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## Computation of transition trajectories and rare events in non-equilibrium systems

#### Centre Blaise Pascal, ENS Lyon, France

#### Organisers

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Dynamics of turbulent flows, chemistry of complex molecules, biochemistry, and biophysics often involve a huge number of degrees of freedom, and a large spectrum of temporal and spatial scales. In the dynamics of those systems, microscopic low-intensity disorganized motion can lead to events which are extremely rare for the microscopic dynamics, but are essential for understanding the macroscopic properties. Classical examples are activation processes leading to the transition between two very different states (e.g., conformational change, which is typical in chemistry, polymer physics, biomolecules), or situations of phase transitions in magnetic systems or in solid state physics. The present workshop focuses on anologous phenomena in non-equilibrium-systems. For instance, much less studied examples of rare trajectories leading to drastic global changes are the transitions between two or more attractors in turbulent flows (e.g., polarity reversal for the Earth's magnetic field, weather regimes in meteorology, the Kuroshio current's bistability, or bistability situations in turbulent convection).

Whatever the system studied, such rare events have in common that they appear with very low frequencies compared to the temporal scale at which simulations are usually carried out. Observing them in a numerical simulation using, e.g., molecular dynamics or direct numerical simulation in turbulence, is thus extremely difficult and simply impossible in many instances, such as the study of the Earth's magnetic polarity change in a genuine turbulent regime.

The numerical computation of transitions trajectories has been developed for a long time. One of the drawbacks of classical approaches (e.g., transition path sampling, equilibrium Monte Carlo algorithm, string methods) is that they mainly deal with statistical equilibrium problems. By contrast many applications require methods dealing with non-equilibrium situations (chemistry, biology, turbulence). For instance, in the field of hydrodynamics and turbulence, even if rare events have been studied for a very long time, there exists no specific numerical algorithm that can be used to compute them, and the recent interest for those approaches is related to the possibility to deal with non-equilibrium problems.

An essential originality of our workshop is to gather people who recently developed approaches that can be applied to non-equilibrium problems as well as equilibrium problems (e.g., multilevel splitting, numerical computation of instantons using Freidlin-Wentzell's large deviation theory, Giardina-Tailleur-Kurchan algorithm). These new tools have mainly been developed within the statistical mechanics community, sometimes in relation to applications, sometimes in order to address theoretical issues. Some of the recent non-equilibrium methods (e.g., the Fokker-Planck and thermodynamics of histories approaches) have also been developed in relation with applications (e.g., glassy behavior, phase transitions).

The efficiency of some of these non-equilibrium algorithms have recently been studied in a precise way by mathematicians to delineate their limitations and use them in an optimal way.

The scientific aim of this workshop is gather people that recently devised dedicated algorithms that allow the computation of transition trajectories and rare events, with an emphasize on applications to non-equilibrium problems.

Whereas specific specialized conferences or workshops have been organized recently in some of the interested communities, the novelty of our program is to gather different communities:

i. theoretical statistical mechanics and mathematics,

ii. chemistry, polymer physics, biocomputation,

iii. turbulence and hydrodynamics,

iv. mathematics.

The specific focus of the workshop is to deal with methodologies and algorithms. This should enable ideas to propagate beyond the original community that promoted them, and favor the emergence of novel concepts, by getting these communities to meet and interact. Since some of the ideas are currently developing fast in some of the communities, but do not exist yet in others, the organization of this program now seems especially appropriate.

Some of the phenomena of interest are currently studied in a number of experimental groups (there currently exists an unprecedented interest in turbulence, and a lasting interest in biophysics and chemistry).

This meeting is expected to have a two-stage program. In the first part, we plan to organize a two-day introductory school at which specialists of some of the numerical methods mentioned earlier will present their work to a large audience of researchers, post-docs and PhD students. There will follow a three-day workshop, at which researchers will present up-to-date results in order to favor discussions aimed at developing new methods. The aim of this application to CECAM is to fund the three-day workshop only. Complementary funds have already been found through other applications (ANR, IXXI, GDR) or have been applied for (ENS-Lyon, CNRS) in order to organize the introductory school.

