# Workshop Programme

## for period 29 October - 1 November 2013

### Non-equilibrium Statistical Mechanics and the Theory of Extreme Events in Earth Science

29 October - 1 November 2013

Timetable

 Tuesday 29 October 08:30-08:50 Registration 08:50-09:00 Welcome by John Toland, Director of the Institute 09:00-09:35 Pollicott, M (University of Warwick) Geodesic flows: Mixing, zeta functions and resonances Sem 1 Historically important examples of chaotic'' dynamical systems are Anosov flows, in particular, and geodesic flows on negatively curved manifolds. In particular, they provide a concrete setting to explore a wealth of interesting topics: (i) mixing rates (which can be studied using zeta function and resonances); (ii) large deviations and fluctuation theorems (Gallavotti-Cohen theorem in non-equilibrium statistical mechanics); and (iii) escape rates (the rate at which mass escapes from an open system) and extremes. 09:35-10:10 Galatolo, S (Università di Pisa) Rigorous computation of invariant measures and Lyapunov exponents Sem 1 Co-author: Isaia Nisoli (Universitade Federal Rio de Janeiro) We will consider the problem of computation of invariant measures and other aspects related to the statistical behavior of the dynamics up to certified errors. In this way the output of a computation represent some rigorous quantitative estimation on the behavior of the dynamics under study, going towards more reliable tools for the simulation of dynamical models. We will show some general approach which can be applied in several cases of systems having some hyperbolic behavior, including maps with indifferent fixed points. Time permitting we will also consider a class piecewise hyperbolic maps related to the Lorenz attractor. 10:10-10:45 Imkeller, P (Humboldt-Universität zu Berlin) Paleo-climatic time series: statistics and dynamics Sem 1 Co-authors: Arnaud Debussche (ENS Cachan), Jan Gairing (HU Berlin), Claudia Hein (HU Berlin), Michael Högele (U Potsdam), Ilya Pavlyukevich (U Jena) Dynamical systems of the reaction-diffusion type with small noise have been instrumental to explain basic features of the dynamics of paleo-climate data. For instance, a spectral analysis of Greenland ice time series performed at the end of the 1990s representing average temperatures during the last ice age suggest an $\alpha-$stable noise component with an $\alpha\sim 1.75.$ We model the time series as a dynamical system perturbed by $\alpha$-stable noise, and develop an efficient testing method for the best fitting $\alpha$. The method is based on the observed $p$-variation of the residuals of the time series, and their asymptotic $\frac{\alpha}{p}$-stability established in local limit theorems.\par\smallskip Generalizing the solution of this model selection problem, we are led to a class of reaction-diffusion equations with additive $\alpha$-stable L\'evy noise, a stochastic perturbation of the Chafee-Infante equation. We study exit and transition between meta-stable states of their solutions. Due to the heavy-tail nature of an $\alpha$-stable noise component, the results differ strongly from the well known case of purely Gaussian perturbations. 10:45-11:10 Morning Coffee 11:10-11:45 Kuna, T (University of Reading) Typical behaviour of extremes of chaotic dynamical systems for general observables Sem 1 In this talk we discuss the distribution of extreme events for a chaotic dynamical system for a general class of observables. More precisely, we link directly the distribution of events over threshold to the local geometrical structure on the surface of the attractor. It is shown how this can provide us with information about the local stable and unstable dimensions. Using Ruelle's response theory, we discuss the sensitivity of the parameters of the distribution under perturbations. This is a joint work with Vlaerio Lucarini, Davide Faranda and Jeroen Wouters. 11:45-12:20 Wang, S (Indiana University) Interplay between Mathematics and Physics Sem 1 Co-author: Tian Ma (Sichuan University) In this talk, we shall present three first principles and a few examples, demonstrating the symbiotic interplay between theoretical physics and advanced mathematics. We start with a general principle that dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic and random. We shall illustrate this principle with a few examples in both equilibrium and non-equilibrium phase transitions, including the metastable oscillation mechanism of the El Nino Southern Oscillation (ENSO) and the existence of 3rd-order transitions beyond the Andrews critical point. Then we present two basic principles: the principle of interaction dynamics (PID) and the principle of representation invariance (PRI), to study the nature's fundamental interactions/forces. Intuitively, PID takes the variation of the action functional under energy-momentum conservation constraint. PRI requires that physical laws be independent of representations of the gauge groups. These two principles give rise to a unified field model for four interactions, which can be naturally decoupled to study individual interactions. With PID, for example, we derive new gravitational field equations with a vector field $\Phi_\mu$, which can be considered as a spin-1 massless bosonic particle field. The field equations induce a natural duality between the graviton (spin-2 massless bosonic particle) and this spin-1 massless bosonic particle, leading to a unified theory for dark matter and dark energy. In addition, the PID offers a completely different and much simpler way of introducing Higgs fields. 