We present a non-perturbative, mean-field theory for the Fermi-Pasta-Ulam-Tsingou model with quartic interaction, capturing the quasiperiodic features shown by the system at all energies in the thermodynamic limit. Starting from the true Hamiltonian H of the system with N degrees of freedom, we introduce a mean-field Hamiltonian H such that the difference hN=(H-H)/N, considered as a random variable with respect to the Gibbs measure, tends to zero as N ->infinity, in probabilistic sense. The dynamics of the mean-field Hamiltonian H consists of N independent oscillation modes with renormalized frequencies Omega k=omega k1+gamma(epsilon), omega k being the frequency of the kth normal mode of the linearized system, whereas gamma(epsilon) is an explicit function of the specific energy epsilon of the system. Analytical predictions drawn from the effective Langevin equations ruling the dynamics of such oscillation modes are successfully compared with the numerical data from the original Hamiltonian dynamics. Such a simple decomposition of the true dynamics into N effective normal modes holds at all energy scales, i.e. from the quasi-integrable regime to the strongly chaotic one.

The "butterfly effect", introduced over half a century ago by Edward Lorenz, has shifted from a cornerstone of dynamical systems to a popular metaphor, yet its true physical manifestation in fully developed turbulence spans a spectrum of phenomena from standard chaotic sensitivity to the recently established concept of Eulerian spontaneous stochasticity. This paper presents an attempt at a systematic synthesis that brings these different but interconnected ideas together within the unifying framework of the Finite Size Lyapunov Exponent (FSLE). The FSLE describes the growth rate of perturbations as a function of their scale, enabling a comprehensive characterization of the multiscale physics of turbulent flows. Using the FSLE and the Sabra shell model, extended to include thermal noise, we bridge the classical, small-scale Lyapunov regime with predictability at large scales and its interpretation in terms of Eulerian spontaneous stochasticity. Moreover, using the FSLE and the Kraichnan model, we also illustrate the closely related phenomenon of Lagrangian spontaneous stochasticity. To complete the spectrum of butterfly effects, we also examine the "literal butterfly" scenario of localized, sub-dissipative perturbations. Ultimately, this synthesis clarifies the physical mechanisms that dictate the fundamental boundaries of forecasting in high-Reynolds-number flows.

In their seminal work, Fermi, Pasta, Ulam and Tsingou explored the connection between statistical mechanics and dynamical properties, such as chaos and ergodicity. Even today, seventy years later, the topic is not fully understood: while most results of statistical mechanics require the ergodic hypothesis to be rigorously proved, there are many indications that these predictions, both in and out of equilibrium, hold even in the absence of a rigorous form of ergodicity. Motivated by the above considerations, in this work we reconsider the point of view that the relevant ingredients for the validity of statistical mechanics are the large number of degrees of freedom and the choice of extensive observables, while the details of the dynamics do not play an essential role. This is the idea behind Khinchin's famous proof of the typicality of macroscopic observables at equilibrium. We extend this perspective to the context of non equilibrium, by investigating the thermalization properties of both harmonic (integrable) and nonharmonic (chaotic) oscillator chains initially prepared in out-of-equilibrium conditions. In integrable systems, thermalization occurs, or not, depending on the observable. In the chaotic regime, instead, thermalization is reached by any observable, although the relaxation timescale might be larger than the observation time.

We establish an analytical criterion for dynamical thermalization within harmonic systems, applicable to both classical and quantum models. Specifically, we prove that thermalization of various observables-such as particle energies in physically relevant random quadratic Hamiltonians-is typical for large systems (N >> 1) with initial conditions drawn from the microcanonical distribution. Moreover, we show that thermalization can also arise from nontypical initial conditions, where only a finite fraction of the normal modes is excited. A different choice of initial conditions, such as all the initial energy localized in a single particle, instead leads to energy equipartition without thermalization. Since the models we consider are integrable, our findings provide a general dynamical basis for an approach to thermalization that bypasses chaos and ergodicity, focusing instead on the physical requirement that thermodynamic observables depend on a large number of normal modes, and they build a bridge between the classical and quantum theories of thermalization.

Although not as wide, and popular, as that of quantum mechanics, the investigation of fundamental aspects of statistical mechanics constitutes an important research field in the building of modern physics. Besides the interest for itself, both for physicists and philosophers, and the obvious pedagogical motivations, there is a further, compelling reason for a thorough understanding of the subject. The fast development of models and methods at the edge of the established domain of the field requires indeed a deep reflection on the essential aspects of the theory, which are at the basis of its success. These elements should never be disregarded when trying to expand the domain of statistical mechanics to systems with novel, little known features. It is thus important to (re)consider in a careful way the main ingredients involved in the foundations of statistical mechanics. Among those, a primary role is covered by the dynamical aspects (e.g. presence of chaos), the emergence of collective features for large systems, and the use of probability in the building of a consistent statistical description of physical systems. With this goal in mind, in the present review we aim at providing a consistent picture of the state of the art of the subject, both in the classical and in the quantum realm. In particular, we will highlight the similarities of the key technical and conceptual steps with emphasis on the relevance of the many degrees of freedom, to justify the use of statistical ensembles in the two domains.

