arXiv subjects

Jeroen Wouters

Publications and source records attributed to Jeroen Wouters.

At least 19 recordsLinked to original sources

arXiv2021

On spurious detection of linear response and misuse of the fluctuation-dissipation theorem in finite time series

Using a sensitive statistical test we determine whether or not one can detect the breakdown of linear response given observations of deterministic dynamical systems. A goodness-of-fit statistics is developed for a linear statistical model of the observations, based on results on central limit theorems for deterministic dynamical systems, and used to detect linear response breakdown. We apply the method to discrete maps which do not obey linear response and show that the successful detection of breakdown depends on the length of the time series, the magnitude of the perturbation and on the choice of the observable. We find that in order to reliably reject the assumption of linear response for typical observables sufficiently large data sets are needed. Even for simple systems such as the logistic map, one needs of the order of $10^6$ observations to reliably detect the breakdown, if less observations are available one may be falsely led to conclude that linear response theory is valid. The amount of data required is larger the smaller the applied perturbation. For judiciously chosen observables the necessary amount of data can be drastically reduced, but requires detailed {\em{a priori}} knowledge about the invariant measure which is typically not available for complex dynamical systems. Furthermore we explore the use of the fluctuation-dissipation theorem (FDT) in cases with limited data length or coarse-graining of observations. The FDT, if applied naively to a system without linear response, is shown to be very sensitive to the details of the sampling method, resulting in erroneous predictions of the response.

nlin.CD↗

arXiv2021

Applications of large deviation theory in geophysical fluid dynamics and climate science

The climate system is a complex, chaotic system with many degrees of freedom and variability on a vast range of temporal and spatial scales. Attaining a deeper level of understanding of its dynamical processes is a scientific challenge of great urgency, especially given the ongoing climate change and the evolving climate crisis. In statistical physics, complex, many-particle systems are studied successfully using the mathematical framework of Large Deviation Theory (LDT). A great potential exists for applying LDT to problems relevant for geophysical fluid dynamics and climate science. In particular, LDT allows for understanding the fundamental properties of persistent deviations of climatic fields from the long-term averages and for associating them to low-frequency, large scale patterns of climatic variability. Additionally, LDT can be used in conjunction with so-called rare events algorithms to explore rarely visited regions of the phase space and thus to study special dynamical configurations of the climate. These applications are of key importance to improve our understanding of high-impact weather and climate events. Furthermore, LDT provides powerful tools for evaluating the probability of noise-induced transitions between competing metastable states of the climate system or of its components. This in turn essential for improving our understanding of the global stability properties of the climate system and of its predictability of the second kind in the sense of Lorenz. The goal of this review is manifold. First, we want to provide an introduction to the derivation of large deviation laws in the context of stochastic processes. We then relate such results to the existing literature showing the current status of applications of LDT in climate science and geophysical fluid dynamics. Finally, we propose some possible lines of future investigations.

cond-mat.stat-mech↗

arXiv2019

Deviations from Gaussianity in deterministic discrete time dynamical systems

In this paper we examine the deviations from Gaussianity for two types of random variable converging to a normal distribution, namely sums of random variables generated by a deterministic discrete time map and a linearly damped variable driven by a deterministic map. We demonstrate how Edgeworth expansions provide a universal description of the deviations from the limiting normal distribution. We derive explicit expressions for these asymptotic expansions and provide numerical evidence of their accuracy.

nlin.CD↗

arXiv2019

Edgeworth expansions for slow-fast systems with finite time scale separation

We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of slow-fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel-Dyson series is used to asymptotically determine the corrections at any desired order of the time scale parameter $\varepsilon$. The corrections involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in $\varepsilon$ and provide explicit expressions for the first two orders. At a formal level, the expressions derived are valid in the case when the fast dynamics is stochastic as well as when the fast dynamics is entirely deterministic. We corroborate our analytical results with numerical simulations and show that our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time scale separation.