12:30-13:30 Lunch at Wolfson Court 13:40-14:15 Branicki, M (New York University) Quantifying uncertainty and improving reduced-order predictions of partially observed turbulent dynamical systems Sem 1 Co-author: A. J. Majda (Courant Institute, NYU) The issue of mitigating model error in reduced-order prediction of high-dimensional dynamics is particularly important when dealing with turbulent geophysical systems with rough energy spectra and intermittency near the resolution cut-off of the corresponding numerical models. I will discuss a new framework which allows for information-theoretic quantification of uncertainty and mitigation of model error in imperfect stochastic/statistical predictions of non-Gaussian, multi-scale dynamics. In particular, I will outline the utility of this framework in derivation of a sufficient condition for improving imperfect predictions via a popular but heuristic Multi Model Ensemble approach. Time permitting, the role and validity of 'fluctuation-dissipation' arguments for improving imperfect predictions of externally perturbed non-autonomous turbulent systems will also be addressed. 14:15-14:50 Vaienti, S (Centre de Physique Théorique, Marseille) Extreme value theory for randomly perturbed systems: getting the local dimensions Sem 1 We present some new results for extreme values distributions in dynamical systems perturbed "via" random transformations and with observational noise. In both cases the linear scaling parameters of the Gumbel law will allow to get informations on the local behavior respectively of the stationary measure (random transformations), and of the invariant measure (observational noise). This collects work done with Aytac, Faranda, Freitas, Lucarini and Turchetti. 14:50-15:25 Faranda, D (CEA/Saclay) A new recurrences based technique for detecting robust extrema in long temperature records Sem 1 Co-author: Sandro Vaienti (University of Marseille) By using new techniques originally developed for the analysis of extreme values of dynamical systems, several long records of temperatures at different locations are analysed by showing that they have the same recurrence time statistics of a chaotic dynamical system perturbed with dynamical noise and by instrument errors. The technique provides a criterion to discriminate whether the recurrence of a certain temperature belongs to the natural climate variability or can be considered as a real extreme event with respect to a specific time scale fixed as parameter. The method gives a self-consistent estimation of the convergence. 15:25-15:50 Afternoon Tea 15:50-16:25 Didenkulova, I (Tallinn Technical University) Extreme sea waves in the coastal zone Sem 1 16:25-17:00 Vollmer, J (Max-Planck-Institut fur Dynamics and Self-Organisation) Dew droplets and cloud droplets: droplet growth, size distributions, and corrections to scaling Sem 1 I present the results of comprehensive laboratory experiments and numerical studies addressing droplet growth and droplet size distributions in systems where droplets grow due to sustained supersaturation of their environment. Both, for droplets condensing on a substrate (like dew) and droplets entrained in an external flow (like in clouds), we observe remarkable shortcomings of classical scaling theories addressing these growth processes. The origins of the discrepancies are identified, and appropriate extensions of the theories are discussed. 17:00-18:00 Welcome Wine Reception