In the study of stochastic processes, identifying the parity under time-reversal is essential to verify detailed balance, and to compute the entropy production rate (which is, otherwise, ambiguously defined). While in many cases the correct time-reversal symmetry is suggested by physical arguments, for generic processes the identification is not trivial: as a result, systems at thermal equilibrium may be mistakenly interpreted as non-equilibrium ones. We focus on the reversible deterministic dynamics of a slow variable coupled to many degrees of freedom acting as a thermal bath. We show that the time-reversal symmetry of the slow variable is preserved when passing to an effective stochastic description, independently of the nature of the bath. In turn, for generic 2-dimensional continuous Markov processes, we provide a criterion to identify the time-reversal parity rules under which the dynamics is at equilibrium (if any). The case of the Lotka-Volterra model is discussed as an example.

Current research in statistical mechanics mostly concerns the investigation of out-of-equilibrium, irreversible processes, which are ubiquitous in nature and still far from being theoretically understood. Even the precise characterization of irreversibility is the object of an open debate: while in the context of Hamiltonian systems the one-century-old proposal by M. Smoluchowski looks still valid (a process appears irreversible when the initial state has a recurrence time that is long compared to the time of observation [1]), in dissipative systems, particularly in the case of stochastic processes, the problem is more involved, and quantifying the "degree of irreversibility" is a pragmatic need. The most employed strategies rely on the estimation of entropy production: this quantity, although mathematically well-defined, is often difficult to compute, especially when analyzing experimental data. Moreover, being a global observable, entropy production fails to capture specific aspects of irreversibility in extended systems, such as the role of different currents and their spatial development. This review aims to address various conceptual and technical challenges encountered in the analysis of irreversibility, including the role of the coarse-graining procedure and the treatment of data in the absence of complete information. The discussion will be mostly based on simple models, analytically treatable, and supplemented by examples of complex, more realistic non-equilibrium systems.

Random exchange kinetic models are widely employed to describe the conservative dynamics of large interacting systems. Due to their simplicity and generality, they are quite popular in several fields, from statistical mechanics to biophysics and economics. Here, we study a version where bounds on the individual shares of a globally conserved quantity are introduced. We analytically show that this dynamic allows stationary states with population inversion, described by Boltzmann statistics at negative absolute temperature, if the conserved quantity has the physical meaning of an energy. The proposed model therefore provides a privileged system for the study of thermalization towards a negative temperature state. First, the genuine equilibrium nature of the stationary state is verified by checking the detailed balance condition. Then, an H-theorem is proven, ensuring that such equilibrium condition is reached by a monotonic increase in the Boltzmann entropy. We also provide analytical and numerical evidence that a large intruder in contact with the system thermalizes, suggesting a practical way to design a thermal bath at negative temperature.

The statistical properties of turbulent flows are fundamentally different from those of systems at equilibrium due to the presence of an energy flux from the scales of injection to those where energy is dissipated by the viscous forces: a scenario dubbed "direct energy cascade." From a statistical mechanics point of view, the cascade picture prevents the existence of detailed balance, which holds at equilibrium, e.g., in the inviscid and unforced case. Here, we aim at characterizing the nonequilibrium properties of turbulent cascades in a shell model of turbulence by studying an asymmetric time-correlation function and the relaxation behavior of an energy perturbation, measured at scales smaller or larger than the perturbed one. We contrast the behavior of these two observables in both nonequilibrium (forced and dissipated) and equilibrium (inviscid and unforced) cases. Finally, we show that equilibrium and nonequilibrium physics coexist in the same system, namely, at scales larger and smaller, respectively, of the forcing scale.

Chapter 3 presents an introduction and sections on: quantum many-body systems and emergent phenomena; the search for new materials; manipulating photons and atoms: photonics and nanophysics; extreme light; systems with numerous degrees of freedom;

Abstract The statistical properties of turbulent flows are fundamentally different from those of systems at equilibrium due to the presence of an energy flux from the scales of injection to those where energy is dissipated by the viscous forces: a scenario dubbed “direct energy cascade”. Here, we aim at characterizing the non-equilibrium properties of turbulent cascades in a shell model of turbulence by studying an asymmetric time-correlation function and the relaxation behavior of an energy perturbation, measured at scales smaller or larger than the perturbed one. We shall contrast the behavior of these two observables in both non-equilibrium (forced and dissipated) and equilibrium (inviscid and unforced) cases. Finally, we shall show that equilibrium and non-equilibrium physics coexist in the same system, namely at scales larger and smaller, respectively, of the forcing scale.

