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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., Comment: 94 pages, 9 figures

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., Comment: 79 pages, 19 figures

In this paper we present a discussion of the basic aspects of the well-known problem of prediction and inference in physics, with specific attention to the role of models, the use of data and the application of recent developments in artificial intelligence. By focussing in the time evolution of dynamic system, it is shown that main difficulties in predictions arise due to the presence of few factors as: the occurrence of chaotic dynamics, the existence of many variables with very different characteristic time-scales and the lack of an accurate understanding of the underlying physical phenomena. It is shown that a crucial role is assigned to the preliminary identification of the proper variables, their selection and the identification of an appropriate level of description (coarse-graining procedure).

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 ($N \gg 1$) with initial conditions drawn from the microcanonical distribution. Moreover, we show that thermalization can also arise from non-typical initial conditions, where only a finite fraction of the normal modes is excited. Our findings provide a general dynamical basis for an approach to thermalization that bypasses chaos and ergodicity, focusing instead on observables dependent on a large number of normal modes, and build a bridge between the classical and quantum theories of thermalization., Comment: 20 pages (5 main + 2 references + 13 Supplemental Material)

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 the globally conserved quantity are introduced. We analytically show that this dynamics allows stationary states with population inversion, described by Boltzmann statistics at negative absolute temperature. Their genuine equilibrium nature is verified by checking the detailed balance condition. An H-theorem is proven: the Boltzmann entropy monotonically increases during the dynamics. Finally, we 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. These results open new research perspectives, creating a bridge between negative temperature statistical descriptions and kinetic models with bounds., Comment: 5 pages, 2 figures

We consider the Prandtl-Tomlinson model in the case of a constant driving force and in the presence of thermal fluctuations. We show that the system dynamics is well reproduced by a simplified description obtained through a Markov process, even in the case of potentials with several minima. After estimating the chain parameters by numerical simulation, we compute the average velocity and friction at varying driving force and temperature. Then we take advantage of this approach for calculating the entropy produced by the system and, in the case of a single minimum potential, to derive its explicit relation with the external force and the mobility at low temperatures. We observe that the coefficient relating the entropy production to the force is not a monotonic function of the temperature., Comment: 15 pages, 12 figures

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 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., Comment: REVTeX 4.2, 15 pages, 10 figures

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., Comment: 10 pages, 2 figures

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 thermodynamics uncertainty relations., Comment: 30 pages, 11 figures. 20 pages main text, 8 pages appendices

The characterization of the distance from equilibrium is a debated problem in particular in the treatment of experimental signals. If the signal is a 1-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 significative differences with respect to an equilibrium process. Exploiting the presence of two relevant time-scales 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., Comment: 5 pages, 4 figures

The detection of cause-effect relationships from the analysis of paleoclimatic records is a crucial step to disentangle the main mechanisms at work in the climate system. Here, we show that the approach based on the generalized Fluctuation-Dissipation Relation, complemented by the analysis of the Transfer Entropy, allows the causal links to be identified between temperature, CO2 concentration and astronomical forcing during the glacial cycles of the last 800 kyr based on Antarctic ice core records. When considering the whole spectrum of time scales, the results of the analysis suggest that temperature drives CO2 concentration, or that are both driven by the common astronomical forcing. However, considering only millennial-scale fluctuations, the results reveal the presence of more complex causal links, indicating that CO2 variations contribute to driving the changes of temperature on such time scales. The results also evidence a slow temporal variability in the strength of the millennial-scale causal links between temperature and CO2 concentration., Comment: 13 pages, 5 figures + Supplemental material (10 pages, 7 figures)

In this paper, we offer to the reader an essential review of the theory of Fluctuation-Dissipation Relations (FDR), from the first formulations due to Einstein and Onsager, to the recent developments in the framework of stochastic thermodynamics of non-equilibrium system. We focus on two general approaches, somehow complementary, where out-of-equilibrium contributions to the FDR are expressed in terms of different quantities, related either to the stationary distribution or to the transition rates of the system. In particular, we discuss applications of the FDR in the general field of causation and inference, and in the contexts of non-equilibrium systems, such as spin models, granular media and active matter., Comment: 29 pages, 3 figures, chapter in the book "Nonequilibrium Thermodynamics and Fluctuation Kinetics" (editors L\'eon Brenig, Nikolai Brilliantov, Mustapha Tlidi) Springer 2022

