Nonequilibrium statistical mechanics of the turbulent energy cascade: Irreversibility and response functions.

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.

Correlation, response and entropy approaches to allosteric behaviors: a critical comparison on the ubiquitin case.

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.

Extracting causation from millennial-scale climate fluctuations in the last 800 kyr.

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, CO[Formula: see text] 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 CO[Formula: see text] 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 CO[Formula: see text] 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 CO[Formula: see text] concentration.

Excess and loss of entropy production for different levels of coarse graining.

We investigate the effect of coarse graining on the thermodynamic 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 friction coefficient, the two dynamics have the same thermodynamics properties, while, for smaller friction 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 friction coefficient.

Response and flux of information in extended nonequilibrium dynamics.

It is well known that entropy production is a proxy to the detection of nonequilibrium, 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.

Effective models and predictability of chaotic multiscale systems via machine learning.

Understanding and modeling the dynamics of multiscale systems is a problem of considerable interest both for theory and applications. For unavoidable practical reasons, in multiscale systems, there is the need to eliminate from the description the fast and small-scale degrees of freedom and thus build effective models for only the slow and large-scale degrees of freedom. When there is a wide scale separation between the degrees of freedom, asymptotic techniques, such as the adiabatic approximation, can be used for devising such effective models, while away from this limit there exist no systematic techniques. Here, 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.

Derivation of a Langevin equation in a system with multiple scales: The case of negative temperatures.

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 generalize previous studies by considering the case in which the coarse graining leads to (i) a renormalization 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 excellent agreement with numerical simulations.

Statistical mechanics of systems with long-range interactions and negative absolute temperature.

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 the Fluctuation-Dissipation Theorem is verified through a numerical study, also at negative temperature and in the region where the two ensembles are nonequivalent.

Langevin equations from experimental data: The case of rotational diffusion in granular media.

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.

Reaction fronts in persistent random walks with demographic stochasticity.

Standard reaction-diffusion systems are characterized by infinite velocities and no persistence in the movement of individuals, two conditions that are violated when considering living organisms. Here we consider a discrete particle model in which individuals move following a persistent random walk with finite speed and grow with logistic dynamics. We show that, when the number of individuals is very large, the individual-based model is well described by the continuous reactive Cattaneo equation (RCE), but for smaller values of the carrying capacity important finite-population effects arise. The effects of fluctuations on the propagation speed are investigated both considering the RCE with a cutoff in the reaction term and by means of numerical simulations of the individual-based model. Finally, a more general Lévy walk process for the transport of individuals is examined and an expression for the front speed of the resulting traveling wave is proposed.

Active escape dynamics: The effect of persistence on barrier crossing.

We study a system of non-interacting active particles, propelled by colored noises, characterized by an activity time τ, 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 τ is sufficiently large, equilibrium-like approximations break down in the barrier crossing region. In the model under investigation, it emerges as 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 τ, 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.

DCA for genome-wide epistasis analysis: the statistical genetics perspective.

Direct coupling analysis (DCA) is a now widely used method to leverage statistical information from many similar biological systems to draw meaningful conclusions on each system separately. DCA has been applied with great success to sequences of homologous proteins, and also more recently to whole-genome population-wide sequencing data. We here argue that the use of DCA on the genome scale is contingent on fundamental issues of population genetics. DCA can be expected to yield meaningful results when a population is in the quasi-linkage equilibrium (QLE) phase studied by Kimura and others, but not, for instance, in a phase of clonal competition. We discuss how the exponential (Potts model) distributions emerge in QLE, and compare couplings to correlations obtained in a study of about 3000 genomes of the human pathogen Streptococcus pneumoniae.

Thermodynamics and Statistical Mechanics of Small Systems.

A challenging frontier in modern statistical physics is concerned with systems with a small number of degrees of freedom, far from the thermodynamic limit.[...].

The Role of Data in Model Building and Prediction: A Survey Through Examples.

