Functional integral derivation of the kinetic equation of two-dimensional point vortices

We present a brief derivation of the kinetic equation describing the secular evolution of point vortices in two-dimensional hydrodynamics, by relying on a functional integral formalism. We start from Liouville's equation which describes the exact dynamics of a two-dimensional system of point vortices. At the order ${1/N}$, the evolution of the system is characterised by the first two equations of the BBGKY hierarchy involving the system's 1-body distribution function and its 1-body correlation function. Thanks to the introduction of auxiliary fields, these two evolution constraints may be rewritten as a functional integral. When functionally integrated over the 2-body correlation function, this rewriting leads to a new constraint coupling the 1-body distribution function and the two auxiliary fields. Once inverted, this constraint provides, through a new route, the closed non-linear kinetic equation satisfied by the 1-body distribution function. Such a method sheds new lights on the origin of these kinetic equations complementing the traditional derivation methods.Comment: 7 pages, 1 figur

Secular diffusion in discrete self-gravitating tepid discs I : analytic solution in the tightly wound limit

The secular evolution of an infinitely thin tepid isolated galactic disc made of a finite number of particles is described using the inhomogeneous Balescu-Lenard equation. Assuming that only tightly wound transient spirals are present in the disc, a WKB approximation provides a simple and tractable quadrature for the corresponding drift and diffusion coefficients. It provides insight into the physical processes at work during the secular diffusion of a self-gravitating discrete disc and makes quantitative predictions on the initial variations of the distribution function in action space. When applied to the secular evolution of an isolated stationary self-gravitating Mestel disc, this formalism predicts initially the importance of the corotation resonance in the inner regions of the disc leading to a regime involving radial migration and heating. It predicts in particular the formation of a "ridge like" feature in action space, in agreement with simulations, but over-estimates the timescale involved in its appearance. Swing amplification is likely to resolve this discrepancy. In astrophysics, the inhomogeneous Balescu-Lenard equation and its WKB limit may also describe the secular diffusion of giant molecular clouds in galactic discs, the secular migration and segregation of planetesimals in proto-planetary discs, or even the long-term evolution of population of stars within the Galactic center.Comment: 22 pages, 12 figure

Resonant thickening of self-gravitating discs: imposed or self-induced orbital diffusion in the tightly wound limit

The secular thickening of a self-gravitating stellar galactic disc is investigated using the dressed collisionless Fokker-Planck equation and the inhomogeneous multicomponent Balescu-Lenard equation. The thick WKB limits for the diffusion fluxes are found using the epicyclic approximation, while assuming that only radially tightly wound transient spirals are sustained by the disc. This yields simple quadratures for the drift and diffusion coefficients, providing a clear understanding of the positions of maximum vertical orbital diffusion within the disc, induced by fluctuations either external or due to the finite number of particles. These thick limits also offer a consistent derivation of a thick disc Toomre parameter, which is shown to be exponentially boosted by the ratio of the vertical to radial scale heights. Dressed potential fluctuations within the disc statistically induce a vertical bending of a subset of resonant orbits, triggering the corresponding increase in vertical velocity dispersion. When applied to a tepid stable tapered disc perturbed by shot noise, these two frameworks reproduce qualitatively the formation of ridges of resonant orbits towards larger vertical actions, as found in direct numerical simulations, but overestimates the time-scale involved in their appearance. Swing amplification is likely needed to resolve this discrepancy, as demonstrated in the case of razor-thin discs. Other sources of thickening are also investigated, such as fading sequences of slowing bars, or the joint evolution of a population of giant molecular clouds within the disc.Comment: 31 pages, 19 figure

Dressed diffusion and friction coefficients in inhomogeneous multicomponent self-gravitating systems

General self-consistent expressions for the coefficients of diffusion and dynamical friction in a stable, bound, multicomponent self-gravitating and inhomogeneous system are derived. They account for the detailed dynamics of the colliding particles and their self-consistent dressing by collective gravitational interactions. The associated Fokker-Planck equation is shown to be fully consistent with the corresponding inhomogeneous Balescu-Lenard equation and, in the weak self-gravitating limit, to the inhomogeneous Landau equation. Hence it provides an alternative derivation to both and demonstrates their equivalence. The corresponding stochastic Langevin equations are presented: they can be a practical alternative to numerically solving the inhomogeneous Fokker-Planck and Balescu-Lenard equations. The present formalism allows for a self-consistent description of the secular evolution of different populations covering a spectrum of masses, with a proper accounting of the induced secular mass segregation, which should be of interest to various astrophysical contexts, from galactic centers to protostellar discs.Comment: 27 pages, 1 figur

Effective homology and periods of complex projective hypersurfaces

We provide an algorithm to compute an effective description of the homology of complex projective hypersurfaces relying on Picard-Lefschetz theory. Next, we use this description to compute high-precision numerical approximations of the periods of the hypersurface. This is an improvement over existing algorithms as this new method allows for the computation of periods of smooth quartic surfaces in an hour on a laptop, which could not be attained with previous methods. The general theory presented in this paper can be generalised to varieties other than just hypersurfaces, such as elliptic fibrations as showcased on an example coming from Feynman graphs. Our algorithm comes with a SageMath implementation.Comment: 35 page

Thermal regime of the NW shelf of the Gulf of Mexico. 1) Thermal and pressure fields

National audienceThe thermal field of the Gulf of Mexico (GoM) is analyzed from a comprehensive temperature-depth database of about 8500 Bottom Hole Temperatures and Reservoir Temperatures. Our stochastic analysis reveals a widespread, systematic sharp thermal gradient increase between 2500 and 4000 m. The analysis of the pressure regime indicates a systematic correlation between the pressure and temperature fields

Nano mechanical and microstructural investigation of damage mechanisms in copper wire bonds

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VLTI/VINCI diameter constraints on the evolutionary status of delta Eri, xi Hya, eta Boo.

other location: http://www.obs-nice.fr/pichon/science.html ; Accepted for publication in Astron. Astrophys.International audienceUsing VLTI/VINCI angular diameter measurements, we constrain the evolutionary status of three asteroseismic targets: the stars $\delta$ Eri, $\xi$ Hya, $\eta$ Boo. Our predictions of the mean large frequency spacing of these stars are in agreement with published observational estimations. Looking without success for a companion of $\delta$ Eri we doubt on its classification as an RS CVn star