We study the anomalous transport in systems of random walks (RW's) on comb-like lattices with fractal sidebranches, showing subdiffusion, and in a system of Brownian particles driven by a random shear along the x-direction, showing a superdiffusive behavior. In particular, we discuss whether scaling and universality are present or not in the shapes of the particle distribution along the preferential transport direction (x-axis)., Comment: latex: 13 pages, 9 pdf figures, regular article

Data from experiments and theoretical arguments are the two pillars sustaining the job of modelling physical systems through inference. In order to solve the inference problem, the data should satisfy certain conditions that depend also upon the particular questions addressed in a research. Here we focus on the characterization of systems in terms of a distance from equilibrium, typically the entropy production (time-reversal asymmetry) or the violation of the Kubo fluctuation-dissipation relation. We show how general, counter-intuitive and negative for inference, is the problem of the impossibility to estimate the distance from equilibrium using a series of scalar data which have a Gaussian statistics. This impossibility occurs also when the data are correlated in time, and that is the most interesting case because it usually stems from a multi-dimensional linear Markovian system where there are many time-scales associated to different variables and, possibly, thermal baths. Observing a single variable (or a linear combination of variables) results in a one-dimensional process which is always indistinguishable from an equilibrium one (unless a perturbation-response experiment is available). In a setting where only data analysis (and not new experiments) is allowed, we propose - as a way out - the combined use of different series of data acquired with different parameters. This strategy works when there is a sufficient knowledge of the connection between experimental parameters and model parameters. We also briefly discuss how such results emerge, similarly, in the context of Markov chains within certain coarse-graining schemes. Our conclusion is that the distance from equilibrium is related to quite a fine knowledge of the full phase space, and therefore typically hard to approximate in real experiments., Comment: 23 pages - 8 figures

Recent numerical results showed that thermalization of Fourier modes is achieved in short time-scales in the Toda model, despite its integrability and the absence of chaos. Here we provide numerical evidence that the scenario according to which chaos is irrelevant for thermalization is realized even in the simplest of all classical integrable system: the harmonic chain. We study relaxation from an atypical condition given with respect to "random" modes, showing that a thermal state with equilibrium properties is attained in short times. Such a result is independent from the orthonormal base used to represent the chain state, provided it is random., Comment: 8 pages, 5 figures

We investigate the effect of coarse-graining on the energetics properties of a system, focusing on entropy production. As a case of study, we consider a one-dimensional colloidal particle in contact with a thermal bath, moving in a sinusoidal potential and driven out of equilibrium by a small constant force. Different levels of coarse-graining are evaluated: at first, we compare the results in the underdamped dynamics with those in the overdamped one (first coarse-graining). For large values of the viscosity, the two dynamics have the same energetic properties, while, for smaller viscosity values, the overdamped approximation produces an excess of entropy production with respect to that of the underdamped dynamics. Moreover, for further smaller values of the drag coefficient, the excess of entropy production turns into a loss. These regimes are explained by evaluating the jump statistics, observing that the inertia is able to induce multiple jumps and affect the average jump rate. The periodic shape of the potential allows us to approximate the continuous dynamics via a Markov chain, after the introduction of a suitable time and space discretization (second level of coarse-graining). This discretization procedure is implemented starting both from the underdamped and the overdamped evolution and is analyzed for different values of the viscosity.

