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.
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.
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.
The Connection between Discrete and Continuous Time Descriptions of Gaussian Continuous Processes
F. Ferretti, V. Chard`es, T. Mora, A. M. Walczak, I. Giardina
Learning the continuous equations of motion from discrete observations is a common task in all areas of physics. However, not any discretization of a Gaussian continuous-time stochastic process can be adopted in parametric inference. We show that discretizations yielding consistent estimators have the property of ‘invariance under coarse-graining’, and correspond to fixed points of a renormalization group map on the space of autoregressive moving average (ARMA) models (for linear processes). This result explains why combining differencing schemes for derivatives reconstruction and local-in-time inference approaches does not work for time series analysis of second or higher order stochastic differential equations, even if the corresponding integration schemes may be acceptably good for numerical simulations.
A novel control mechanism for natural flocks
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 within natural flocks are moderate, as it is natural to expect given the aerodynamic, energetic and biomechanical constraints of flight. Yet the spatial correlations of such fluctuations have a range as wide as the entire group: the speed change of a bird influences and is influenced by that of birds on the other side of the flock. Long-range correlations and limited speed fluctuations set conflicting constraints on the mechanism controlling the individual speed of each bird, as the factors boosting correlations also tend to amplify speed fluctuations and vice versa. Here, building on experimental field data on starling flocks with group sizes spanning an unprecedented interval of over two orders of magnitude, we develop a novel type of speed control mechanism that generates scale-free correlations yet allowing for biologically plausible values of the flocks’ speed. The bare-bones hallmark of this new control is a strongly nonlinear speed-restoring force that ignores small deviations of the individual speed from its natural reference value, while ferociously suppressing larger speed fluctuations. Numerical simulations of self-propelled particles flocks in the same size range as the empirical data fully support our theoretical results. The match of field empirical data, theory and simulations over such an extensive range of sizes suggests that the novel speed control we propose is the most economic theoretical mechanism able to reproduce all key experimental traits of real flocks.
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.
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.
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.″
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.