Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit N→+∞, where N is the number of particles. These correlations are responsible for the “collisional” evolution of the system beyond the Vlasov regime due to finite N effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size N and try to provide a coherent discussion of all the numerical results obtained for these systems.     

Escape of stars from gravitational clusters in the Chandrasekhar model

We study the evaporation of stars from globular clusters using the simplified Chandrasekhar model [S. Chandrasekhar, Dynamical friction. II. The rate of escape of stars from clusters and the evidence for the operation of dynamical friction, Astrophys. J. 97 (1943) 263]. This is an analytically tractable model giving reasonable agreement with more sophisticated models that require complicated numerical integrations. In the Chandrasekhar model: (i) the stellar system is assumed to be infinite and homogeneous (ii) the evolution of the velocity distribution of stars f(v,t) is governed by a Fokker-Planck equation, the so-called Kramers-Chandrasekhar equation (iii) the velocities |v| that are above a threshold value R>0 (escape velocity) are not counted in the statistical distribution of the system. In fact, high velocity stars leave the system, due to free evaporation or to the attraction of a neighboring galaxy (tidal effects). Accordingly, the total mass and energy of the system decrease in time. If the star dynamics is described by the Kramers-Chandrasekhar equation, the mass decreases to zero exponentially rapidly. Our goal is to obtain non-perturbative analytical results that complement the seminal studies of Chandrasekhar, Michie and King valid for large times t→+∞ and large escape velocities R→+∞. In particular, we obtain an exact semi-explicit solution of the Kramers-Chandrasekhar equation with the absorbing boundary condition f(R,t)=0. We use it to obtain an explicit expression of the mass loss at any time t when R→+∞. We also derive an exact integral equation giving the exponential evaporation rate λ(R), and the corresponding eigenfunction f_λ(v), when t→+∞ for any sufficiently large value of the escape velocity R. For R→+∞, we obtain an explicit expression of the evaporation rate that refines the Chandrasekhar results. More generally, our results can have applications in other contexts where the Kramers equation applies, like the classical diffusion of particles over a barrier of potential (Kramers problem).     

The Role of Instrumental Variables in Causal Inference Based on Independence of Cause and Mechanism

Causal inference methods based on conditional independence construct Markov equivalent graphs and cannot be applied to bivariate cases. The approaches based on independence of cause and mechanism state, on the contrary, that causal discovery can be inferred for two observations. In our contribution, we pose a challenge to reconcile these two research directions. We study the role of latent variables such as latent instrumental variables and hidden common causes in the causal graphical structures. We show that methods based on the independence of cause and mechanism indirectly contain traces of the existence of the hidden instrumental variables. We derive a novel algorithm to infer causal relationships between two variables, and we validate the proposed method on simulated data and on a benchmark of cause-effect pairs. We illustrate by our experiments that the proposed approach is simple and extremely competitive in terms of empirical accuracy compared to the state-of-the-art methods.     

A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme.     

La formule des traces pour les algèbres de Lie

Nous établissons un analogue pour les algèbres de Lie de la formule des traces d'Arthur-Selberg. Soit G un groupe réductif connexe défini sur et son algèbre de Lie. On considère et deux familles de distributions sur les points adéliques de , chacune in dexée par les classes de -conjugaison semi-simple dans : la première est formée des analogues des termes du c?té géométrique de la formule des traces pour les groupes et la seconde de leurs transformées de Fourier. On montre que pour toute fontion f dans la classe de Schwartz et que ces deux sommes convergent absolument. C'est cette égalité qui est un analogue de la formule d'Arthur-Selberg. Une telle formule peut étre utile pour des problèmes d'analyse harmonique locale. Pour terminer, nous exprimons les termes associés aux classes de conjugaison semi-simples régulières à l'aide d'intégrales orbitales pondérées. Received: 28 December 1998 / Revised version: 9 May 2001 / Published online: 19 October 2001     

Cyclometallated platinum(II) complexes incorporating ethynyl-flavone ligands: switching between triplet and singlet emission induced by selective binding of Pb2+ ions

Platinum-ethynylflavone complexes featuring various polyether arms display (3)IL phosphorescence associated with the appended flavone perturbed by the platinum centre (tau approximately 20 mus), but switch dramatically to flavone-localised (1)IL fluorescence (tau approximately 2 ns) upon selective binding of Pb(2+).     

The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case

Publications mathématiques de l'IHÉS - In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups $U_{n}\times U_{n+1}$ in all the...     

Kinetic theory of stellar systems, two-dimensional vortices and HMF model

We discuss the kinetic theories of stellar systems, two-dimensional vortices and Hamiltonian mean field model, stressing their analogies and differences. We describe the evolution of the system as a whole and discuss the timescale of relaxation towards the Boltzmann distribution predicted by statistical mechanics. We also consider the relaxation of a “test” particle in a bath of “field” particles and analyze it with the aid of a Fokker–Planck equation involving a term of diffusion counterbalanced by a friction or a drift.     

Analysis of a model of phosphorus uptake by plant roots

In this paper, we consider a model of phosphorus uptake by plant roots, governed by a quasilinear parabolic equation. We first study the well posedness of the associated Cauchy problem. Then, we consider a shape optimization problem: how to deform the shape of the root in order to increase phosphorus uptake. Finally, we give some numerical results of the shape optimization process.     

Influence of bromine substitution pattern on the singlet oxygen generation efficiency of two-photon absorbing chromophores

A molecular engineering strategy based on rational variations of the bromine substitution pattern in two-photon absorbing singlet oxygen sensitizers allows studying the relations that exist between the positioning of an inter-system crossing promoter on the charge-transfer chromophore and its ability to generate singlet oxygen.