Nature Physics
Natural swarms in 3.99 dimensions
A. Cavagna, L. Di Carlo, I. Giardina, T.S. Grigera, S. Melillo, L. Parisi, G. Pisegna, M. Scandolo
The renormalization group is a key set of ideas and quantitative tools of statistical physics that allow for the calculation of universal quantities that encompass the behaviour of different kinds of collective systems. Extension of the predictive power of the renormalization group to collective biological systems would greatly strengthen the effort to put physical biology on a firm basis. Here we present a step in that direction by calculating the dynamical critical exponent z of natural swarms of insects using the renormalization group to order ϵ = 4 − d. We report the emergence of a novel fixed point, where both activity and inertia are relevant. In three dimensions, the critical exponent at the new fixed point is z = 1.35, in agreement with both experiments (1.37 ± 0.11) and numerical simulations (1.35 ± 0.04). Our results probe the power of the renormalization group for the quantitative description of collective behaviour, and suggest that universality may also play a decisive role in strongly correlated biological systems.
Ecological Indicators
Testing for stationary dynamics in the Barro Colorado Island forest
A. Cavagna, H. Fort, T. S. Grigera
We analyse population dynamics in Barro Colorado Island (Panama) using census data of a 50 ha forest plot spanning 35 years, and address the question whether this community is in a stationary state. Individual species abundances show large fluctuations, but assessing stationariety requires discriminating random fluctuations from actual trends. This requires evaluating mean quantities as well as the structure (i.e. the correlations) of the fluctuations around this mean. We argue that a species average is the best surrogate for the theoretically required but unfeasible history average. We define the overlap, a species-averaged measure of composition similarity, which reveals that the BCI population dynamics is stationary but not static, displaying fluctuations with a characteristic time of around 15 years, two orders of magnitude less than previously estimated.


Kardar–Parisi–Zhang universality in a one-dimensional polariton condensate
Q. Fontaine, D. Squizzato, F. Baboux, I. Amelio, A. Lemaître, M. Morassi, I. Sagnes, L. Le Gratiet, A. Harouri, M. Wouters, I. Carusotto, A. Amo, M. Richard, A. Minguzzi, L. Canet, S. Ravets, J. Bloch
Revealing universal behaviours is a hallmark of statistical physics. Phenomena such as the stochastic growth of crystalline surfaces and of interfaces in bacterial colonies, and spin transport in quantum magnets all belong to the same universality class, despite the great plurality of physical mechanisms they involve at the microscopic level. More specifically, in all these systems, space–time correlations show power-law scalings characterized by universal critical exponents. This universality stems from a common underlying effective dynamics governed by the nonlinear stochastic Kardar–Parisi–Zhang (KPZ) equation. Recent theoretical works have suggested that this dynamics also emerges in the phase of out-of-equilibrium systems showing macroscopic spontaneous coherence. Here we experimentally demonstrate that the evolution of the phase in a driven-dissipative one-dimensional polariton condensate falls in the KPZ universality class. Our demonstration relies on a direct measurement of KPZ space–time scaling laws, combined with a theoretical analysis that reveals other key signatures of this universality class. Our results highlight fundamental physical differences between out-of-equilibrium condensates and their equilibrium counterparts, and open a paradigm for exploring universal behaviours in driven open quantum systems.
Physical Review E
Renormalization group study of marginal ferromagnetism
A. Cavagna, A. Culla, T.S. Grigera
When studying the collective motion of biological groups, a useful theoretical framework is that of ferromagnetic systems, in which the alignment interactions are a surrogate of the effective imitation among the individuals. In this context, the experimental discovery of scale-free correlations of speed fluctuations in starling flocks poses a challenge to common statistical physics wisdom, as in the ordered phase of standard ferromagnetic models with O(n) symmetry, the modulus of the order parameter has finite correlation length. To make sense of this anomaly, a ferromagnetic theory has been proposed, where the bare confining potential has zero second derivative (i.e., it is marginal) along the modulus of the order parameter. The marginal model exhibits a zero-temperature critical point, where the modulus correlation length diverges, hence allowing us to boost both correlation and collective order by simply reducing the temperature. Here, we derive an effective field theory describing the marginal model close to the T = 0 critical point and calculate the renormalization group equations at one loop within a momentum shell approach. We discover a nontrivial scenario, as the cubic and quartic vertices do not vanish in the infrared limit, while the coupling constants effectively regulating the exponents ν and η have upper critical dimension dc = 2, so in three dimensions the critical exponents acquire their free values, ν = 1/2 and η = 0. This theoretical scenario is verified by a Monte Carlo study of the modulus susceptibility in three dimensions, where the standard finite-size scaling relations have to be adapted to the case of d>dc. The numerical data fully confirm our theoretical results.
