014, is to make progress on that question, and to understand whether a thermodynamic limit of the model previously proposed can be identified.
Reduced models
We propose in [13] a procedure for replacing a complex, reactive potential of REBO type by a simple harmonic approximation, in regions where the system is close to equilibrium. The parameters of the harmonic approximation are chosen so that the phonon spectrum is exactly reproduced. We are currently testing the ability of the soobtained hybrid model to predict the fracture of graphene.