The goal of Science is to understand phenomena and systems in order to predict their development and gain control over them. In the scientific process of knowledge elaboration, a crucial role is played by models which, in the language of quantitative sciences, mean abstract mathematical or algorithmical representations. This short review discusses a few key examples from Physics, taken from dynamical systems theory, biophysics, and statistical mechanics, representing three paradigmatic procedures to build models and predictions from available data. In the case of dynamical systems we show how predictions can be obtained in a virtually model-free framework using the methods of analogues, and we briefly discuss other approaches based on machine learning methods. In cases where the complexity of systems is challenging, like in biophysics, we stress the necessity to include part of the empirical knowledge in the models to gain the minimal amount of realism. Finally, we consider many body systems where many (temporal or spatial) scales are at play-and show how to derive from data a dimensional reduction in terms of a Langevin dynamics for their slow components.

Chaotic Lagrangian models for turbulent relative dispersion.

A deterministic multiscale dynamical system is introduced and discussed as a prototype model for relative dispersion in stationary, homogeneous, and isotropic turbulence. Unlike stochastic diffusion models, here trajectory transport and mixing properties are entirely controlled by Lagrangian chaos. The anomalous "sweeping effect," a known drawback common to kinematic simulations, is removed through the use of quasi-Lagrangian coordinates. Lagrangian dispersion statistics of the model are accurately analyzed by computing the finite-scale Lyapunov exponent (FSLE), which is the optimal measure of the scaling properties of dispersion. FSLE scaling exponents provide a severe test to decide whether model simulations are in agreement with theoretical expectations and/or observation. The results of our numerical experiments cover a wide range of "Reynolds numbers" and show that chaotic deterministic flows can be very efficient, and numerically low-cost, models of turbulent trajectories in stationary, homogeneous, and isotropic conditions. The mathematics of the model is relatively simple, and, in a geophysical context, potential applications may regard small-scale parametrization issues in general circulation models, mixed layer, and/or boundary layer turbulence models as well as Lagrangian predictability studies.

Frequency-control of protein translocation across an oscillating nanopore.

The translocation of a lipid binding protein (LBP) is studied using a phenomenological coarse-grained computational model that simplifies both chain and pore geometry. We investigated via molecular dynamics the interplay between transport and unfolding in the presence of a nanopore whose section oscillates periodically in time with a frequency ω, a motion often referred to as the radial breathing mode (RBM). We found that the LPB when mechanically pulled into the vibrating nanopore exhibits a translocation dynamics that in some frequency range is accelerated and shows a frequency locking to the pore dynamics. The main effect of pore vibrations is the suppression of stalling events of the translocation dynamics, hence, proper frequency tuning allows both regularization and control of the overall transport process. Finally, the interpretation of the simulation results is easily achieved by resorting to a first passage theory of elementary driven-diffusion processes.

Linear and anomalous front propagation in systems with non-Gaussian diffusion: The importance of tails.

We investigate front propagation in systems with diffusive and subdiffusive behavior. The scaling behavior of moments of the diffusive problem, both in the standard and in the anomalous cases, is not enough to determine the features of the reactive front. In fact, the shape of the bulk of the probability distribution of the transport process, which determines the diffusive properties, is important just for preasymptotic behavior of front propagation, while the precise shape of the tails of the probability distribution determines asymptotic behavior of front propagation.

Linear and nonlinear thermodynamics of a kinetic heat engine with fast transformations.

We investigate a kinetic heat engine model composed of particles enclosed in a box where one side acts as a thermostat and the opposite side is a piston exerting a given pressure. Pressure and temperature are varied in a cyclical protocol of period τ: their relative excursions, δ and ε, respectively, constitute the thermodynamic forces dragging the system out of equilibrium. The analysis of the entropy production of the system allows us to define the conjugated fluxes, which are proportional to the extracted work and the consumed heat. In the limit of small δ and ε the fluxes are linear in the forces through a τ-dependent Onsager matrix whose off-diagonal elements satisfy a reciprocal relation. The dynamics of the piston can be approximated, through a coarse-graining procedure, by a Klein-Kramers equation which-in the linear regime-yields analytic expressions for the Onsager coefficients and the entropy production. A study of the efficiency at maximum power shows that the Curzon-Ahlborn formula is always an upper limit which is approached at increasing values of the thermodynamic forces, i.e., outside of the linear regime. In all our analysis the adiabatic limit τ→∞ and the the small-force limit δ,ε→0 are not directly related.