In this short review we propose a critical assessment of the role of chaos for the thermalization of Hamiltonian systems with high dimensionality. We discuss this problem for both classical and quantum systems. A comparison is made between the two situations: some examples from recent and past literature are presented which support the point of view that chaos is not necessary for thermalization. Finally, we suggest that a close analogy holds between the role played by Kinchin's theorem for high-dimensional classical systems and the role played by Von Neumann's theorem for many-body quantum systems., Comment: 11 pages, 3 figures, to appear as a contribution in "EPS Grand Challenges: Physics for Society at the Horizon 2050"

We introduce a general formulation of the fluctuation-dissipation relations (FDR) holding also in far-from-equilibrium stochastic dynamics. A great advantage of this version of the FDR is that it does not require the explicit knowledge of the stationary probability density function. Our formula applies to Markov stochastic systems with generic noise distributions: when the noise is additive and Gaussian, the relation reduces to those known in the literature; for multiplicative and non-Gaussian distributions (e.g. Cauchy noise) it provides exact results in agreement with numerical simulations. Our formula allows us to reproduce, in a suitable small-noise limit, the response functions of deterministic, strongly non-linear dynamical models, even in the presence of chaotic behavior: this could have important practical applications in several contexts, including geophysics and climate. As a case of study, we consider the Lorenz '63 model, which is paradigmatic for the chaotic properties of deterministic dynamical systems.

It is well known that entropy production is a proxy to the detection of non-equilibrium, i.e. of the absence of detailed balance; however, due to the global character of this quantity, its knowledge does not allow to identify spatial currents or fluxes of information among specific elements of the system under study. In this respect, much more insight can be gained by studying transfer entropy and response, which allow quantifying the relative influence of parts of the system and the asymmetry of the fluxes. In order to understand the relation between the above-mentioned quantities, we investigate spatially asymmetric extended systems. First, we consider a simplified linear stochastic model, which can be studied analytically; then, we include nonlinear terms in the dynamics. Extensive numerical investigation shows the relation between entropy production and the above-introduced degrees of asymmetry. Finally, we apply our approach to the highly nontrivial dynamics generated by the Lorenz '96 model for Earth oceanic circulation., Comment: 9 pages, 6 figures

Do negative absolute temperatures matter physics and specifically Statistical Physics? We provide evidence that we can certainly answer positively to this vexata quaestio. The great majority of models investigated by statistical mechanics over almost one century and a half exhibit positive absolute temperature, because their entropy is a nondecreasing function of energy. Since more than half a century ago it has been realized that this may not be the case for some physical systems as incompressible fluids, nuclear magnetic chains, lasers, cold atoms and optical waveguides. We review these examples and discuss their peculiar thermodynamic properties, which have been associated to the presence of thermodynamic regimes, characterized by negative absolute temperatures. As reported in this review, the ambiguity inherent the definition of entropy has recurrently raised a harsh debate about the possibility of considering negative temperature states as genuine thermodynamic equilibrium ones. Here we show that negative absolute temperatures are consistent with equilibrium as well as with non-equilibrium thermodynamics. In particular, thermometry, thermodynamics of cyclic transformations, ensemble equivalence, fluctuation-dissipation relations, response theory and even transport processes can be reformulated to include them, thus dissipating any prejudice about their exceptionality, typically presumed as a manifestation of transient metastable effects., Comment: 64 pages, 19 figures

In spite of the (correct) common-wisdom statement correlation does not imply causation, a proper employ of time correlations and of fluctuation-response theory allows to understand the causal relations between the variables of a multi-dimensional linear Markov process. It is shown that the fluctuation-response formalism can be used both to find the direct causal links between the variables of a system and to introduce a degree of causation, cumulative in time, whose physical interpretation is straightforward. Although for generic non-linear dynamics there is no simple exact relationship between correlations and response functions, the described protocol can still give a useful proxy also in presence of weak nonlinear terms.

We provide here an explicit example of Khinchin's idea that the validity of equilibrium statistical mechanics in high dimensional systems does not depend on the details of the dynamics. This point of view is supported by extensive numerical simulation of the one-dimensional Toda chain, an integrable non-linear Hamiltonian system where all Lyapunov exponents are zero by definition. We study the relaxation to equilibrium starting from very atypical initial conditions and focusing on energy equipartion among Fourier modes, as done in the original Fermi-Pasta-Ulam-Tsingou numerical experiment. We find evidence that in the general case, i.e., not in the perturbative regime where Toda and Fourier modes are close to each other, there is a fast reaching of thermal equilibrium in terms of a single temperature. We also find that equilibrium fluctuations, in particular the behaviour of specific heat as function of temperature, are in agreement with analytic predictions drawn from the ordinary Gibbs ensemble, still having no conflict with the established validity of the Generalized Gibbs Ensemble for the Toda model. Our results suggest thus that even an integrable Hamiltonian system reaches thermalization on the constant energy hypersurface, provided that the considered observables do not strongly depend on one or few of the conserved quantities. This suggests that dynamical chaos is irrelevant for thermalization in the large-$N$ limit, where any macroscopic observable reads of as a collective variable with respect to the coordinate which diagonalize the Hamiltonian. The possibility for our results to be relevant for the problem of thermalization in generic quantum systems, i.e., non-integrable ones, is commented at the end., Comment: 6 pages, 4 figures + Appendix

