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

**January 8-11, 2013, ****University of Reading**

## Abstracts

**Christian Beck: **

**Environmental Superstatistics****Abstract:** Complex systems in driven nonequilibrium situations often consist of a superposition of various dynamics on well-separated time scales. Often one or several parameters (such as the local inverse temperature beta) fluctuate on a rather large time scale, much larger than the relaxation time of the local system, and one can then apply so-called superstatistical techniques. The basic idea is to have a superposition of two statistics, one given by ordinary statistical mechanics and the other one given by that of the fluctuating parameter. The resulting marginal distributions typically have fat tails, which can be understood by methods borrowed from large dviation theory. The superstatistics concept has been recently applied to variety of of complex systems, in particular to the dynamics or tracer particles in turbulent flows, to traffic delay statistics, cancer survival statisics, and share price fluctuations. After a short review of the field I will decribe some new results on environmentally relevant superstatistics, dealing with typical temperature distributions observed on planet Earth, as well as rainfall/flooding statistics.

**Freddy Bouchet: Large deviations, instantons and the statistical mechanics of atmosphere jetsLarge deviations, instantons and the statistical mechanics of atmosphere jets **

**Abstract:** We will discuss the recent applications of theoretical physics and field theory approaches (large deviations, instantons, non-equilibrium free energies, stochastic averaging) in order to describe the statistical mechanics of the simplest models relevant to atmosphere dynamics. We consider the formation of large scale structures (zonal jets and vortices), in geostrophic turbulence forced by random forces. We study the limit of a time scale separation between inertial dynamics on one hand, and the effect of forces and dissipation on the other hand. We prove that stochastic averaging can be performed in this problem, which is unusual in turbulent systems. It is then possible to integrate out all fast turbulent degrees of freedom, and to get explicitly an equation that describes the slow evolution of zonal jets. This equation is a stochastic differential equation for a one dimensional field (the zonal velocity), with multiplicative noise. The average is governed by a non-linear Fokker-Planck equation. This equation describes the attractors for the dynamics (alternating zonal jets, the number of which depends on the force correlation function), and the relaxation towards those attractors. We describe regimes where the system has several attractors for the same force correlation function. Starting either from this Fokker-Planck equation or from the initial dynamics, we use path integral formalism and instanton theory in order to make explicit analytic predictions about large deviations and rare events. We discuss the computation of transition probabilities between two such attractors. This rare event, is extremely important, as the dynamics and large scale velocity field then undergo macroscopic qualitative changes. We will also discuss briefly large deviation approaches to closely related problems : the equilibrium statistical mechanics of the shallow water equations, the equilibrium statistical mechanics of axisymmetric Euler equations, and computations of non-equilibrium free-energies for systems with mean-field interactions.

**Matteo Colangeli: Fluctuation Relations and Fluctuation-Dissipation Theorems in chaotic dissipative maps **

**Abstract:** In a recent paper [Colangeli, M., Rondoni, L., Physica D 241, 681 (2011)] it was argued that the Fluctuation Relation for the phase space contraction rate could suitably be extended to irreversible dissipative systems. I will review those arguments, by discussing the properties of a simple irreversible nonequilibrium baker model. I will also consider the issue about the extension of the Fluctuation-Dissipation Theorem to dissipative deterministic dynamical systems, which enjoy a nonvanishing average phase space contraction rate. As noted by Ruelle, the statistical features of the perturbation and, in particular, of the relaxation, cannot be understood solely in terms of the unperturbed dynamics on the attractor. Nevertheless, I will show that the singular character of the steady state does not constitute a serious limitation in the case of systems with many degrees of freedom. The reason is that, in statistical mechanics, one typically deals with projected dynamics, and these are associated with regular probability distributions in the corresponding lower dimensional spaces.

**Roberto Deidda: Space-time rainfall downscaling with multifractal models **

**Abstract:** It is well know that rainfall fields display fluctuations in space and time that increase as the scale of observation decreases. Multifractal theory represents a solid base to characterize scale-invariance properties observed in rainfall fields as well as to develop downscaling models able to reproduce observed statistics. The availability of such downscaling tools allows forecasting of floods in small basins by coupling meteorological and hydrological models working on different space-time grid resolution. In this talk multifractal theory will be reviewed highlighting the most relevant aspects for rainfall downscaling (e.g., the concept of scale-invariance in rainfall fields displaying space-time self-similarity or self-affinity, the role of orography). The main results of the scale-invariance analysis of rainfall retrieved by remote sensors will be discussed. Finally the application of multifractal models for rainfall downscaling will be presented and some new ideas for ensemble verification will be argued.