nlin.CD↗

arXiv2019

A Large Deviation Theory-based Analysis of Heat Waves and Cold Spells in a Simplified Model of the General Circulation of the Atmosphere

We study temporally persistent and spatially extended extreme events of temperature anomalies, i.e. heat waves and cold spells, using large deviation theory. To this end, we consider a simplified yet Earth-like general circulation model of the atmosphere and numerically estimate large deviation rate functions of near-surface temperature in the mid-latitudes. We find that, after a re-normalisation based on the integrated auto-correlation, the rate function one obtains at a given latitude by looking, locally in space, at long time averages agrees with what is obtained, instead, by looking, locally in time, at large spatial averages along the latitude. This is a result of scale symmetry in the spatial-temporal turbulence and of the fact that advection is primarily zonal. This agreement hints at the universality of large deviations of the temperature field. Furthermore, we discover that the obtained rate function is able to describe spatially extended and temporally persistent heat waves or cold spells, if we consider temporal averages of spatial averages over intermediate spatial scales. Finally, we find out that large deviations are relatively more likely to occur when looking at these spatial averages performed over intermediate scales, thus pointing to the existence of weather patterns associated to the low-frequency variability of the atmosphere. Extreme value theory is used to benchmark our results.

cond-mat.stat-mech↗

arXiv2018

Stochastic model reduction for slow-fast systems with moderate time-scale separation

We propose a stochastic model reduction strategy for deterministic and stochastic slow-fast systems with finite time-scale separation. The stochastic model reduction relaxes the assumption of infinite time-scale separation of classical homogenization theory by incorporating deviations from this limit as described by an Edgeworth expansion. A surrogate system is constructed the parameters of which are matched to produce the same Edgeworth expansions up to any desired order of the original multi-scale system. We corroborate our analytical findings by numerical examples, showing significant improvements to classical homogenized model reduction.

cond-mat.stat-mech↗

arXiv2017

Computation of extreme heat waves in climate models using a large deviation algorithm

Studying extreme events and how they evolve in a changing climate is one of the most important current scientific challenges. Starting from complex climate models, a key difficulty is to be able to run long enough simulations in order to observe those extremely rare events. In physics, chemistry, and biology, rare event algorithms have recently been developed to compute probabilities of events that cannot be observed in direct numerical simulations. Here we propose such an algorithm, specifically designed for extreme heat or cold waves, based on statistical physics approaches. This gives an improvement of more than two orders of magnitude in the sampling efficiency. We describe the dynamics of events that would not be observed otherwise. We show that European extreme heat waves are related to a global teleconnection pattern involving North America and Asia. This tool opens a full range of studies, so far impossible, aimed at assessing quantitatively climate change impacts.

physics.ao-ph↗

arXiv2017

Response Formulae for $n$-point Correlations in Statistical Mechanical Systems and Application to a Problem of Coarse Graining

Predicting the response of a system to perturbations is a key challenge in mathematical and natural sciences. Under suitable conditions on the nature of the system, of the perturbation, and of the observables of interest, response theories allow to construct operators describing the smooth change of the invariant measure of the system of interest as a function of the small parameter controlling the intensity of the perturbation. In particular, response theories can be developed both for stochastic and chaotic deterministic dynamical systems, where in the latter case stricter conditions imposing some degree of structural stability are required. In this paper we extend previous findings and derive general response formulae describing how n-point correlations are affected by perturbations to the vector flow. We also show how to compute the response of the spectral properties of the system to perturbations. We then apply our results to the seemingly unrelated problem of coarse graining in multiscale systems: we find explicit formulae describing the change in the terms describing parameterisation of the neglected degrees of freedom resulting from applying perturbations to the full system. All the terms envisioned by the Mori-Zwanzig theory - the deterministic, stochastic, and non-Markovian terms - are affected at 1st order in the perturbation. The obtained results provide a more comprehesive understanding of the response of statistical mechanical systems to perturbations and contribute to the goal of constructing accurate and robust parameterisations and are of potential relevance for fields like molecular dynamics, condensed matter, and geophysical fluid dynamics. We envision possible applications of our general results to the study of the response of climate variability to anthropogenic and natural forcing and to the study of the equivalence of thermostatted statistical mechanical systems.