Nature Communications
Marginal speed confinement resolves the conflict between correlation and control in collective behaviour
A. Cavagna, A. Culla, X. Feng, I. Giardina, T.S. Grigera, W. Kion-Crosby, S. Melillo, G. Pisegna, L. Postiglione, P. Villegas
Speed fluctuations of individual birds in natural flocks are moderate, due to the aerodynamic and biomechanical constraints of flight. Yet the spatial correlations of such fluctuations are scale-free, namely they have a range as wide as the entire group, a property linked to the capacity of the system to collectively respond to external perturbations. Scale-free correlations and moderate fluctuations set conflicting constraints on the mechanism controlling the speed of each agent, as the factors boosting correlation amplify fluctuations, and vice versa. Here, using a statistical field theory approach, we suggest that a marginal speed confinement that ignores small deviations from the natural reference value while ferociously suppressing larger speed fluctuations, is able to reconcile scale-free correlations with biologically acceptable group’s speed. We validate our theoretical predictions by comparing them with field experimental data on starling flocks with group sizes spanning an unprecedented interval of over two orders of magnitude.
Physical Review E
Signatures of irreversibility in microscopic models of flocking
F. Ferretti, S. Grosse-Holz, C. Holmes, J. L. Shivers, I. Giardina, M. Thierry, A. M. Walczak
Flocking in d = 2 is a genuine nonequilibrium phenomenon for which irreversibility is an essential ingredient. We study a class of minimal flocking models whose only source of irreversibility is self-propulsion and use the entropy production rate (EPR) to quantify the departure from equilibrium across their phase diagrams. The EPR is maximal in the vicinity of the order-disorder transition, where reshuffling of the interaction network is fast. We show that signatures of irreversibility come in the form of asymmetries in the steady-state distribution of the flock’s microstates. These asymmetries occur as consequences of the time-reversal symmetry breaking in the considered self-propelled systems, independently of the interaction details. In the case of metric pairwise forces, they reduce to local asymmetries in the distribution of pairs of particles. This study suggests a possible use of pair asymmetries both to quantify the departure from equilibrium and to learn relevant information about aligning interaction potentials from data.
Physical Review E
Renormalization group approach to connect discrete- and continuous-time descriptions of Gaussian processes
F. Ferretti, V. Chardès, T. Mora, A. M. Walczak, I. Giardina
Discretization of continuous stochastic processes is needed to numerically simulate them or to infer models from experimental time series. However, depending on the nature of the process, the same discretization scheme may perform very differently for the two tasks, if it is not accurate enough. Exact discretizations, which work equally well at any scale, are characterized by the property of invariance under coarse-graining. Motivated by this observation, we build an explicit renormalization group (RG) approach for Gaussian time series generated by autoregressive models. We show that the RG fixed points correspond to discretizations of linear SDEs, and only come in the form of first order Markov processes or non-Markovian ones. This fact provides an alternative explanation of why standard delay-vector embedding procedures fail in reconstructing partially observed noise-driven systems. We also suggest a possible effective Markovian discretization for the inference of partially observed underdamped equilibrium processes based on the exploitation of the Einstein relation.
Physical Review Research
Inference of time irreversibility from incomplete information: Linear systems and its pitfalls
D. Lucente, A. Baldassarri, A. Puglisi, A. Vulpiani, M. Viale
Data from experiments and theoretical arguments are the two pillars sustaining the job of modeling 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, counterintuitive 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 timescales 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.
