Large-scale collective properties of self-propelled rods

We study, in two space dimensions, the large-scale properties of collections of constant-speed polar point particles interacting locally by nematic alignment in the presence of noise. This minimal approach to self-propelled rods allows one to deal with large numbers of particles, revealing a phenomenology previously unseen in more complicated models, and moreover distinctively different from both that of the purely polar case (e.g. the Vicsek model) and of active nematics.Comment: Submitted to Phys. Rev. Let

Synchronization of spatio-temporal chaos as an absorbing phase transition: a study in 2+1 dimensions

The synchronization transition between two coupled replicas of spatio-temporal chaotic systems in 2+1 dimensions is studied as a phase transition into an absorbing state - the synchronized state. Confirming the scenario drawn in 1+1 dimensional systems, the transition is found to belong to two different universality classes - Multiplicative Noise (MN) and Directed Percolation (DP) - depending on the linear or nonlinear character of damage spreading occurring in the coupled systems. By comparing coupled map lattice with two different stochastic models, accurate numerical estimates for MN in 2+1 dimensions are obtained. Finally, aiming to pave the way for future experimental studies, slightly non-identical replicas have been considered. It is shown that the presence of small differences between the dynamics of the two replicas acts as an external field in the context of absorbing phase transitions, and can be characterized in terms of a suitable critical exponent.Comment: Submitted to Journal of Statistical Mechanics: Theory and Experimen

Fast moving of a population of robots through a complex scenario

Swarm robotics consists in using a large number of coordinated autonomous robots, or agents, to accomplish one or more tasks, using local and/or global rules. Individual and collective objectives can be designed for each robot of the swarm. Generally, the agents' interactions exhibit a high degree of complexity that makes it impossible to skip nonlinearities in the model. In this paper, is implemented both a collective interaction using a modified Vicsek model where each agent follows a local group velocity and the individual interaction concerning internal and external obstacle avoidance. The proposed strategies are tested for the migration of a unicycle robot swarm in an unknown environment, where the effectiveness and the migration time are analyzed. To this aim, a new optimal control method for nonlinear dynamical systems and cost functions, named Feedback Local Optimality Principle - FLOP, is applied

Emergence of chaotic behaviour in linearly stable systems

Strong nonlinear effects combined with diffusive coupling may give rise to unpredictable evolution in spatially extended deterministic dynamical systems even in the presence of a fully negative spectrum of Lyapunov exponents. This regime, denoted as ``stable chaos'', has been so far mainly characterized by numerical studies. In this manuscript we investigate the mechanisms that are at the basis of this form of unpredictable evolution generated by a nonlinear information flow through the boundaries. In order to clarify how linear stability can coexist with nonlinear instability, we construct a suitable stochastic model. In the absence of spatial coupling, the model does not reveal the existence of any self-sustained chaotic phase. Nevertheless, already this simple regime reveals peculiar differences between the behaviour of finite-size and that of infinitesimal perturbations. A mean-field analysis of the truly spatially extended case clarifies that the onset of chaotic behaviour can be traced back to the diffusion process that tends to shift the growth rate of finite perturbations from the quenched to the annealed average. The possible characterization of the transition as the onset of directed percolation is also briefly discussed as well as the connections with a synchronization transition.Comment: 30 pages, 8 figures, Submitted to Journal of Physics

Continuous theory of active matter systems with metric-free interactions

We derive a hydrodynamic description of metric-free active matter: starting from self-propelled particles aligning with neighbors defined by "topological" rules, not metric zones, -a situation advocated recently to be relevant for bird flocks, fish schools, and crowds- we use a kinetic approach to obtain well-controlled nonlinear field equations. We show that the density-independent collision rate per particle characteristic of topological interactions suppresses the linear instability of the homogeneous ordered phase and the nonlinear density segregation generically present near threshold in metric models, in agreement with microscopic simulations.Comment: Submitted to Physical Review Letter

Characterizing dynamics with covariant Lyapunov vectors

A general method to determine covariant Lyapunov vectors in both discrete- and continuous-time dynamical systems is introduced. This allows to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents.Comment: 4 pages, 3 figures, submitted to Physical Review letter

Nonequilibrium wetting of finite samples

As a canonical model for wetting far from thermal equilibrium we study a Kardar-Parisi-Zhang interface growing on top of a hard-core substrate. Depending on the average growth velocity the model exhibits a non-equilibrium wetting transition which is characterized by an additional surface critical exponent theta. Simulating the single-step model in one spatial dimension we provide accurate numerical estimates for theta and investigate the distribution of contact points between the substrate and the interface as a function of time. Moreover, we study the influence of finite-size effects, in particular the time needed until a finite substrate is completely covered by the wetting layer for the first time.Comment: 17 pages, 8 figures, revisio

Universal mean-field upper bound for the generalization gap of deep neural networks

Modern deep neural networks (DNNs) represent a formidable challenge for theorists: according to the commonly accepted probabilistic framework that describes their performance, these architectures should overfit due to the huge number of parameters to train, but in practice they do not. Here we employ results from replica mean field theory to compute the generalization gap of machine learning models with quenched features, in the teacher-student scenario and for regression problems with quadratic loss function. Notably, this framework includes the case of DNNs where the last layer is optimized given a specific realization of the remaining weights. We show how these results-combined with ideas from statistical learning theory-provide a stringent asymptotic upper bound on the generalization gap of fully trained DNN as a function of the size of the dataset P. In particular, in the limit of large P and N-out (where N-out is the size of the last layer) and N-out << P, the generalization gap approaches zero faster than 2N(out)/P, for any choice of both architecture and teacher function. Notably, this result greatly improves existing bounds from statistical learning theory. We test our predictions on a broad range of architectures, from toy fully connected neural networks with few hidden layers to state-of-the-art deep convolutional neural networks

Collective Lyapunov modes

We show, using covariant Lyapunov vectors in addition to standard Lyapunov analysis, that there exists a set of collective Lyapunov modes in large chaotic systems exhibiting collective dynamics. Associated with delocalized Lyapunov vectors, they act collectively on the trajectory and hence characterize the instability of its collective dynamics. We further develop, for globally-coupled systems, a connection between these collective modes and the Lyapunov modes in the corresponding Perron-Frobenius equation. We thereby address the fundamental question of the effective dimension of collective dynamics and discuss the extensivity of chaos in presence of collective dynamics.Comment: 24 pages, 15 figures; display error correction of Fig.11 and other minor changes (v2