Dynamics of turbulent flows, chemistry of complex molecules, biochemistry, and biophysics often involve a huge number of degrees of freedom and a large spectrum of temporal and spatial scales. In the dynamics of these systems, microscopic low-intensity disorganized motion can lead to events which are extremely rare for the microscopic dynamics, but are essential for understanding the macroscopic properties. Classical examples are activation processes leading to the transition between two very different states (e.g., conformational change, which is typical in chemistry, polymer physics, biomolecules), or situations of phase transitions in magnetic systems or in solid state physics. The present workshop focuses on anologous phenomena in non-equilibrium systems. For instance, much less studied examples of rare trajectories leading to drastic global changes are the transitions between two or more attractors in turbulent flows (e.g., polarity reversal for the Earth's magnetic field, weather regimes in meteorology, the Kuroshio current's bistability, or bistability situations in turbulent convection). Whatever the system studied, such rare events have in common that they appear with very low frequencies compared to the temporal scale at which simulations are usually carried out. Observing them in a numerical simulation using, e.g., molecular dynamics or direct numerical simulation in turbulence, is thus extremely difficult and simply impossible in many instances, such as the study of the Earth's magnetic polarity change in a genuine turbulent regime. The numerical computation of transition trajectories has been developed for a long time. One of the drawbacks of classical approaches (e.g., transition path sampling, equilibrium Monte Carlo algorithm, string methods) is that they mainly deal with statistical equilibrium problems. By contrast, many applications require methods dealing with non-equilibrium situations (chemistry, biology, turbulence). For instance, in the field of hydrodynamics and turbulence, even if rare events have been studied for a very long time, there exists no specific numerical algorithm that can be used to compute them, and the recent interest for those approaches is related to the possibility to deal with non-equilibrium problems.

An essential originality of our workshop is to gather people who recently developed approaches that can be applied to non-equilibrium problems as well as equilibrium problems (e.g., multilevel splitting, numerical computation of instantons using Freidlin-Wentzell's large deviation theory, Giardina-Tailleur-Kurchan algorithm). These new tools have mainly been developed within the statistical mechanics community, sometimes in relation to applications, sometimes in order to address theoretical issues. Some of the recent non-equilibrium methods (e.g., the Fokker-Planck and thermodynamics-of-histories approaches) have also been developed in relation with applications (e.g., glassy behavior, phase transitions). The efficiency of some of these non-equilibrium algorithms have recently been studied in a precise way by mathematicians to delineate their limitations and use them in an optimal way.

The scientific aim of this workshop is to gather people that recently devised dedicated algorithms that allow the computation of transition trajectories and rare events, with an emphasis on applications to non-equilibrium problems. Whereas specific specialized conferences or workshops have been organized recently in some of the interested communities, the novelty of our program is to gather different communities: i. theoretical statistical mechanics and mathematics, ii. chemistry, polymer physics, biocomputation, iii. turbulence and hydrodynamics, iv. mathematics.The specific focus of the workshop is to deal with methodologies and algorithms. This should enable ideas to propagate beyond the original community that promoted them, and favor the emergence of novel concepts, by getting these communities to meet and interact. Since some of the ideas are currently developing fast in some of the communities, but do not exist yet in others, the organization of this program now seems especially appropriate. Some of the phenomena of interest are currently studied in a number of experimental groups (there currently exists an unprecedented interest in turbulence, and a lasting interest in biophysics and chemistry).

This meeting is expected to have a two-stage program. In the first part, we plan to organize a two-day introductory school where specialists of some of the numerical methods mentioned earlier will present their work to a large audience of researchers, post-docs and PhD students. A three-day workshop will follow, where researchers will present up-to-date results in order to favor discussions aimed at developing new methods. The aim of this application to CECAM is to fund the three-day workshop only. Complementary funds have already been found through other applications (ANR, IXXI, GDR) or have been applied for (ENS de Lyon, CNRS) in order to organize the introductory school.

