Eric Bertin

Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis

Considering a gas of self-propelled particles with binary interactions, we derive the hydrodynamic equations governing the density and velocity fields from the microscopic dynamics, in the framework of the associated Boltzmann equation. Explicit expressions for the transport coefficients are given, as a function of the microscopic parameters of the model. We show that the homogeneous state with zero hydrodynamic velocity is unstable above a critical density (which depends on the microscopic parameters), signalling the onset of a collective motion. Comparison with numerical simulations on a standard model of self-propelled particles shows that the phase diagram we obtain is robust, in the sense that it depends only slightly on the precise definition of the model. While the homogeneous flow is found to be stable far from the transition line, it becomes unstable with respect to finite-wavelength perturbations close to the transition, implying a non-trivial spatio-temporal structure for the resulting flow. We find solitary wave solutions of the hydrodynamic equations, quite similar to the stripes reported in direct numerical simulations of self-propelled particles.

Boltzmann and hydrodynamic description for self-propelled particles.

We study analytically the emergence of spontaneous collective motion within large bidimensional groups of self-propelled particles with noisy local interactions, a schematic model for assemblies of biological organisms. As a central result, we derive from the individual dynamics the hydrodynamic equations for the density and velocity fields, thus giving a microscopic foundation to the phenomenological equations used in previous approaches. A homogeneous spontaneous motion emerges below a transition line in the noise-density plane. Yet, this state is shown to be unstable against spatial perturbations, suggesting that more complicated structures should eventually appear.

Glucose metabolism and β-cell mass in adult offspring of rats protein and/or energy restricted during the last week of pregnancy

An association between low birth weight and later impaired glucose tolerance was recently demonstrated in several human populations. Although fetal malnutrition is probably involved, the biological bases of such a relationship are not yet clear, and animal studies on the matter are scarce. The present study was aimed to identify, in adult (8-wk) female offspring, the effects of reduced protein and/or energy intake strictly limited to the last week of pregnancy. Thus we have tested three protocols of gestational malnutrition: a low-protein isocaloric diet (5 instead of 15%), with pair feeding to the mothers receiving the control diet; a restricted diet (50% of the control diet); and a low-protein restricted diet (50% of low-protein diet). Only the low-protein diet protocols, independent of total energy intake, led to a lower birth weight. The adult offspring female rats in the three deprived groups exhibited no decrease in body weight and no major impairment in glucose tolerance, glucose utilization, or glucose production (basal state and hyperinsulinemic clamp studies). However, pancreatic insulin content and β-cell mass were significantly decreased in the low-protein isocaloric diet group compared with the two energy-restricted groups. Such impairment of β-cell mass development induced by protein deficiency limited to the last part of intrauterine life could represent a situation predisposing to impaired glucose tolerance.

Nonlinear field equations for aligning self-propelled rods.

We derive a set of minimal and well-behaved 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 density-segregated, banded solutions, allowing us to develop a more complete analytic picture of the problem at the nonlinear level.

Generalized extreme value statistics and sum of correlated variables

We show that generalized extreme value statistics—the statistics of the kth largest value among a large set of random variables—can be mapped onto a problem of random sums. This allows us to identify classes of non-identical and (generally) correlated random variables with a sum distributed according to one of the three (k-dependent) asymptotic distributions of extreme value statistics, namely the Gumbel, Frechet and Weibull distributions. These classes, as well as the limit distributions, are naturally extended to real values of k, thus providing a clear interpretation to the onset of Gumbel distributions with non-integer index k in the statistics of global observables. This is one of the very few known generalizations of the central limit theorem to non-independent random variables. Finally, in the context of a simple physical model, we relate the index k to the ratio of the correlation length to the system size, which remains finite in strongly correlated systems.

Global Fluctuations and Gumbel Statistics

We explain how the statistics of global observables in correlated systems can be related to extreme value problems and to Gumbel statistics. This relationship then naturally leads to the emergence of the generalized Gumbel distribution Ga(x), with a real index a, in the study of global fluctuations. To illustrate these findings, we introduce an exactly solvable nonequilibrium model describing an energy flux on a lattice, with local dissipation, in which the fluctuations of the global energy are precisely described by the generalized Gumbel distribution.

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 Ito calculus approaches. The stochastic partial differential equations thus obtained are shown to recover the key terms argued in EPL 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.

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.

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, bandlike structures in a parameter region including the linear onset of nematic order. These results are obtained from the study of both the well-known (deterministic) hydrodynamic equations describing these systems and of the self-propelled particle model they were derived from. We prove, in particular, that the chaos stems from the generic instability of the band solution of the hydrodynamic equations. Revisiting the status of the strong fluctuations and long-range correlations in the particle model, we show that the giant number fluctuations observed in the chaotic phase are a trivial consequence of density segregation. However anomalous, curvature-driven number fluctuations are present in the homogeneous quasiordered nematic phase and characterized by a nontrivial scaling exponent.

Free volume distributions and compactivity measurement in a bidimensional granular packing

We investigate experimentally the statistics of the free volume inside bidimensional granular packings, for two different kinds of grains and two levels of compaction. With the aim to gain new insight into the statistical description of granular media, we measure the free volume distribution in clusters of increasing size. Our main result is that the logarithm of the free volume distribution scales in a nonextensive way with the cluster size, while still enabling the extraction of two intensive parameters. We discuss the interpretation of these parameters, and the consequences of the nonextensivity on the possible measurements of Edwards compactivity.We investigate experimentally the distribution of the free volume inside a bidimensional granular packing associated either with single grains or with clusters of grains. This is done for two different types of grains and two levels of compaction. The logarithm of the free volume distribution scales in an on-extensive way with the cluster size for cluster sizes up to a few hundred grains. Having factorized this size dependence, we characterize the distributions by two intensive parameters. We discuss the interpretation o ft heseparameters and their possible relation to Edwards compactivity.