cond-mat.stat-mech↗

arXiv2016

Parametrization of stochastic multiscale triads

We discuss applications of a recently developed method for model reduction based on linear response theory of weakly coupled dynamical systems. We apply the weak coupling method to simple stochastic differential equations with slow and fast degrees of freedom. The weak coupling model reduction method results in general in a non-Markovian system, we therefore discuss the Markovianization of the system to allow for straightforward numerical integration. We compare the applied method to the equations obtained through homogenization in the limit of large time scale separation between slow and fast degrees of freedom. We look at ensemble spread from a fixed initial condition, correlation functions and exit times from domain. The weak coupling method gives better results in all test cases, albeit with a higher numerical cost.

cond-mat.stat-mech↗

arXiv2016

Rare event computation in deterministic chaotic systems using genealogical particle analysis

In this paper we address the use of rare event computation techniques to estimate small over-threshold probabilities of observables in determin-istic dynamical systems. We demonstrate that the genealogical particle analysis algorithms can be successfully applied to a toy model of atmospheric dynamics, the Lorenz '96 model. We furthermore use the Ornstein-Uhlenbeck system to illustrate a number of implementation issues. We also show how a time-dependent objective function based on the fluctuation path to a high threshold can greatly improve the performance of the estimator compared to a fixed-in-time objective function.

cond-mat.stat-mech↗

arXiv2014

Mathematical and Physical Ideas for Climate Science

The climate is a forced and dissipative nonlinear system featuring non-trivial dynamics of a vast range of spatial and temporal scales. The understanding of the climate's structural and multiscale properties is crucial for the provision of a unifying picture of its dynamics and for the implementation of accurate and efficient numerical models. We present some recent developments at the intersection between climate science, mathematics, and physics, which may prove fruitful in the direction of constructing a more comprehensive account of climate dynamics. We describe the Nambu formulation of fluid dynamics, and the potential of such a theory for constructing sophisticated numerical models of geophysical fluids. Then, we focus on the statistical mechanics of quasi-equilibrium flows in a rotating environment, which seems crucial for constructing a robust theory of geophysical turbulence. We then discuss ideas and methods suited for approaching directly the non-equilibrium nature of the climate system. First, we describe some recent findings on the thermodynamics of climate and characterize its energy and entropy budgets, and discuss related methods for intercomparing climate models and for studying tipping points. These ideas can also create a common ground between geophysics and astrophysics by suggesting general tools for studying exoplanetary atmospheres. We conclude by focusing on non-equilibrium statistical mechanics, which allows for a unified framing of problems as different as the climate response to forcings, the effect of altering the boundary conditions or the coupling between geophysical flows, and the derivation of parametrizations for numerical models.

physics.ao-ph↗

arXiv2014

On using Extreme Values to detect global stability thresholds in multi-stable systems: The case of transitional plane Couette flow

Extreme Value Theory (EVT) is exploited to determine the global stability threshold $R_g$ of plane Couette flow --the flow of a viscous fluid in the space between two parallel plates-- whose laminar or turbulent behavior depends on the Reynolds number R. Even if the existence of a global stability threshold has been detected in simulations and experiments, its numerical value has not been unequivocally defined. $R_g$ is the value such that for $R > R_g$ turbulence is sustained, whereas for $R < R_g$ it is transient and eventually decays. We address the problem of determining $R_g$ by using the extremes - maxima and minima - of the perturbation energy fluctuations. When $R_g$, both the positive and negative extremes are bounded. As the critical Reynolds number is approached from above, the probability of observing a very low minimum increases causing asymmetries in the distributions of maxima and minima. On the other hand, the maxima distribution is unaffected as the fluctuations towards higher values of the perturbation energy remain bounded. This tipping point can be detected by fitting the data to the Generalized Extreme Value (GEV) distribution and by identifying $R_g$ as the value of $R$ such that the shape parameter of the GEV for the minima changes sign from negative to positive. The results are supported by the analysis of theoretical models which feature a bistable behavior.

