BARRÉ , Julien

National audience

We first introduce the percolation problems associated with the graph theoretical concepts of (k,l)-sparsity, and make contact with the physical concepts of ordinary and rigidity percolation. We then devise a renormalization transformation for (k,l)-percolation problems, and investigate its domain of validity. In particular, we show that it...

In the presence of long range interactions, physics is very peculiar: energy is no more additive, phase separation in the usual sense is impossible, dynamics is necessarily coherent on a global scale... These peculiarities are independent of the origin of the long range interaction involved, which may be of many different types: gravitationnal,...

National audience

Systems with long-range interactions often relax towards statistical equilibrium over timescales that diverge with N , the number of particles. A recent work [S. Gupta and D. Mukamel, J. Stat. Mech.: Theory Exp. P03015 (2011)] analyzed a model system comprising N globally coupled classical Heisenberg spins and evolving under classical spin...

Many physical systems are governed by long range interactions, the main example being self-gravitating stars. Long range interaction implies a lack of additivity for the energy. As a consequence, the usual thermodynamic limit is not appropriate. However, by contrast with many claims, the statistical mechanics of such systems is a well...

We consider the one-dimensional Vlasov equation with an attractive cosine potential, and its non homogeneous stationary states that are decreasing functions of the energy. We show that in the Sobolev space W 1,p (p > 2) neighborhood of such a state, all stationary states that are decreasing functions of the energy are stable. This is in sharp...

We investigate the bifurcation of a homogeneous stationary state of Vlasov-Newton equation in one dimension, in presence of a small dissipation mod-eled by a Fokker-Planck operator. Depending on the relative size of the dissipation and the unstable eigenvalue, we find three different regimes: for a very small dissipa-tion, the system behaves as...

We briefly review the classical approach to equilibrium and out of equilibrium statistical mechanics of long range interacting systems, for which the energy is not additive, and emphasize some new results. At equilibrium, we explain the thermodynamic consequences of the lack of additivity, like the generic occurrence of statistical ensemble...

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We investigate the density large deviation function for a multidimensional conservation law in the vanishing viscosity limit, when the probability concentrates on weak solutions of a hyperbolic conservation law conservation law. When the conductivity and dif-fusivity matrices are proportional, i.e. an Einstein-like relation is satisfied, the...

International audience

In self-gravitating stars, two dimensional or geophysical flows and in plasmas, long range interactions imply a lack of additivity for the energy; as a consequence, the usual thermodynamic limit is not appropriate. However, by contrast with many claims, the equilibrium statistical mechanics of such systems is a well understood subject. In this...

Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction q of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the...

We investigate the asymptotic behavior of a perturbation around a spatially non homogeneous stable stationary state of a one-dimensional Vlasov equation. Under general hypotheses, after transient exponential Landau damping, a perturbation evolving according to the linearized Vlasov equation decays algebraically with the exponent -2 and a well...

We study the stability in finite times of the tra jectories of interacting particles. Our aim is to show that in average and uniformly in the number of particles, two tra jectories whose initial positions in phase space are close, remain close enough at later times. For potential less singular than the classical electrostatic kernel, we are...

Landau damping is a fundamental phenomenon in plasma physics, which also plays an important role in astrophysics, and sometimes under different names, in fluid dynamics and other fields. Its theoretical discussion in the framework of the Vlasov equation often assumes that the reference stationary state is homoge-neous in space. However, Landau...

International audience

We study the asymptotic regime of strong electric fields that leads from the Vlasov–Poisson system to the Incompressible Euler equations. We also deal with the Vlasov–Poisson–Fokker– Planck system which induces dissipative effects. The originality consists in considering a situation with a finite total charge confined by a strong external...

Self-propelled particle (SPP) systems are intrinsically out of equi-librium systems, where each individual particle converts energy into work to move in a dissipative medium. When interacting through a velocity alignment mechanism, and with the medium acting as a momentum sink, even momen-tum is not conserved. In this scenario, a mapping into...

A phenomenological theory is proposed to analyze the asymptotic dynamics of perturbed inviscid Kol-mogorov shear flows in two dimensions. The phase diagram provided by the theory is in qualitative agree-ment with numerical observations, which include three phases depending on the aspect ratio of the domain and the size of the perturbation: a...

We show that the quasi-stationary states observed in the N-particle dynamics of the Hamiltonian mean-field (HMF) model are nothing but Vlasov stable homogeneous (zero magnetization) states. There is an infinity of Vlasov stable homogeneous states corresponding to different initial momentum distributions. Tsallis q-exponentials in momentum,...

We consider extended slow-fast systems of N interacting diffusions. The typical behavior of the empirical density is described by a nonlinear McKean–Vlasov equation depending on ε, the scaling parameter separating the time scale of the slow variable from the time scale of the fast variable. Its atypical behavior is encapsulated in a large N...