**Ana Cristina Moreira de Freitas: Extremal index, hitting time statistics and periodicity **

**Abstract:** We consider discrete time dynamical systems and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory and vice versa. We exploit the connection between these two approaches both in the absence and presence of clustering. Clustering means that the occurrence of rare events has a tendency to appear concentrated in time. The strength of the clustering is quantified by the Extremal Index, which takes values between 0 and 1. We associate the existence of an Extremal Index less than 1 to the occurrence of periodic phenomena.

Jorge Miguel Milhazes de Freitas:*An extremal dichotomy for expanding maps *

**Abstract:** We consider systems for which there exists decay of correlations against $L1$ observables. Examples include expanding and piecewise expanding systems. For such systems we consider rare events consisting on the entrance into very small neighbourhoods of some chosen points $z$ on the phase space. We will see that there is a dichotomy regarding the extremal behaviour of these systems, depending on whether the point $z$ is periodic or not. Namely, we will see that if the point $z$ is periodic then we have an Extremal Index (EI) equal to 1 (which means no clustering of rare events) and the point processes counting the occurrence of rare events converge to a standard Poisson process. On the other hand, if the point $z$ is periodic we obtain an EI less than 1 (which means the occurrence of clustering) and the rare events point processes converge to a compound Poisson process.

**Petra Friederichs: A Bayesian hierarchical model for wind gust prediction **

**Abstract:** A postprocessing for wind gust forecasts given by a mesoscale limited area numerical weather prediction (NWP) model is presented. Extreme value theory constitutes the basis data level. A process layer for the parameters of a generalized extreme value distribution (GEV) is introduced using a Bayesian hierarchical model (BHM). The process parameters, on the one hand, model the spatial response surfaces of the GEV parameters as Gaussian random fields, and, on the other hand, incorporate the information of the COMSO-DE forecasts. The spatial BHM provides area wide forecasts of wind gusts in terms of a conditional GEV. It models the marginal distribution of the spatial gust process and provides not only forecasts of the conditional GEV at locations without observations, but also uncertainty information about the estimates. At this stage, the BHM ignores the conditional dependence between gusts at neighboring locations. However, an outline is given how this will be incorporated in a subsequent study.

**Davide Gabrielli: Large fluctuations of particle systems, an overview **

**Abstract:** I wil give a general presentation of some large deviations results for stochastic interacting particle systems. In particular I will discuss a joint dynamic large deviation principle for the empirical current and measure from the hydrodynamic diffusive rescaling. Then I will discuss the variational problem associated to the computation of the quasipotential. Finally I will discuss its relation with the work and a Clausius inequality for stationary non equilibrium states. Examples like the exclusion process the zero range process and the KMP process will be discussed.

**Giovanni Gallavotti:**

*Universal large fluctuations and time reversal***Abstract:** In systems in which a time reversal symmetry the anomalous fluctuations obey a probabilistic law which reflects the symmetry and can be regarded as an extension beyond the linear regime of Onsager reciprocity relations. I will discuss the fluctuation relation and comment on the possible consequences in case in which time reversal is a broken symmetry because of frictional dissipation.

**Guido Gentile:**

*Attractiveness of periodic orbits in parametrically forced systems with time-increasing friction***Abstract:** An interesting problem about dissipative periodically forced systems is how time-varying friction affects the dynamics. I will report about some numerical results and conjectures which suggest that, if the damping coefficient increases in time up to a final constant value, then the basins of attraction of the leading resonances are larger than they would have been if the coefficient had been fixed at that value since the beginning. I will briefly discuss also the possible relevance of the results for the spin-orbit model, in particular for the capture of Mercury into the 3:2 resonance. [Joint work with Michele Bartuccelli and Jonathan Deane.]

