Statistical features of systems driven by non-Gaussian processes: theory & practice

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.Comment: 32 pages, 11 figures. 22 pages main text, 10 pages appendice

Nonsymmetric Interactions Trigger Collective Swings in Globally Ordered Systems

Many systems in nature, from ferromagnets to flocks of birds, exhibit ordering phenomena on the large scale. In condensed matter systems, order is statistically robust for large enough dimensions, with relative fluctuations due to noise vanishing with system size. Several biological systems, however, are less stable and spontaneously change their global state on relatively short time scales. Here we show that there are two crucial ingredients in these systems that enhance the effect of noise, leading to collective changes of state on finite time scales and off-equilibrium behavior: the nonsymmetric nature of interactions between individuals, and the presence of local heterogeneities in the topology of the network. Our results might explain what is observed in several living systems and are consistent with recent experimental data on bird flocks and other animal groups

Silent Flocks

Experiments find coherent information transfer through biological groups on length and time scales distinctly below those on which asymptotically correct hydrodynamic theories apply. We present here a new continuum theory of collective motion coupling the velocity and density fields of Toner and Tu to the inertial spin field recently introduced to describe information propagation in natural flocks of birds. The long-wavelength limit of the new equations reproduces Toner-Tu theory, while at shorter wavelengths (or, equivalently, smaller damping), spin fluctuations dominate over density fluctuations and second sound propagation of the kind observed in real flocks emerges. We study the dispersion relation of the new theory and find that when the speed of second sound is large, a gap sharply separates first from second sound modes. This gap implies the existence of `silent' flocks, namely medium-sized systems across which neither first nor second sound can propagate

Short-range interaction vs long-range correlation in bird flocks

Bird flocks are a paradigmatic example of collective motion. One of the prominent experimental traits discovered about flocks is the presence of long range velocity correlations between individuals, which allow them to influence each other over the large scales, keeping a high level of group coordination. A crucial question is to understand what is the mutual interaction between birds generating such nontrivial correlations. Here we use the Maximum Entropy (ME) approach to infer from experimental data of natural flocks the effective interactions between birds. Compared to previous studies, we make a significant step forward as we retrieve the full functional dependence of the interaction on distance and find that it decays exponentially over a range of a few individuals. The fact that ME gives a short-range interaction even though its experimental input is the long-range correlation function, shows that the method is able to discriminate the relevant information encoded in such correlations and single out a minimal number of effective parameters. Finally, we show how the method can be used to capture the degree of anisotropy of mutual interactions.Comment: 21 pages, 7 figures, 1 tabl

Social interactions dominate speed control in driving natural flocks toward criticality

Flocks of birds exhibit a remarkable degree of coordination and collective response. It is not just that thousands of individuals fly, on average, in the same direction and at the same speed, but that even the fluctuations around the mean velocity are correlated over long distances. Quantitative measurements on flocks of starlings, in particular, show that these fluctuations are scale-free, with effective correlation lengths proportional to the linear size of the flock. Here we construct models for the joint distribution of velocities in the flock that reproduce the observed local correlations between individuals and their neighbors, as well as the variance of flight speeds across individuals, but otherwise have as little structure as possible. These minimally structured, or maximum entropy models provide quantitative, parameter-free predictions for the spread of correlations throughout the flock, and these are in excellent agreement with the data. These models are mathematically equivalent to statistical physics models for ordering in magnets, and the correct prediction of scale-free correlations arises because the parameters - completely determined by the data - are in the critical regime. In biological terms, criticality allows the flock to achieve maximal correlation across long distances with limited speed fluctuations

Out-of-Equilibrium Non-Gaussian Behavior in Driven Granular Gases

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.Comment: 5 pages, 4 figure

Emergence of collective changes in travel direction of starling flocks from individual birds fluctuations

One of the most impressive features of moving animal groups is their ability to perform sudden coherent changes in travel direction. While this collective decision can be a response to an external perturbation, such as the presence of a predator, recent studies show that such directional switching can also emerge from the intrinsic fluctuations in the individual behaviour. However, the cause and the mechanism by which such collective changes of direction occur are not fully understood yet. Here, we present an experimental study of spontaneous collective turns in natural flocks of starlings. We employ a recently developed tracking algorithm to reconstruct three-dimensional trajectories of each individual bird in the flock for the whole duration of a turning event. Our approach enables us to analyze changes in the individual behavior of every group member and reveal the emergent dynamics of turning. We show that spontaneous turns start from individuals located at the elongated edges of the flocks, and then propagate through the group. We find that birds on the edges deviate from the mean direction of motion much more frequently than other individuals, indicating that persistent localized fluctuations are the crucial ingredient for triggering a collective directional change. Finally, we quantitatively show that birds follow equal radius paths during turning allowing the flock to change orientation and redistribute risky locations among group members. The whole process of turning is a remarkable example of how a self-organized system can sustain collective changes and reorganize, while retaining coherence.Comment: 18 pages, 2 Videos adde