New Journal of Physics
Evidence of fluctuation-induced first-order phase transition in active matter
D. Lucente, A. Baldassarri, A. Puglisi, A. Vulpiani, M. Viale
We investigate the effects of density fluctuations on the near-ordering phase of a flock by studying the Malthusian Toner–Tu theory. Because of the birth/death process, characteristic of this Malthusian model, density fluctuations are partially suppressed. We show that unlike its incompressible counterpart, where the absence of the density fluctuations renders the ordering phase transition similar to a second-order phase transition, in the Malthusian theory density fluctuations may turn the phase from continuous to first-order. We study the model using a perturbative renormalization group approach. At one loop, we find that the renormalization group flow drives the system in an unstable region, suggesting a fluctuation-induced first-order phase transition.
Informational Entropy Threshold as a Physical Mechanism for Explaining Tree-Like Decision Making in Humans
J. Cristín, V. Méndez, D. Campos
While approaches based on physical grounds (such as the drift-diffusion model—DDM) have been exhaustively used in psychology and neuroscience to describe perceptual decision making in humans, similar approaches to complex situations, such as sequential (tree-like) decisions, are still scarce. For such scenarios that involve a reflective prospection of future options, we offer a plausible mechanism based on the idea that subjects can carry out an internal computation of the uncertainty about the different options available, which is computed through the corresponding Shannon entropy. When the amount of information gathered through sensory evidence is enough to reach a given threshold in the entropy, this will trigger the decision. Experimental evidence in favor of this entropy-based mechanism was provided by exploring human performance during navigation through a maze on a computer screen monitored with the help of eye trackers. In particular, our analysis allows us to prove that (i) prospection is effectively used by humans during such navigation tasks, and an indirect quantification of the level of prospection used is attainable; in addition, (ii) the distribution of decision times during the task exhibits power-law tails, a feature that our entropy-based mechanism is able to explain, unlike traditional (DDM-like) frameworks.


Journal of Statistical Physics
Dynamical Renormalization Group for Mode-Coupling Field Theories with Solenoidal Constraint
A. Cavagna, L. Di Carlo, I. Giardina, T.S. Grigera, G. Pisegna, M. Scandolo
The recent inflow of empirical data about the collective behaviour of strongly correlated biological systems has brought field theory and the renormalization group into the biophysical arena. Experiments on bird flocks and insect swarms show that social forces act on the particles’ velocity through the generator of its rotations, namely the spin, indicating that mode-coupling field theories are necessary to reproduce the correct dynamical behaviour. Unfortunately, a theory for three coupled fields—density, velocity and spin—has a prohibitive degree of intricacy. A simplifying path consists in getting rid of density fluctuations by studying incompressible systems. This requires imposing a solenoidal constraint on the primary field, an unsolved problem even for equilibrium mode-coupling theories. Here, we perform an equilibrium dynamic renormalization group analysis of a mode-coupling field theory subject to a solenoidal constraint; using the classification of Halperin and Hohenberg, we can dub this case as a solenoidal Model G. We demonstrate that the constraint produces a new vertex that mixes static and dynamical coupling constants, and that this vertex is essential to grant the closure of the renormalization group structure and the consistency of dynamics with statics. Interestingly, although the solenoidal constraint leads to a modification of the static universality class, we find that it does not change the dynamical universality class, a result that seems to represent an exception to the general rule that dynamical universality classes are narrower than static ones. Our results constitute a solid stepping stone in the admittedly large chasm towards developing an off-equilibrium mode-coupling theory of biological groups.
Physical Review E
Vicsek Model by Time-Interlaced Compression: a Dynamical Computable Information Density
A. Cavagna, P.M.Chaikin, D. Levine, S. Martiniani, A. Puglisi, M. Viale
Collective behavior, both in real biological systems as well as in theoretical models, often displays a rich combination of different kinds of order. A clear-cut and unique definition of “phase” based on the standard concept of order parameter may therefore be complicated, and made even trickier by the lack of thermodynamic equilibrium. Compression-based entropies have been proved useful in recent years in describing the different phases of out-of-equilibrium systems. Here, we investigate the performance of a compression-based entropy, namely the Computable Information Density (CID), within the Vicsek model of collective motion. Our entropy is defined through a crude coarse-graining of the particle positions, in which the key role of velocities in the model only enters indirectly through the velocity-density coupling. We discover that such entropy is a valid tool in distinguishing the various noise regimes, including the crossover between an aligned and misaligned phase of the velocities, despite the fact that velocities are not used by this entropy. Furthermore, we unveil the subtle role of the time coordinate, unexplored in previous studies on the CID: a new encoding recipe, where space and time locality are both preserved on the same ground, is demonstrated to reduce the CID. Such an improvement is particularly significant when working with partial and/or corrupted data, as it is often the case in real biological experiments.
