Comment on Ihle, "Towards a quantitative kinetic theory of polar active matter"
A comment on the preprint "Towards a quantitative kinetic theory of polar active matter" by T. Ihle, arXiv:1401.8056.
cond-mat.soft↗
arXiv subjects
Francesco Ginelli
Publications and source records attributed to Francesco Ginelli.
Boltzmann-Ginzburg-Landau approach for continuous descriptions of generic Vicsek-like models
We describe a generic theoretical framework, denoted as the Boltzmann-Ginzburg-Landau approach, to derive continuous equations for the polar and/or nematic order parameters describing the large scale behavior of assemblies of point-like active particles interacting through polar or nematic alignment rules. Our study encompasses three main classes of dry active systems, namely polar particles with 'ferromagnetic' alignment (like the original Vicsek model), nematic particles with nematic alignment ("active nematics"), and polar particles with nematic alignment ("self-propelled rods"). The Boltzmann-Ginzburg-Landau approach combines a low-density description in the form of a Boltzmann equation, with a Ginzburg-Landau-type expansion close to the instability threshold of the disordered state. We provide the generic form of the continuous equations obtained for each class, and comment on the relationships and differences with other approaches.
cond-mat.stat-mech↗
Large-scale chaos and fluctuations in active nematics
We show that "dry" active nematics, e.g. collections of shaken elongated granular particles, exhibit large-scale spatiotemporal chaos made of interacting dense, ordered, band-like structures in a parameter region including the linear onset of nematic order. These results are obtained from the study of the relatively simple and well-known (deterministic) hydrodynamic equations describing these systems in a dilute limit, and of a self-propelled particle Vicsek-like model for this class of active matter. In this last case, revisiting the status of the strong fluctuations and long-range correlations now considered as landmarks of orientationally-ordered active phases, we show that the giant number fluctuations observed in the chaotic phase are a trivial consequence of density segregation. However anomalous density fluctuations are present in the homogeneous quasi-ordered nematic phase and characterized by a non-trivial scaling exponent.
cond-mat.soft↗
Dynamical maximum entropy approach to flocking
We derive a new method to infer from data the out-of-equilibrium alignment dynamics of collectively moving animal groups, by considering the maximum entropy distribution consistent with temporal and spatial correlations of flight direction. When bird neighborhoods evolve rapidly, this dynamical inference correctly learns the parameters of the model, while a static one relying only on the spatial correlations fails. When neighbors change slowly and detailed balance is satisfied, we recover the static procedure. We demonstrate the validity of the method on simulated data. The approach is applicable to other systems of active matter.
cond-mat.stat-mech↗
Mesoscopic theory for fluctuating active nematics
The term active nematics designates systems in which apolar elongated particles spend energy to move randomly along their axis and interact by inelastic collisions in the presence of noise. Starting from a simple Vicsek-style model for active nematics, we derive a mesoscopic theory, complete with effective multiplicative noise terms, using a combination of kinetic theory and Itô calculus approaches. The stochastic partial differential equations thus obtained are shown to recover the key terms argued in EPL \textbf{62} (2003) 196 to be at the origin of anomalous number fluctuations and long-range correlations. Their deterministic part is studied analytically, and is shown to give rise to the long-wavelength instability at onset of nematic order (see arXiv:1011.5408). The corresponding nonlinear density-segregated band solution is given in a closed form.
cond-mat.stat-mech↗
Polar vs. apolar alignment in systems of polar self-propelled particles
The symmetry of the alignment mechanism in systems of polar self-propelled particles determines the possible macroscopic large-scale patterns that can emerge. Here we compare polar and apolar alignment. These systems share some common features like giant number fluctuations in the ordered phase and self-segregation in the form of bands near the onset of orientational order. Despite these similarities, there are essential differences like the symmetry of the ordered phase and the stability of the bands.
physics.bio-ph↗
Covariant Lyapunov vectors
The recent years have witnessed a growing interest for covariant Lyapunov vectors (CLVs) which span local intrinsic directions in the phase space of chaotic systems. Here we review the basic results of ergodic theory, with a specific reference to the implications of Oseledets' theorem for the properties of the CLVs. We then present a detailed description of a "dynamical" algorithm to compute the CLVs and show that it generically converges exponentially in time. We also discuss its numerical performance and compare it with other algorithms presented in literature. We finally illustrate how CLVs can be used to quantify deviations from hyperbolicity with reference to a dissipative system (a chain of Hénon maps) and a Hamiltonian model (a Fermi-Pasta-Ulam chain).
nlin.CD↗
Simple and faithful nonlinear field equations for aligning self-propelled rods
We derive a set of minimal yet complete nonlinear field equations describing the collective properties of self-propelled rods from a simple microscopic starting point, the Vicsek model with nematic alignment. Analysis of their linear and nonlinear dynamics shows good agreement with the original microscopic model. In particular, we derive an explicit expression for the fronts forming density-segregated, banded solutions, allowing us to develop a more complete analytic picture of the problem at the nonlinear level.
cond-mat.soft↗
Boundary information inflow enhances correlation in flocking
The most conspicuous trait of collective animal behaviour is the emergence of highly ordered structures. Less obvious to the eye, but perhaps more profound a signature of self-organization, is the presence of long-range spatial correlations. Experimental data on starling flocks in 3d show that the exponent ruling the decay of the velocity correlation function, C(r) ~ 1/r^γ, is extremely small, γ<< 1. This result can neither be explained by equilibrium field theory, nor by off-equilibrium theories and simulations of active systems. Here, by means of numerical simulations and theoretical calculations, we show that a dynamical field applied to the boundary of a set of Heisemberg spins on a 3d lattice, gives rise to a vanishing exponent γ, as in starling flocks. The effect of the dynamical field is to create an information inflow from border to bulk that triggers long range spin wave modes, thus giving rise to an anomalously long-ranged correlation. The biological origin of this phenomenon can be either exogenous - information produced by environmental perturbations is transferred from boundary to bulk of the flock - or endogenous - the flock keeps itself in a constant state of dynamical excitation that is beneficial to correlation and collective response.