**Michael Ghil: The Complex Physics of Climate Change and Climate Sensitivity: A Grand Unification **

**Abstract:** Recent estimates of climate evolution over the coming century still differ by several degrees. This uncertainty motivates in part the work presented in this lecture. The complex physics of climate change arises from the large number of components of the climate system, as well as from the wealth of processes occurring in each of the components and across them. This complexity has given rise to countless attempts to model each component and process, as well as to two overarching approaches to apprehend the complexity as a whole: deterministically nonlinear and stochastically linear. Call them the Ed Lorenz and the Klaus Hasselmann approach, for short. We propose a 'grand unification' of these two approaches that relies on the theory of random dynamical systems (RDS). In particular, we apply this theory to the problem of climate sensitivity, and study the random attractors of nonlinear, stochastically perturbed systems, as well as the time-dependent probability densities associated with these attractors. The random attractors so obtained are visually spectacular objects that generalize the strange attractors of the Lorenz approach. Results are presented for several simple climate models, from the classical Lorenz convection model to El Niño-Southern Oscillation models. Their attractors carry probability densities with nice physical properties. Implications of these properties for climate predictability on interannual and decadal time scales are discussed. The RDS setting allows one to examine the interaction of internal climate variability with the forcing, whether natural or anthropogenic, and to provide a definition of climate sensitivity that takes into account the climate system's non-equilibrium behavior. Such a definition is of the essence in studying systematically the sensitivity of global climate models (GCMs) to the uncertainties in tens of semi-empirical parameters; it is given here in terms of the response of the appropriate probability densities to changes in the parameters and compared with numerical results for a somewhat simplified GCM. This talk presents the results of joint work with A. Bracco, M. D. Chekroun, D. Kondrashov, J. C. McWilliams, J. D. Neelin, E. Simonnet, S. Wang, and I. Zaliapin.

**Julia Gundermann: Crooks' Fluctuation Theorem for a Process on a 2D Fluid Field **

**Abstract:** We investigate the behavior of a two-dimensional inviscid and incompressible flow when pushed out of dynamical equilibrium. We use the 2D vorticity equation with spectral truncation on a rectangular domain. For sufficiently large number of degrees of freedom, the equilibrium statistics of the flow can be described through a canonical ensemble approach with two conserved quantities, energy and enstrophy. To perturb the system out of equilibrium, we change the shape of the domain according to a protocol, which changes the kinetic energy but leaves the enstrophy constant. We interpret this as doing work to the system. Evolving along a forward and its corresponding backward process, we show that the statistics of the work performed satisfies Crooks' relation $P_f(W)/P_b(-W) = e^{\beta (W-\Delta F)}$. The parameters $\Delta F$ and $\beta$ are given by the formal analogy with the canonical ensemble as the free energy difference and, respectively, the Lagrangian multiplier representing the inverse temperature $1/k_B T$. In collaboration with Jochen Br\"ocker and Holger Kantz

**Rosemary Harris: Fluctuations in stochastic systems with long-range memory **

**Abstract:**

**Mark Holland: On the convergence to extreme value distributions in non-uniformly hyperbolic dynamical systems. **

**Abstract:** (Joint with P. Rabassa and A. Sterk). We consider the problem of determining when a time series of observations on a dynamical system converges to a generalized extreme value distribution, and develop theoretical conditions that need to be checked in order to ensure this convergence takes place. The conditions to be checked are amenable to numerical studies, especially for high dimensional systems. We mention a range of examples where these conditions can be checked (or are known to hold) theoretically.

**Holger Kantz: Quantitative approaches to the statistics of extreme events in atmospheric dynamics **

**Abstract:** We discuss various statistical aspects of extreme events based on empirical and theoretical grounds. We consider specifically extreme events caused by the dynamics of the atmosphere such as wind gusts, precipitation events and large temperature anomalies. Relevant quantities are the event rate/return time as a function of event magnitude for very rare events. Whereas data analysis shows that marginal distributions of most types of extremes do not have fat tails and hence extremes are not by orders of magnitudes larger than the standard deviation, correlations in the succession of extremes render large deviation results invalid. One potential theoretical approach to the estimation of the frequency of very rare events is offered by non-equilibrium fluctuation theorems, and we present preliminary results of a study of a two-dimensional hydrodynamical flow. Finally, we discuss predictability and performance of extreme event prediction and show that conclusions depend much stronger on the choice of the performance criterion than anticipated.