IEEE Transactions on Instrumentation and Measurement
CoMo: A novel co-moving 3D camera system
A. Cavagna, X. Feng, S. Melillo, L. Parisi, L. Postiglione, P. Villegas
Motivated by the theoretical interest in reconstructing long 3D trajectories of individual birds in large flocks, we developed CoMo, a co-moving camera system of two synchronized high speed cameras coupled with rotational stages, which allow us to dynamically follow the motion of a target flock. With the rotation of the cameras we overcome the limitations of standard static systems that restrict the duration of the collected data to the short interval of time in which targets are in the cameras common field of view, but at the same time we change in time the external parameters of the system, which have then to be calibrated frame-by-frame. We address the calibration of the external parameters measuring the position of the cameras and their three angles of yaw, pitch and roll in the system home configuration (rotational stage at an angle equal to 0 ◦) and combining this static information with the time dependent rotation due to the stages. We evaluate the robustness and accuracy of the system by comparing reconstructed and measured 3D distances in what we call 3D tests, which show a relative error of the order of 1%. The novelty of the work presented in this paper is not only on the system itself, but also on the approach we use in the tests, which we show to be a very powerful tool in detecting and fixing calibration inaccuracies and that, for this reason, may be relevant for a broad audience.
Physical Review Research
Equilibrium to off-equilibrium crossover in homogeneous active matter
A. Cavagna, L. Di Carlo, I. Giardina, T.S. Grigera, G. Pisegna
We study the crossover between equilibrium and off-equilibrium dynamical universality classes in the Vicsek model near its ordering transition. Starting from the incompressible hydrodynamic theory of Chen et al. [Critical phenomenon of the order-disorder transition in incompressible active fluids, New J. Phys. 17, 042002 (2015)], we show that increasing the activity leads to a renormalization group (RG) crossover between the equilibrium ferromagnetic fixed point, with dynamical critical exponent z = 2, and the off-equilibrium active fixed point, with z = 1.7 (in d = 3). We run simulations of the classic Vicsek model in the near-ordering regime and find that critical slowing down indeed changes with activity, displaying two exponents that are in remarkable agreement with the RG prediction. The equilibrium to off-equilibrium crossover is ruled by a characteristic length scale, beyond which active dynamics takes over. The larger the activity is, the smaller is such a length scale, suggesting the existence of a general trade-off between activity and the system’s size in determining the dynamical universality class of active matter.
Royal Society
Joint assessment of density correlations and fluctuations for analysing spatial tree patterns
P. Villegas, A. Cavagna, M. Cencini, H. Fort, T.S. Grigera
Inferring the processes underlying the emergence of observed patterns is a key challenge in theoretical ecology. Much effort has been made in the past decades to collect extensive and detailed information about the spatial distribution of tropical rainforests, as demonstrated, e.g. in the 50 ha tropical forest plot on Barro Colorado Island, Panama. These kinds of plots have been crucial to shed light on diverse qualitative features, emerging both at the single-species or the community level, like the spatial aggregation or clustering at short scales. Here, we build on the progress made in the study of the density correlation functions applied to biological systems, focusing on the importance of accurately defining the borders of the set of trees, and removing the induced biases. We also pinpoint the importance of combining the study of correlations with the scale dependence of fluctuations in density, which are linked to the well-known empirical Taylor’s power law. Density correlations and fluctuations, in conjunction, provide a unique opportunity to interpret the behaviours and, possibly, to allow comparisons between data and models. We also study such quantities in models of spatial patterns and, in particular, we find that a spatially explicit neutral model generates patterns with many qualitative features in common with the empirical ones.


Physical Review X
Building general Langevin models from discrete data sets
F. Ferretti, V. Chardès, T. Mora, A. M. Walczak, I. Giardina
Experiments revealed inertial signatures in the effective dynamics of natural flocks of starlings and swarms of midges at their observational space-time scales. A quantitative assessment of the damping regime in which these systems operate may bring relevant information for the understanding of their dynamic properties and function. Learning from empirical observations is a long-standing goal in physics, but even today it can be often harder than expected. Building a stochastic model with memory from discrete time series is a challenging task, due to the non-Markovian character of the observed process. In the search for a robust inference method for our birds’ data, we developed a novel maximum likelihood scheme for non-Markovian dynamics with linear dissipation. Its potential application is not restricted to simple underdamped equilibrium models, but it can be implemented also in out-of-equilibrium and large size systems.