cond-mat.soft↗
Competing ferromagnetic and nematic alignment in self-propelled polar particles
We study a Vicsek-style model of self-propelled particles where ferromagnetic and nematic alignment compete in both the usual "metric" version and in the "metric-free" case where a particle interacts with its Voronoi neighbors. We show that the phase diagram of this out-of-equilibrium XY model is similar to that of its equilibrium counterpart: the properties of the fully-nematic model, studied before in [F. Ginelli, F. Peruani, M. Baer, and H. Chaté, Phys. Rev. Lett. 104, 184502 (2010)], are thus robust to the introduction of a modest bias of interactions towards ferromagnetic alignment. The direct transitions between polar and nematic ordered phases are shown to be discontinuous in the metric case, and continuous, belonging to the Ising universality class, in the metric-free version.
cond-mat.soft↗
Chaos in the Hamiltonian mean field model
We study the dynamical properties of the canonical ordered phase of the Hamiltonian mean-field (HMF) model, in which $N$ particles, globally-coupled via pairwise attractive interactions, form a rotating cluster. Using a combination of numerical and analytical arguments, we first show that the largest Lyapunov exponent remains strictly positive in the infinite-size limit, converging to its asymptotic value with $1/\ln N$ corrections. We then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered, clustered phase and the disordered phase present at high energy densities. We also show that the full spectrum of Lyapunov exponents consists of a bulk component converging to the (zero) value taken by a test oscillator forced by the mean field, plus subextensive bands of ${\cal O}(\ln N)$ exponents taking finite values. We finally investigate the robustness of these results by studying a "2D" extension of the HMF model where each particle is endowed with 4 degrees of freedom, thus allowing the emergence of chaos at the level of single particle. Altogether, these results illustrate the subtle effects of global (or long-range) coupling, and the importance of the order in which the infinite-time and infinite-size limits are taken: for an infinite-size HMF system represented by the Vlasov equation, no chaos is present, while chaos exists and subsists for any finite system size.
nlin.CD↗
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.
cond-mat.soft↗
Hyperbolic decoupling of tangent space and effective dimension of dissipative systems
We show, using covariant Lyapunov vectors, that the tangent space of spatially-extended dissipative systems is split into two hyperbolically decoupled subspaces: one comprising a finite number of frequently entangled "physical" modes, which carry the physically relevant information of the trajectory, and a residual set of strongly decaying "spurious" modes. The decoupling of the physical and spurious subspaces is defined by the absence of tangencies between them and found to take place generally; we find evidence in partial differential equations in one and two spatial dimensions and even in lattices of coupled maps or oscillators. We conjecture that the physical modes may constitute a local linear description of the inertial manifold at any point in the global attractor.
nlin.CD↗
Extensive and Sub-Extensive Chaos in Globally-Coupled Dynamical Systems
Using a combination of analytical and numerical techniques, we show that chaos in globally-coupled identical dynamical systems, be they dissipative or Hamiltonian, is both extensive and sub-extensive: their spectrum of Lyapunov exponents is asymptotically flat (thus extensive) at the value $λ_0$ given by a single unit forced by the mean-field, but sandwiched between sub-extensive bands containing typically $\mathcal{O}(\log N)$ exponents whose values vary as $λ\simeq λ_\infty + c/\log N$ with $λ_\infty \neq λ_0$.
nlin.CD↗
Relevance of Metric-Free Interactions in Flocking Phenomena
We show that the collective properties of self-propelled particles aligning with their "topological" (Voronoi) neighbors are qualitatively different from those of usual models where metric interaction ranges are used. This relevance of metric-free interactions, shown in a minimal setting, indicate that realistic models for the cohesive motion of cells, bird flocks, and fish schools may have to incorporate them, as suggested by recent observations.
cond-mat.stat-mech↗
Lyapunov analysis captures the collective dynamics of large chaotic systems
We show, using generic globally-coupled systems, that the collective dynamics of large chaotic systems is encoded in their Lyapunov spectra: most modes are typically localized on a few degrees of freedom, but some are delocalized, acting collectively on the trajectory. For globally-coupled maps, we show moreover a quantitative correspondence between the collective modes and some of the so-called Perron-Frobenius dynamics. Our results imply that the conventional definition of extensivity must be changed as soon as collective dynamics sets in.
nlin.CD↗
French Roadmap for complex Systems 2008-2009
This second issue of the French Complex Systems Roadmap is the outcome of the Entretiens de Cargese 2008, an interdisciplinary brainstorming session organized over one week in 2008, jointly by RNSC, ISC-PIF and IXXI. It capitalizes on the first roadmap and gathers contributions of more than 70 scientists from major French institutions. The aim of this roadmap is to foster the coordination of the complex systems community on focused topics and questions, as well as to present contributions and challenges in the complex systems sciences and complexity science to the public, political and industrial spheres.
nlin.AO↗
Hyperbolicity and the effective dimension of spatially-extended dissipative systems
We show, using covariant Lyapunov vectors, that the chaotic solutions of spatially extended dissipative systems evolve within a manifold spanned by a finite number of physical modes hyperbolically isolated from a set of residual degrees of freedom, themselves individually isolated from each other. In the context of dissipative partial differential equations, our results imply that a faithful numerical integration needs to incorporate at least all physical modes and that increasing the resolution merely increases the number of isolated modes.
nlin.CD↗