**Juergen Kurths: Network of Networks and the Climate System **

**Abstract:** In collaboration with J. Donges, R. Donner, N. Malik, N. Marwan, H. Schultz and Y. Zou Network of networks is a new direction in complex systems science. One can find such networks in various fields, such as infrastructure (power grids etc.), human brain or Earth system. Basic properties and new characteristics, such as cross-degree, or cross-betweenness will be discussed. This allows us to quantify the structural role of single vertices or whole sub-networks with respect to the interaction of a pair of subnetworks on local, mesoscopic, and global topological scales.

Next, we consider an inverse problem: Is there a backbone-like structure underlying the climate system? For this we propose a method to reconstruct and analyze a complex network from data generated by a spatio-temporal dynamical system. This technique is then applied to 3-dimensional data of the climate system. We interpret different heights in the atmosphere as different networks and the whole as a network of networks. This approach enables us to uncover relations to global circulation patterns in oceans and atmosphere. The global scale view on climate networks offers promising new perspectives for detecting dynamical structures based on nonlinear physical processes in the climate system.

This concept is applied to Indian Monsoon data in order to characterize the regional occurrence of strong rain events and its impact on predictability.

References

Arenas, A., A. Diaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Phys. Reports 2008, 469, 93.

Donges, J., Y. Zou, N. Marwan, and J. Kurths, Europhys. Lett. 2009, 87, 48007.

Donner, R., Y. Zou, J. Donges, N. Marwan, and J. Kurths, Phys. Rev. E 2010, 81, 015101(R ).

Mokhov, I. I., D. A. Smirnov, P. I. Nakonechny, S. S. Kozlenko, E. P. Seleznev, and J. Kurths, Geophys. Res. Lett. 2011, 38, L00F04.

Malik, N., B. Bookhagen, N. Marwan, and J. Kurths, Climate Dynamics, 2012, 39, 971.

Donges, J., H. Schultz, N. Marwan, Y. Zou, J. Kurths, Eur. J. Phys. B 2011, 84, 635-651.

Donges, J., R. Donner, M. Trauth, N. Marwan, H.J. Schellnhuber, and J. Kurths, PNAS 2011, 108, 20422-20427.

Runge, J. , J. Heitzig, V. Petoukhov, J. Kurths, Phys. Rev. Lett. 2012, 108, 258701.

**Frank Kwasniok: Regime-dependent modelling of extremes in the extra-tropical atmospheric circulation **

**Abstract:** 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 extreme value or 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 simultaneously from data. 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 dynamical framework of a three-level quasigeostrophic atmospheric model with realistic mean state, variability and teleconnection patterns. The methodology can also be applied to data from GCM simulations, predicting future extremes.

**Valerio Lucarini: Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems **

**Abstract:** In this paper we derive by direct calculation the statistical properties of extremes of general observables for chaotic system, taking as mathematical framework that given by Axiom A dynamical systems possessing an invariant SRB measure. We prove that the extremes of so-called physical observables are distributed according to the classical Generalised Pareto Distribution (GPD) and derive explicit expressions for the scaling parameter $\sigma$ and the shape parameter $\xi$. In particular, we derive that $\xi$ has a universal expression which does not depend on the chosen observables, with $\xi=-1/(d_s+d_u/2+d_n/2)$, where $d_s$, $d_u$, and $d_n$ are the partial dimensions of the attractor of the set in the stable, unstable, and neutral directions, respectively. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what derived recently using the Generalised Extreme Value approach. Combining the results obtained using such physical observable and the properties of the extremes of distance observables considered in previous papers, where a direct link is found between the $\xi$ parameter and the Kaplan-Yorke dimension of the attractor, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by rewriting the expression of the shape parameter as suitable combination of moments of the considered observable and by using the Ruelle response theory, we provide a general framework for describing the sensitivity of the shape parameter with respect to $\epsilon$-perturbations to the flow. The possibility of defining the linear response for $\xi$ with respect to $\epsilon$ provides mathematical arguments for suggesting that the Kaplan-Yorke dimension of the attractor changes with a certain degree of smoothness when small external perturbations are introduced, as typically found in numerical experiments with intermediate to high-dimensional chaotic systems.

**Christian Maes: Does dynamical activity matter? **

**Abstract:** We review aspects of dynamical activity and the frenetic contribution in constructions of nonequilibrium statistical mechanics. Three examples are discussed in more detail: statistical forces, response and the Clausius relation.