math.DS↗

arXiv2013

Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems

In this paper we provide a connection between the geometrical properties of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, 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 writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

cond-mat.stat-mech↗

arXiv2013

Towards a Fluctuation Theorem in an Atmospheric Circulation Model

An investigation of the distribution of finite time trajectory divergence is performed on an Atmospheric Global Circulation Model. The distribution of the largest local Lyapunov exponent shows a significant probability for negative values over time spans up to 10 days. This effect is present for resolutions up to wave numbers l=42 (~ 250km). The probability for a negative local largest Lyapunov exponent decreases over time, similarly to the predictions of the Fluctuation Theorem for entropy production. The model used is hydrostatic with variable numbers of vertical levels and different horizontal resolutions.

physics.ao-ph↗

arXiv2013

Avalanches, Breathers and Flow Reversal in a Continuous Lorenz-96 Model

For the discrete model suggested by Lorenz in 1996 a one-dimensional long wave approximation with nonlinear excitation and diffusion is derived. The model is energy conserving but non-Hamiltonian. In a low order truncation weak external forcing of the zonal mean flow induces avalanche-like breather solutions which cause reversal of the mean flow by a wave-mean flow interaction. The mechanism is an outburst-recharge process similar to avalanches in a sand pile model.

nlin.AO↗

arXiv2012

Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach

In this paper we consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system, which are weakly coupled to the remaining variables. In a previous paper we have used the Ruelle response theory on such a weakly coupled system to construct a surrogate dynamics, such that the expectation value of any observable agrees, up to second order in the coupling strength, to its expectation evaluated on the full dynamics. We show here that such surrogate dynamics agree up to second order to an expansion of the Mori-Zwanzig projected dynamics. This implies that the parametrizations of unresolved processes suited for prediction and for the representation of long term statistical properties are closely related, if one takes into account, in addition to the widely adopted stochastic forcing, the often neglected memory effects.

cond-mat.stat-mech↗

arXiv2012

Fermionic Markov Chains

We study a quantum process that can be considered as a quantum analogue for the classical Markov process. We specifically construct a version of these processes for free Fermions. For such free Fermionic processes we calculate the entropy density. This can be done either directly using Szegö's theorem for asymptotic densities of functions of Toeplitz matrices, or through an extension of said theorem to rates of functions, which we present in this article.

quant-ph↗

arXiv2011

Disentangling multi-level systems: averaging, correlations and memory

We consider two weakly coupled systems and adopt a perturbative approach based on the Ruelle response theory to study their interaction. We propose a systematic way to parametrize the effect of the coupling as a function of only the variables of a system of interest. Our focus is on describing the impacts of the coupling on the long-term statistics rather than on the finite-time behaviour. By direct calculation, we find that, at first order, the coupling can be surrogated by adding a deterministic perturbation to the autonomous dynamics of the system of interest. At second order, there are additionally two separate and very different contributions. One is a term taking into account the second order contributions of the fluctuations in the coupling, which can be parametrized as a stochastic forcing with given spectral properties. The other one is a memory term, coupling the system of interest to its previous history, through the correlations of the second system. If these correlations are known, this effect can be implemented as a perturbation with memory on the single system. In order to treat this case, we present an extension to Ruelle's response theory able to deal with integral operators. We discuss our results in the context of other methods previously proposed to disentangle the dynamics of two coupled systems. We emphasize that our results do not rely on assuming a time scale separation, and, if such a separation exist, can be used equally well to study the statistics of the slow as well as that of the fast variables. By recursively applying the technique proposed here, we can treat the general case of multilevel systems.

cond-mat.stat-mech↗