### Equilibrium approaches and applications in physics, chemistry, and biophysics

The computation of transition trajectories in physics, chemistry of complex molecules, biophysics has been very active over the last decades and has led to the development of several new methods. To list a few: transition path sampling (Bolhuis and col, 2002), Monte Carlo methods (J. Hu, A. Ma, and A. R. Dinner, 2006), the “string” method (E, Ren, and Vanden-Eijnden, 2002).

We plan on inviting leading scientists in those domains, focusing on people that developed numerical algorithms (e.g., P. Bolhuis, D. Chandler, C. Dellago, A. Dinner, C. Schütte, E. Vanden-Eijnden, and others). This community has been at the forefront of the development of the key ideas underlying computations of transition trajectories.

These methods have been applied to many problems in physics (e.g., self assembly in liquids and dynamical arrest in glass formers (L.O. Hedges, R.L. Jack, J.P. Garrahan, D. Chandler, 2009)), chemistry (e.g., auto ionization in water (Bolhuis and col, 2002)), polymer physics and biology (e.g., folding/unfolding of DNA, conformation changes, folding of proteins, biomolecular isomerisation), and continuously give new fascinating results in bio-chemistry and biology (see, e.g., J. Vreede, J. Juraszek, and P.G. Bolhuis 2010) .

Recent theoretical developments include the application of Jarzynski’s theorem to the calculation of reaction rate constants, the calculation of activation energies in systems with unknown reaction coordinate (J. Vreede, J. Juraszek, and P. G. Bolhuis 2010). This community also proposed a number of generalizations of these methods to non-equilibrium problems (see, e.g., A. Dickson, A. Warmflash, and A.R. Dinner 2009, J.C. Latorre, C. Hartmann, and C. Schütte 2010, C. Schütte, F. Noé, J. Lu, M. Sarich, and E. Vanden-Eijnden 2011).

### Theory and rare statistics computations in non-equilibrium statistical mechanics

Recently, a number of different methods have been devised in order to specifically deal with non-equilibrium problems. A first set of methods is based on generalizations of Monte-Carlo methods and importance sampling to non-equilibrium statistics (see, e.g., A. Dickson, A. Warmflash, and A.R. Dinner 2009). A second set, based on go-with-the-winner methods (Grassberger, 2002), uses a reproduction of trajectories of interest which keeps track of probabilities. Among these methods, multilevel splitting (T. Dean, and P. Dupuis, 2009; P. Del Moral, and J. Garnier, 2005) is one of the most versatile and amenable to mathematical analysis. A third class of methods evaluate trajectory statistics from the action provided by the Onsager-Machlup formalism, and leads to a thermodynamic theory of histories or trajectories. In the limit of small noise, the statistics of these trajectories is dominated by the minimizers of an action, which has the role of a large deviation rate function in the Freidlin-Wentzell theory (see Touchette, 2009 for a review).

Monte-Carlo methods resort to simulation for estimating quantities that are out of reach in standard analytical approaches. However, when the event to estimate has very low probability (< 10-6), but is of crucial practical significance, the classical Monte-Carlo procedure, which counts the success rate in a large number or experiments, becomes too imprecise, and hence, demands refinement. Two main methods meet this demand: importance sampling and multilevel splitting. Importance sampling is about replacing the actual probability law by an auxiliary law, which favors the rare events. Then, the ratio between the two laws is used to recover the actual probability of the rare events. The computational cost of this method is hardly larger than that of standard Monte-Carlo, but it is difficult to determine an appropriate auxiliary law. Also, the method gives the probability of a rare event only a posteriori, with no information on how the event shows up for the actual probability law.

