## Final Report Summary - TRANSITION (Large Deviations and Non Equilibrium Phase Transitions for Turbulent Flows, Climate, and the Solar System)

Rare events can be essential because of their impact, for instance, extreme events like hurricanes or heat waves in climate dynamics, or the possible destabilization of the solar system. Other rare events can be important because they completely change the state of a system, for instance abrupt transitions between different climates. Such rare events are hardly understood and poorly studied because of the difficulty to study them in complex dynamics and because of a lack of a proper mathematical framework. The TRANSITION project developed mathematical, theoretical and computational tools in order to compute rare events in complex non-equilibrium dynamical systems, and applied them to phenomena in several fields: climate, turbulence and the large time dynamics of the solar system.

One of the main difficulties for studying extreme events is their rarity. In order to cope with this problem, we have extended and developed algorithms that produce several hundreds or thousands more extreme events, together with their dynamics, and still estimate properly their probabilities. We have demonstrated that those algorithms can be applied to dynamics as complex as turbulent flows and climate dynamics, and open the way for studies that would have been impossible otherwise. Using this approach, we were able to study for the first-time abrupt climate changes for Jupiter’s troposphere jets, demonstrate that transition in turbulent flows unexpectedly follow a phenomenology similar to nucleation with Arrhenius law, compute the dynamics and probability of extreme drags on objects embedded in turbulent flows with foreseen applications in engineering. Application of this methodology are foreseen for the study of abrupt changes of the Earth climate. We expect this approach will change the paradigm for studying climate extremes, for instance heat waves, hurricanes or storms, in the near future.

A key property of rare events is that their dynamics is sometimes predictable, akin to instantons in statistical physics, opening the door for precursor studies. We have demonstrated this phenomenology in transitions in turbulent flows. We have also demonstrated that the destabilization of the solar system through a Mercury-Jupiter resonance over a few hundred-million-year time scale, also follows closely an instanton, in a simplified model of the solar system.

Those application results have been made possible thanks to advances in the mathematical study of extremes. For instance, we have developed a generalization to non-equilibrium dynamics of the Eyring--Kramers formula, that describes the transition rate of rare transitions between metastable states. We have also developed a theory for computing non-equilibrium quasipotentials (or free energies) for complex systems, like for example particle systems with long range interactions.

One of the main difficulties for studying extreme events is their rarity. In order to cope with this problem, we have extended and developed algorithms that produce several hundreds or thousands more extreme events, together with their dynamics, and still estimate properly their probabilities. We have demonstrated that those algorithms can be applied to dynamics as complex as turbulent flows and climate dynamics, and open the way for studies that would have been impossible otherwise. Using this approach, we were able to study for the first-time abrupt climate changes for Jupiter’s troposphere jets, demonstrate that transition in turbulent flows unexpectedly follow a phenomenology similar to nucleation with Arrhenius law, compute the dynamics and probability of extreme drags on objects embedded in turbulent flows with foreseen applications in engineering. Application of this methodology are foreseen for the study of abrupt changes of the Earth climate. We expect this approach will change the paradigm for studying climate extremes, for instance heat waves, hurricanes or storms, in the near future.

A key property of rare events is that their dynamics is sometimes predictable, akin to instantons in statistical physics, opening the door for precursor studies. We have demonstrated this phenomenology in transitions in turbulent flows. We have also demonstrated that the destabilization of the solar system through a Mercury-Jupiter resonance over a few hundred-million-year time scale, also follows closely an instanton, in a simplified model of the solar system.

Those application results have been made possible thanks to advances in the mathematical study of extremes. For instance, we have developed a generalization to non-equilibrium dynamics of the Eyring--Kramers formula, that describes the transition rate of rare transitions between metastable states. We have also developed a theory for computing non-equilibrium quasipotentials (or free energies) for complex systems, like for example particle systems with long range interactions.