We scrutinize the use of machine learning, based on reservoir computing, to build data-driven effective models of multiscale chaotic systems. We show that, for a wide scale separation, machine learning generates effective models akin to those obtained using multiscale asymptotic techniques and, remarkably, remains effective in predictability also when the scale separation is reduced. We also show that predictability can be improved by hybridizing the reservoir with an imperfect model., Comment: 12 pages with 10 figures. Accepted in Physical Review E

Datacentric enthusiasm is growing strong across a variety of domains. Whilst data science asks unquestionably exciting scientific questions, we argue that its contributions should not be extrapolated from the scientific context in which they originate. In particular we suggest that the simple-minded idea to the effect that data can be seen as a replacement for scientific modelling is not tenable. By recalling some well-known examples from dynamical systems we conclude that data science performs at its best when coupled with the subtle art of modelling., Comment: arXiv admin note: text overlap with arXiv:1810.10446

Data Science and Machine learning have been growing strong for the past decade. We argue that to make the most of this exciting field we should resist the temptation of assuming that forecasting can be reduced to brute-force data analytics. This owes to the fact that modelling, as we illustrate below, requires mastering the art of selecting relevant variables. More specifically, we investigate the subtle relation between "data and models" by focussing on the role played by algorithmic complexity, which contributed to making mathematically rigorous the long-standing idea that to understand empirical phenomena is to describe the rules which generate the data in terms which are "simpler" than the data itself. A key issue for the appraisal of the relation between algorithmic complexity and algorithmic learning is to do with a much needed clarification on the related but distinct concepts of compressibility, determinism and predictability. To this end we will illustrate that the evolution law of a chaotic system is compressibile, but a generic initial condition for it is not, making the time series generated by chaotic systems incompressible in general. Hence knowledge of the rules which govern an empirical phenomenon are not sufficient for predicting its outcomes. In turn this implies that there is more to understanding phenomena than learning -- even from data alone -- such rules. This can be achieved only in those cases when we are capable of "good modelling". Clearly, the very idea of algorithmic complexity rests on Turing's seminal analysis of computation. This motivates our remarks on this extremely telling example of analogy-based abstract modelling which is nonetheless heavily informed by empirical facts., Comment: arXiv admin note: substantial text overlap with arXiv:1709.07062

In the framework of the problem of finding proper reaction coordinates (RCs) for complex systems and their effective evolution equations, we consider the case study of a polymer chain in an external double-well potential, experiencing thermally activated dynamics. Langevin effective equations describing the macroscopic dynamics of the system can be inferred from data by using a data-driven approach, once a suitable set of RCs is chosen. We show that, in this case, the validity of such choice depends on the stiffness of the polymer's bonds: if they are sufficiently rigid, we can employ a reduced description based only on the coordinate of the center of mass; whereas, if the stiffness reduces, the one-variable dynamics is no more Markovian and (at least) a second reaction coordinate has to be taken into account to achieve a realistic dynamical description in terms of memoryless Langevin equations.

The problem of effective equations is reviewed and discussed. Starting from the classical Langevin equation, we show how it can be generalized to Hamiltonian systems with non-standard kinetic terms. A numerical method for inferring effective equations from data is discussed; this protocol allows to check the validity of our results. In addition we show that, with a suitable treatment of time series, such protocol can be used to infer effective models from experimental data. We briefly discuss the practical and conceptual difficulties of a pure data-driven approach in the building of models.