Nowadays many tools, e.g. fluctuation relations, are available to characterize the statistical properties of non-equilibrium systems. However, most of these tools rely on the assumption that the driving noise is normally distributed. Here we consider a class of Markov processes described by Langevin equations driven by a mixture of Gaussian and Poissonian noises, focusing on their non-equilibrium properties. In particular, we prove that detailed balance does not hold even when correlation functions are symmetric under time reversal. In such cases, a breakdown of the time reversal symmetry can be highlighted by considering higher order correlation functions. Furthermore, the entropy production may be different from zero even for vanishing currents. We provide analytical expressions for the average entropy production rate in several cases. We also introduce a scale dependent estimate for entropy production, suitable for inference from experimental signals. The empirical entropy production allows us to discuss the role of spatial and temporal resolutions in characterizing non-equilibrium features. Finally, we revisit the Brownian gyrator introducing an additional Poissonian noise showing that it behaves as a two dimensional linear ratchet. It has also the property that when Onsager relations are satisfied its entropy production is positive although it is minimal. We conclude discussing estimates of entropy production for partially accessible systems, comparing our results with the lower bound provided by the thermodynamic uncertainty relations.

The talk is devoted to a discussion of different typologies of models: I- Oversimplified models; II- Models by analogy; III- Large scale models;IV- Models from data. In the class I there is the celebrated Lorenz model; the Lotka-Volterra system is in the class II, and it is at the origin of biomathematics.Among the models in the class III we have the effective equations used, e.g., in meteorology and engineering, where only "relevant variables" are taken into account.In the class IV we find the most interesting (and difficult) problem:the building of models just from datawithout a reference theoretical framework.

Correlation analysis and its close variant principal component analysis are tools widely applied to predict the biological functions of macromolecules in terms of the relationship between fluctuation dynamics and structural properties. However, since this kind of analysis does not necessarily imply causation links among the elements of the system, its results run the risk of being biologically misinterpreted. By using as a benchmark the structure of ubiquitin, we report a critical comparison of correlation-based analysis with the analysis performed using two other indicators, response function and transfer entropy, that quantify the causal dependence. The use of ubiquitin stems from its simple structure and from recent experimental evidence of an allosteric control of its binding to target substrates. We discuss the ability of correlation, response and transfer-entropy analysis in detecting the role of the residues involved in the allosteric mechanism of ubiquitin as deduced by experiments. To maintain the comparison as much as free from the complexity of the modeling approach and the quality of time series, we describe the fluctuations of ubiquitin native state by the Gaussian network model which, being fully solvable, allows one to derive analytical expressions of the observables of interest. Our comparison suggests that a good strategy consists in combining correlation, response and transfer entropy, such that the preliminary information extracted from correlation analysis is validated by the two other indicators in order to discard those spurious correlations not associated with true causal dependencies.

In the framework of statistical mechanics the properties of macroscopic systems are deduced starting from the laws of their microscopic dynamics. One of the key assumptions in this procedure is the ergodic property, namely the equivalence between time averages and ensemble averages. This property can be proved only for a limited number of systems; however, as proved by Khinchin [1], weak forms of it hold even in systems that are not ergodic at the microscopic scale, provided that extensive observables are considered. Here we show in a pedagogical way the validity of the ergodic hypothesis, at a practical level, in the paradigmatic case of a chain of harmonic oscillators. By using analytical results and numerical computations, we provide evidence that this non-chaotic integrable system shows ergodic behavior in the limit of many degrees of freedom. In particular, the Maxwell-Boltzmann distribution turns out to fairly describe the statistics of the single particle velocity. A study of the typical time-scales for relaxation is also provided.

Fabio Cecconi1,2,∗, Giulio Costantini, Carlo Guardiani, Marco Baldovin and Angelo Vulpiani 1 CNR-Istituto dei Sistemi Complessi, Via dei Taurini 19, 00185 Rome, Italy 2 INFN-Sezione di Roma1, P.le Aldo Moro, 2, 00185 Rome, Italy 3 CNR-Istituto dei Sistemi Complessi, Piazzale A. Moro 5, 00185 Rome, Italy 4 Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Universit̀a di Roma, Via Eudossiana 18, 00184 Rome, Italy 5 CNRS, LPTMS, Université Paris-Saclay, 530 Rue André Riviére, 91405 Orsay, France 6 Dipartimento di Fisica, Universit̀a di Roma Sapienza, P.le Aldo Moro 5, 00185 Rome, Italy ∗ Author to whom any correspondence should be addressed.

The characterization of the distance from equilibrium is a debated problem in particular in the treatment of experimental signals. If the signal is a one-dimensional time series, such a goal becomes challenging. A paradigmatic example is the angular diffusion of a rotator immersed in a vibro-fluidized granular gas. Here, we experimentally observe that the rotator's angular velocity exhibits significant differences with respect to an equilibrium process. Exploiting the presence of two relevant timescales and non-Gaussian velocity increments, we quantify the breakdown of time-reversal asymmetry, which would vanish in the case of a 1D Gaussian process. We deduce a new model for the massive probe, with two linearly coupled variables, incorporating both Gaussian and Poissonian noise, the latter motivated by the rarefied collisions with the granular bath particles. Our model reproduces the experiment in a range of densities, from dilute to moderately dense, with a meaningful dependence of the parameters on the density. We believe the framework proposed here opens the way to a more consistent and meaningful treatment of out-of-equilibrium and dissipative systems.