Multilevel splitting is about keeping the actual probability law, while multiplying trajectories which come near the event of interest. This way, the information on the event is obtained: we have its probability, its history, and we can evaluate functionals related to this event. The drawbacks are however that the computational cost is larger than that of standard Monte-Carlo by a factor of the order of the logarithm of the event's probability, and that it is difficult to determine what “getting close to the event” means.

Recent applications of “go with the winners” methods in physics range from the computation of some properties of the percolation phase transition involving events as rare as 10^(-300) (D. A. Adams, L. M. Sander, and R. M. Ziff). Another interesting result has been the computation of rare quasi-integrable or extremely chaotic trajectories for Hamiltonian systems (J. Tailleur and J. Kurchan, 2007) using a variant of a method suited for the evaluation of additive variables. The same method has also been used to compute reaction trajectories in simple bistable systems (F. Cérou and col., 2011).

More recently, huge progress has also been achieved in the mathematical study of Monte-Carlo methods and multilevel splitting methods (T. Dean and P. Dupuis, 2009; P. Del Moral and J. Garnier, 2005).

In the case of a dynamical system with random forces, such as stochastic differential equations, the Onsager-Machlup formalism provides an action describing the probability of each trajectory. In the low-noise limit, a WKB-type approach leads to minimizing the action (instanton theory) to evaluate the probability of rare events. The mathematics community studied many of those methods using the large deviation theory of Freidlin and Wentzell, which provides a rigorous basis to the physicists' theory of instantons.

### Turbulence and hydrodynamics

Study of rare events is a classical and essential topic of hydrodynamics and turbulence. For instance, the study of intermittency in the statistics of turbulence is a 40-year-old subject (see U. Frisch, 1995), which influenced many other fields of physics, natural science and finance in the quest for rare events. Another example of essential rare events in turbulence are the rare transitions between multiple attractors in turbulent dynamos (Earth's magnetic field reversal, see M. Berhanu and col. 2007), turbulent convection, or hundreds of other turbulent flows (see for instance E.R. Weeks and col., 1997; F. Bouchet, and E. Simonnet, 2009) .

People from the turbulence community have developed a very good science and wonderful works analyzing empirically raw data or making phenomenological models (for instance multifractal formalism (U. Frisch, 1997), or solving the Kraichnan model for passive tracer dynamics (G. Falkovich, K. Gawedzki, and M. Vergassola, 2001). However, quite amazingly, they have never developed any specific numerical tool or algorithm to compute rare events from the known dynamics. The main reason is probably the lack of non-equilibrium methods. As discussed above, several groups are currently developing such tools.

Theoretical investigations of rare events or transition trajectories have been performed for turbulence problems. The method of action minimization within the Onsager–Machlup formalism was applied, for example, to Burger's equation (E. Balkovsky and col., 1997) and passive scalar transport (G. Falkovich, K. Gawedzki, and M. Vergassola, 2001) in order to describe rare events related to intermittency. However, obtaining theoretical results using this approach proved extremely difficult for relevant models of real turbulent flows.

The design of numerical algorithms which can solve this problem through numerical computation is currently under development. Results have been obtained recently in the context of bistability phenomena in two-dimensional turbulence governed by the 2D Navier–Stokes equations with stochastic forcing (F. Bouchet and A. Venaille, 2011). The bistability of magnetic field reversal in VKS experiments was also studied through instanton theory (F. Petrelis and S. Fauve). The use of multilevel splitting is also currently under study for turbulence problems (E. Simonnet, T. Lelièvre, and F. Cérou).

## References

**France**

Freddy Bouchet ( Ecole Normale SupĂ©riore-Lyon ) - Organiser

Frederic Cerou ( INRIA Research Center, Rennes ) - Organiser

Marianne Corvellec ( Ecole Normale SupĂ©riore-Lyon ) - Organiser

Arnaud Guyader ( University of Rennes 2 ) - Organiser

**United Kingdom**

Hugo Touchette ( Queen Mary, University of London ) - Organiser