 Wednesday 30 October 09:00-09:35 Jona-Lasinio, G (Università degli Studi di Roma La Sapienza) On thermodynamics of stationary states of diffusive systems Coauthors L. Bertini, A. De Sole, D. Gabrielli, C. Landim Sem 1 Thermodynamic transformations connecting nonequilibrium stationary states have the peculiarity of dissipating, to keep the system out of equilibrium, an amount of energy which diverges for a quasi static transformation. By subtracting the divergent part one can define a renormalized work that satisfies a Clausius type inequality and with respect to which quasi static transformations are optimal. A different way of analyzing the energy balance and optimality criteria is to consider transformations over a long but finite time T developing the total work and the dissipated energy in powers of 1/T. The diverging terms cancel and one obtains relations among finite quantities. 09:35-10:10 Beck, C (Queen Mary, University of London) Environmental superstatistics Sem 1 Complex systems in driven nonequilibrium situations often consist of a superposition of several dynamics on well-separated time scales. Sometimes the parameters of the system fluctuate as well, on a much larger time scale than the local dynamics. The resulting marginal distributions typically have fat tails, which can be understood by superstatistical techniques. After a short review of the field I will concentrate on some examples relevant for planet earth: The dynamics of tracer particles in turbulent flows, the surface temperature statistics at various locations on planet earth, and the dynamics of sea levels. 10:10-10:45 Kondrashov, D (University of California, Los Angeles) Data-driven model reduction and climate prediction: nonlinear stochastic, energy-conserving models with memory effects Sem 1 Co-authors: Mickael D. Chekroun (University of California, Los Angeles), Michael Ghil (University of California, Los Angeles) This talk will focus on theoretical understanding and climate applications of a data-driven reduction strategy that leads to low-order stochastic-dynamical models with energy-conserving nonlinearities and conveying memory effects. New opportunities for climate prediction will be illustrated in the framework of "Past Noise Forecasting", by utilizing on the one hand estimated history of the driving noise by the low-order model, and on the other hand the phase of low-frequency variability estimated by advanced time series analysis. 10:45-11:10 Morning Coffee 11:10-11:45 Crommelin, DTC (Centrum voor Wiskunde en Informatica (CWI)) Efficient sampling of rare events by splitting Sem 1 Standard (or crude) Monte Carlo (MC) simulation is known to be inefficient for simulating rare events. For events with low probability, the squared relative error on estimates obtained from straightforward MC simulation is inversely proportional to the number of samples, so that an excessively large number of samples may be required to reach a desired accuracy for the estimation of rare event probabilities. To improve the efficiency of MC sampling for rare events, various techniques have been developed in the past, for applications in e.g. communication networks and reliability analysis. Such techniques can be of interest for studying extremes in geophysical models. I will discuss a technique called multilevel splitting, in which model sample paths are split into multiple copies each time they cross thresholds (or levels) that lead closer to the rare event set. 11:45-12:20 Kwasniok, F (University of Exeter) Regime-dependent modelling of extremes in the extra-tropical atmospheric circulation Sem 1 The talk discusses data-based statistical-dynamical modelling of vorticity and wind speed extremes in the extra-tropical atmospheric circulation. The extreme model is conditional on the large-scale flow, consisting of a collection of local generalised Pareto distributions, each associated with a cluster or regime in the space of large-scale flow variables. The clusters and the parameters of the extreme models are estimated from data, either separately or simultaneously. The large-scale flow is represented by the leading empirical orthogonal functions (EOFs). Also temporal clustering of extremes in the different large-scale regimes is investigated using an inhomogeneous Poisson process model whose rate parameter is conditional on the large-scale flow. The study is performed in the framework of an intermediate complexity atmospheric model with realistic mean state, variability and teleconnection patterns. The methodology can also be applied to data from GCM scenario simulations, predicting future extremes. 12:30-13:30 Lunch at Wolfson Court 13:40-14:15 Maes, C (Katholieke Universiteit Leuven) The modified second fluctuation-dissipation theorem Sem 1 Baths produce friction and random forcing on particles suspended in them. The relation between noise and friction in (generalized) Langevin equations is usually referred to as the second fluctuation-dissipation theorem. We show, beyond formalities, what is the proper nonequilibrium extension, to be applied when the environment is itself active and driven. 14:15-14:50 Colangeli, M (Politecnico di Torino) On the use of Ruelle's formalism in response theory Sem 1 We use Ruelle’s formalism to express the response of a generic observable to a certain perturbation in terms of correlation functions computed with respect to the unperturbed invariant measure, for deterministic as well as stochastic dynamics. We discuss the onset of two relevant terms for the entropy production, comment on the Hamiltonian version of the resulting formulae and also propose a connection with similar results, reported in the literature, allowing to extend the Fluctuation-Dissipation formalism to nonequilibrium steady states. 