Physical Review Letters
Dynamical Renormalization Group approach to the collective behavior of swarms
A.Cavagna, L. Di Carlo, I. Giardina, L. Grandinetti, T. S. Grigera, G. Pisegna
The dynamical scaling hypothesis states that, for systems at the critical point the correlation length is linked to the characteristic relaxation time through the dynamic critical exponent z. Experimental results show that swarms of insects satisfy this property exhibiting an exponent z ≈1.2. Searching for a model able to reproduce this result, we study the critical properties of model with non-dissipative couplings. Using the Dynamical Renormalization Group technique, and working in the fixed network approximation, we find a crossover from a non-dissipative fixed point (z=d/2) to a dissipative ( z=2) fixed point. The interplay between these two fixed points gives rise to a crossover in the critical dynamics of the system. The value z=d/2 in 3d is significantly closer to the experimental value, then the value found numerically in fully dissipative models z =1.7. This result suggests that mode coupling interaction is a key ingredient to build a theory of natural swarms close to the experiments. Further investigations including the self-propelled nature of the particles are necessary to embed the out-of-equilibrium essence of these biological systems.″
Physical Review E
Renormalization group crossover in the critical dynamics of field theories with mode coupling terms
A. Cavagna, L. Di Carlo, I. Giardina, L. Grandinetti, T.S. Grigera, G. Pisegna
Motivated by the collective behavior of biological swarms, we study the critical dynamics of field theories with coupling between order parameter and conjugate momentum in the presence of dissipation. Under a fixed-network approximation, we perform a dynamical renormalization group calculation at one loop in the near-critical disordered region, and we show that the violation of momentum conservation generates a crossover between an unstable fixed point, characterized by a dynamic critical exponent z=d/2, and a stable fixed point with z=2. Interestingly, the two fixed points have different upper critical dimensions. The interplay between these two fixed points gives rise to a crossover in the critical dynamics of the system, characterized by a crossover exponent
κ = 4/d. The crossover is regulated by a conservation length scale R0, given by the ratio between the transport coefficient and the effective friction, which is larger as the dissipation is smaller: Beyond R0, the stable fixed point dominates, while at shorter distances dynamics is ruled by the unstable fixed point and critical exponent, a behavior which is all the more relevant in finite-size systems with weak dissipation. We run numerical simulations in three dimensions and find a crossover between the exponents z = 3/2 and z = 2 in the critical slowdown of the system, confirming the renormalization group results. From the biophysical point of view, our calculation indicates that in finite-size biological groups mode coupling terms in the equation of motion can significantly change the dynamical critical exponents even in the presence of dissipation, a step toward reconciling theory with experiments in natural swarms. Moreover, our result provides the scale within which fully conservative Bose-Einstein condensation is a good approximation in systems with weak symmetry-breaking terms violating number conservation, as quantum magnets or photon gases.
Comptes Rendus Physique
Low-temperature marginal ferromagnetism explains anomalous scale-free correlations in natural flocks
A.Cavagna, A. Culla, L. Di Carlo, I. Giardina, T.S. Grigera
We introduced and studied a novel ferromagnetic model, suitable to reproduce a fascinating statistical property of starlings speeds in their flocking phase. From experimental data one finds that, in a strongly polarized flock, speed’s fluctuations are correlated on a characteristic length that is proportional to the system’s size, i.e. speed correlations are scale-free. We show that, using our “marginal” model, it is possible to reproduce this specific and unusual phenomenology, even in the equilibrium case. The key idea is to consider a bounding potential for the single-particle speed that has zero curvature. A novel zero-temperature critical point emerges and the model develops divergent susceptibility and correlation length of the modulus of the microscopic degrees of freedom, in analogy with experimental data of natural flocks.
Our tasks now are many: first of all, we will investigate more deeply the nature of the “marginal” scale-free behaviour of the model, extracting, in a numerical and a rigorous way, the critical exponents. Then, we will closely compare the statistical field theory, built from our microscopic model, to the experimental data.