**Eric Oliver: Sea surface temperature extremes from climate model projections: A hierarchical Bayesian model approach **

**Abstract:** Predicting how extreme events in the ocean will change in a changing climate is important for many fields including fisheries, ecological habitats, and human recreation. Global climate models provide accurate estimates of the large-scale general circulation as well as central moments such as the mean SST and eddy kinetic energy. However, such models do not provide a realistic representation of extreme events. We present estimates of 50-year return period sea surface temperature (SST) extremes by fitting the Gumbel distribution to observed SSTs (from satellite measurements). We then estimate the observed extremes from ocean climate statistics (taken from downscaled global climate model simulations representing the 1990s). To do this we develop a hierarchical Bayesian model which involves a linear regression of the parameters of the Gumbel distribution on the ocean climate statistics. The resulting linear model is then use to predict the future 50-year return period SST extremes using ocean climate statistics from a model simulation representing the 2060s. The method is demonstrated for the waters off of southeastern Australia. This region is of particular interest as the western Tasman Sea is warming at almost four times the global average rate. The results show that the estimated change in SST extremes between the 2060s and 1990s is due to a combination of changes of the mean SST and changes in higher order statistics including SST variance, SST skewness, eddy kinetic energy and sea level variance.

**Mark Pollicott: Non-equilibrium and Open dynamical systems **

**Abstract:** Firstly, I will describe some mathematical results in fluctuation theory and large deviations theory (joint work with Richard Sharp) and, time permitting, other non-equilibrium phenomena. Secondly, I will describe escape rates from open dynamical systems, both in the context of asymptotically shrinking holes where they are closely connected to extremal theory (joint work with Andrew Ferguson) and also the practical issue of numerical computation in the context of fixed holes (joint work with Oscar Bandtlow and Oliver Jenkinson).

**Pau Rabassa Sans: Extreme value laws in dynamical system under physical observables **

**Abstract:** Classical extreme values theory concerns with the maximum over a collection of random variables. This theory can be applied to a process generated by (chaotic) deterministic system composed with an observable (a cost function). This is the basic idea behind the extreme value theory for chaotic deterministic dynamical systems, which is a rapidly expanding area of research. The observables which are typically studied in the literature are expressed as functions of the distance with respect a point within the attractor. This is at odd with the structure of the observable functions typically encountered in physical applications, such as windspeed or vorticity in atmospheric models. We consider extreme value limit laws for observables which are not necessarily functions of the distance from a density point of the dynamical system. We will discuss which difficulties arise when one tries to extend the extreme limits laws to this type of observables and give sufficient conditions for the existence of a extreme value law under this kind of observables.

**Antonio Speranza: The role of self-nonlinearity in large fluctuations of mid-latitude atmospheric circulation **

**Abstract:** In a series of theories concerning the statistical nature low frequency atmospheric variability (LFAV), self-nonlinearity (Jacobian self-advection) has been assumed to play a key role in maintaining persistent, large amplitude, planetary scale fluctuations of the geopotential field. Making use of a self-nonlinearity parameter deduced from a weekly nonlinear formulation of eigenfunction analysis of observed planetary circulation states, statistical inference has been performed on Hera analysis in search of a specific signature of the postulated role of self-nonlinearity. After a short introduction concerning the definition of LFAV and the theoretical analysis of planetary scale self-nonlinearity, some preliminary results of the above mentioned statistical inference analysis will be proposed.

**Sandro Vaienti: Extreme value theory for randomly perturbed dynamical systems **

**Abstract:** We define the extreme value theory for randomly perturbed dynamical systems. We use a direct approach and a spectral approach using the transfer operator. This collects a work with H Aytac and J Freitas. We succesively appy it to non-chaotic dynamics, and this is in collaboration with D Faranda. V Lucarini and G Turchetti.

**Polina Vytnova: **

*A toy model of fast dynamo***Abstract: **The fast dynamo theory addresses the following question. Given a specific PDE, important in physics, with two parameters, a vector field and a small real constant (Reynolds number), we need to find a smooth vector field with bounded support such that one of solution grows exponentially fast as Reynolds number tends to zero. In the real 3-dimensional space the question is open. On our way to its solution, we construct a discrete model on a real line, which turns out to be an open dynamical system with random holes, and study its properties.

**Nicholas Watkins: Dissipative and Non-dissipative Models of Correlated Extreme Fluctuations in Complex Geosystems **

**Abstract:**