14:50-15:25 Klages, R (Queen Mary, University of London) Anomalous fluctuation relations Sem 1 Co-authors: Aleksei V. Chechkin (Institute for Theoretical Physics NSC KIPT, Kharkov, Ukraine), Peter Dieterich (Institut fuer Physiologie, Medizinische Fakultaet Carl Gustav Carus, Dresden, Germany), Friedrich Lenz (Queen Mary University of London, School of Mathematical Sciences, London, UK) We study Fluctuation Relations (FRs) for Gaussian stochastic systems that are anomalous, in the sense that the diffusive properties strongly deviate from the ones of Brownian motion. For this purpose we use a Langevin approach: We first briefly review the concept of transient work FRs for simple Langevin dynamics generating normal diffusion [1]. We then consider two different types of additive, power law correlated Gaussian noise [2,3]: (1) internal noise with a fluctuation-dissipation relation of the second type (FDR2), and (2) external noise without FDR2. For internal noise we find that FDR2 leads to conventional (normal) forms of transient work FRs. For external noise we obtain various forms of violations of normal FRs, which we call anomalous FRs. We show that our theory is important for understanding experimental results on fluctuations in systems with long-time correlations, such as glassy dynamics [1]. [1] R.Klages, A.V.Chechkin, P.Dieterich, Anomalous fluctuation relations, book chapter in: R.Klages, W.Just, C.Jarzynski (Eds.), Nonequilibrium Statistical Physics of Small Systems, Wiley-VCH, Weinheim (2013) [2] A.V.Chechkin, F.Lenz, R.Klages, J.Stat.Mech. L11001 (2012) [3] A.V.Chechkin, R.Klages, J.Stat.Mech. L03002 (2009) 15:25-15:55 Afternoon Tea 15:55-16:30 Vanneste, J (University of Edinburgh) A large-deviation approach to passive scalar advection, diffusion and reaction Sem 1 Co-authors: Peter H. Haynes (University of Cambridge), Alexandra Tzella (University of Birmingham) The dispersion of a passive scalar in a fluid through the combined action of advection and molecular diffusion can often be described as a diffusive process, with an effective diffusivity that is enhanced compared to the molecular value. This description fails to capture the tails of the scalar concentration in initial-value problems, however. This talk addresses this issue and shows how the theory of large deviation can be applied to capture the concentration tails by solving a family of eigenvalue problems. Two types of flows are considered: shear flows and cellular flows. In both cases, large deviation is shown to generalise classical results (Taylor dispersion for shear flows, homogenisation results for cellular flows). Explicit asymptotic results are obtained in the limit of large Péclet number corresponding to small molecular diffusivity. The implications of the results for the problem of front propagation in reacting flows are also discussed. 19:30-22:00 Conference Dinner at Emmanuel College
 Friday 1 November 09:00-09:35 Ruelle, DP (IHES) Hydrodynamic turbulence as a problem in non-equilibrium statistical mechanics Sem 1 The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems which describe units of fluid of smaller and smaller spatial extent. These units are macroscopic but have few degrees of freedom, and can be studied by the methods of (microscopic) non-equilibrium statistical mechanics. The fluctuations predicted by statistical mechanics correspond to the intermittency observed in turbulent flows. Specifically, we obtain the formula $$\zeta_p={p\over3}-{1\over\ln\kappa}\ln\Gamma({p\over3}+1)$$ for the exponents of the structure functions ($\langle|\Delta_rv|^p\rangle\sim r^{\zeta_p}$). The meaning of the adjustable parameter $\kappa$ is that when an eddy of size $r$ has decayed to eddies of size $r/\kappa$ their energies have a thermal distribution. The above formula, with $(\ln\kappa)^{-1}=.32\pm.01$ is in good agreement with experimental data. This lends support to our physical picture of turbulence, a picture which can thus also be used in related problems. 09:35-10:10 Buttazzo, GM (Università di Pisa) Optimal location problems with routing cost Sem 1 Co-authors: Serena Guarino (University of Pisa (Italy)), Fabrizio Oliviero (University of Pisa (Italy)) A model problem for the location of a given number $N$ of points in a given region $\Omega$ and with a given resources density $\rho(x)$ is considered. The main difference between the usual location problems and the present one is that in addition to the location cost an extra {\it routing cost} is considered, that takes into account the fact that the resources have to travel between the locations on a point-to-point basis. The limit problem as $N\to\infty$ is characterized and some applications to airfreight systems are shown. 