With the aid of simple analytical computations for the Ehrenfest model, we clarify some basic features of macroscopic irreversibility. The stochastic character of the model allows us to give a non-ambiguous interpretation of the general idea that irreversibility is a typical property: for the vast majority of the realizations of the stochastic process, a single trajectory of a macroscopic observable behaves irreversibly, remaining "very close" to the deterministic evolution of its ensemble average, which can be computed using probability theory. The validity of the above scenario is checked through simple numerical simulations and a rigorous proof of the typicality is provided in the thermodynamic limit.

A Hamiltonian model living in a bounded phase space and with long-range interactions is studied. It is shown, by analytical computations, that there exists an energy interval in which the microcanonical entropy is a decreasing convex function of the total energy, meaning that ensemble equivalence is violated in a negative-temperature regime. The equilibrium properties of the model are then investigated by molecular dynamics simulations: first, the caloric curve is reconstructed for the microcanonical ensemble and compared to the analytical prediction, and a generalized Maxwell-Boltzmann distribution for the momenta is observed; then, the nonequivalence between the microcanonical and canonical descriptions is explicitly shown. Moreover, the validity of Fluctuation-Dissipation Theorem is verified through a numerical study, also at negative temperature and in the region where the two ensembles are nonequivalent.

By employing a path integral formulation, we obtain the entropy production rate for a system of active Ornstein-Uhlenbeck particles (AOUP) both in the presence and in the absence of thermal noise. The present treatment clarifies some contraddictions concerning the definition of the entropy production rate in the AOUP model, recently appeared in the literature. We derive explicit formulas for three different cases: overdamped Brownian particle, AOUP with and without thermal noise. In addition, we show that it is not necessary to introduce additional hypotheses concerning the parity of auxiliary variables under time reversal transformation. Our results agree with those based on a previous mesoscopic approach.

We consider the problem of building a continuous stochastic model, i.e. a Langevin or Fokker-Planck equation, through a well-controlled coarse-graining procedure. Such a method usually involves the elimination of the fast degrees of freedom of the "bath" to which the particle is coupled. Specifically, we look into the general case where the bath may be at negative temperatures, as found - for instance - in models and experiments with bounded effective kinetic energy. Here, we generalise previous studies by considering the case in which the coarse-graining leads to (i) a renormalisation of the potential felt by the particle, and (ii) spatially dependent viscosity and diffusivity. In addition, a particular relevant example is provided, where the bath is a spin system and a sort of phase transition takes place when going from positive to negative temperatures. A Chapman-Enskog-like expansion allows us to rigorously derive the Fokker-Planck equation from the microscopic dynamics. Our theoretical predictions show an excellent agrement with numerical simulations.

A model has two main aims: predicting the behavior of a physical system and understanding its nature, that is how it works, at some desired level of abstraction. A promising recent approach to model building consists in deriving a Langevin-type stochastic equation from a time series of empirical data. Even if the protocol is based upon the introduction of drift and diffusion terms in stochastic differential equations, its implementation involves subtle conceptual problems and, most importantly, requires some prior theoretical knowledge about the system. Here we apply this approach to the data obtained in a rotational granular diffusion experiment, showing the power of this method and the theoretical issues behind its limits. A crucial point emerged in the dense liquid regime, where the data reveal a complex multiscale scenario with at least one fast and one slow variable. Identifying the latter is a major problem within the Langevin derivation procedure and led us to introduce innovative ideas for its solution.

We study a system of non-interacting active particles, propelled by colored noises, characterized by an activity time $\tau$, and confined by a double-well potential. A straightforward application of this system is the problem of barrier crossing of active particles, which has been studied only in the limit of small activity. When $\tau$ is sufficiently large, equilibrium-like approximations break down in the barrier crossing region. In the model under investigation, it emerges a sort of "negative temperature" region, and numerical simulations confirm the presence of non-convex local velocity distributions. We propose, in the limit of large $\tau$, approximate equations for the typical trajectories which successfully predict many aspects of the numerical results. The local breakdown of detailed balance and its relation with a recent definition of non-equilibrium heat exchange is also discussed., Comment: 5 figures