10:10-10:45 Bouchet, F (ENS - Lyon) Phase transitions and large deviations in geophysical fluid dynamics Sem 1 Geophysical turbulent flows (atmosphere and climate dynamics, the Earth core dynamics) often undergo very rapid transitions. Those abrupt transitions change drastically the nature of the flow and are of paramount importance, for instance in climate. By contrast with most theoretical models of phase transitions, for turbulent flows it is difficult to characterize clearly the attractors (they are not simple fixed points of a deterministic dynamics or statistical equilibrium states) and the trajectories that lead to transitions from one attractor to the others. I will review recent researches in this subject, including experimental and numerical studies of turbulent flows. Most of the talk will focus on theoretical works in the framework of the 2D stochastic quasi-geostrophic Navier-Stokes equations, the quasi-geostrophic equations, and the stochastic Vlasov equations. We will discuss predictions of phase transitions, validity of large deviation results of the Freidlin-Wentzell type, or more involved approaches when the Freidlin-Wentzell approach is not valid. The results involve several works that have been done in collaborations with J. Laurie, M. Mathur, C. Nardini, E. Simonnet, J. Sommeria, T. Tangarife, H. Touchette, and O. Zaboronski. 10:45-11:10 Morning Coffee 11:10-11:45 Eckhardt, B (Philipps-Universität Marburg) Turbulence transition in shear flows: coherent structures, edge states and all that Sem 1 Pipe flow, plane Couette flow and several other shear flows show a transition to turbulence for flow rates where the linear profile is still stable. The turbulent dynamics is transient, so that the transition is related to the formation of a chaotic saddle in the state space of the system. The saddle is supported by exact coherent states and their heteroclinic connections. I will summarize the common features that appear across all these shear flows, sketch the numerical techniques used to identify and track the relevant structures in the state space of the system and point out possible applications beyond fluid mechanics. 11:45-12:20 Lucarini, V (Universität Hamburg/University of Reading) Noise, Fluctuation, and Response in Geophysical Fluid Dynamics Sem 1 Response theory provides formidable methods for addressing many problems in statistical mechanics. Recently, it has been proposed as a gateway for various challenges in geophysical fluid dynamics, such as the provision of a rigorous conceptual framework for computing climate response to a variety of forcings and for deriving effective equations for coarse-grained variables, thus paving the way for constructing accurate parametrization of unresolved processes in numerical models. In this contribution, we first would like to present some new results showing how one can use response theory to compute the impact of adding stochastic forcing to deterministic chaotic systems. Then, we will discuss the applicability of the fluctuation-dissipation theorem in the context of non-equilibrium systems, focusing on the role played by the choice of observable. Finally, we will present some applications of response theory in geophysical fluid dynamical systems, ranging from low-order models such as the Lorenz 63 and Lorenz 96 models to General Circulation Models of the atmosphere. 12:20-12:55 Vanden-Eijnden, E (Courant Institute of Mathematical Sciences) Nonequilibrium statistical mechanics of climate variability: modelling issues and applications to data assimilation techniques Sem 1 Stochastic models and computational tools for the study of transitions between different metastable states (or regimes) in climate system are discussed using the barotropic quasi-geostrophic (QG) equation as a test case. Specifically, a stochastic partial differential equation (SPDE) is obtained by adding appropriate forcing and damping terms to the QG equation to make this equation dynamically consistent with the predictions of equilibrium statistical mechanics, while allowing to study nonequilibrium phenomena such as transitions between different regimes. In the small noise regime, the most likely states of the invariant measure for this SPDE coincide with the selective decay states and we establish conditions under which these states are not unique, implying the existence of different climate regimes. We also analyze the mechanism and rate of the dynamical transitions between these regimes by computing the most likely paths connecting them. Finally we will discuss how the se results can be used in the context of data assimilation procedure based on Kalman or ensemble filters to improve the efficiency of these methods in the presence of regime shifts. 12:55-13:30 Lunch at Wolfson Court