Research

You will find here a brief summary of the various projects I have been involved in.

Radial dependence of the WKB basis elements

Secular resonant dressed orbital diffusion - I. Method and WKB limit for tepid discs

We derived the secular diffusion equation of a self-gravitating collisionless system induced by external stochastic perturbations. In the case of a tepid galactic disc, relying on the WKB assumption that only tightly wound transient spirals are sustained by the disc, we obtained a simple quadrature for the diffusion coefficients, providing a straightforward understanding of the loci of maximal diffusion.

Prediction of the secular diffusion flux exhibiting a narrow ridge of resonant orbits

We recovered the main orbital signatures of the secular evolution of an isolated self-gravitating stellar disc. The shot-noise-driven formation of narrow ridges of resonant orbits is recovered in the WKB limit of tightly wound transient spirals, in agreement with numerical simulations. This justifies the relevance of the dressed Fokker-Planck formalism in angle-action variables to describe the secular evolution of such systems.

Velocity flux in action space induced by Poisson shot noise

Self-gravity, Resonances, and Orbital Diffusion in Stellar Disks

Fluctuations in a stellar system's gravitational field cause the orbits of stars to evolve. The resulting evolution of the system can be computed with the orbit-averaged Fokker-Planck equation. We presented the formalism that enables one to compute the diffusion tensor from a given source of noise when the system's gravitational dynamical response to that noise is included. This formalism, which recovers the formations of narrow ridges of enhanced density in action space, appears as the ideal framework in which to study the long-term evolution of all kinds of stellar discs.

Illustration of a discrete resonant encounter between two stars, captured by the Balescu-Lenard equation. In the appropriate rotating frame, orbits are closed and resonate. In the long-run, this deforms orbits.

We described the secular evolution of an infinitely thin isolated discrete self-gravitating stellar disc using the inhomogeneous Balescu-Lenard equation. Assuming that only tightly wound transient spirals are present in the disc, a WKB approximation provides a simple quadrature for the corresponding drift and diffusion coefficients. When applied to the secular evolution of an isolated stationary discrete self-gravitating Mestel disc, it predicts 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 needed to resolve this discrepancy.

Comparison of the resonance ridges predicted via the Balescu-Lenard equation (top) and observed in the numerical simulations of Sellwood (2012) (bottom)

We investigated the secular evolution of an infinitely thin discrete self-gravitating stella disc using the inhomogeneous Balescu-Lenard equation in terms of angle-action variables. We implemented numerically the matrix method to capture the induced graviational polarisation. The position/shape of the induced resonant ridge are found to be in very good agreement with the numerical simulations, as well as the diffusion timescales. Quantitative comparisons with N-body simulations also yield consistent scalings with the number of particles.

Functional rewriting of the two first equations of the BBGKY hierarchy

Functional integral approach to the kinetic theory of inhomogeneous systems

We present a derivation of the kinetic equation describing the secular evolution of spatially inhomogeneous systems with long-range interactions, the so-called inhomogeneous Landau equation, by relying on a functional integral formalism.

Illustration of an axisymmetric system of 2D point vortices

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.

Illustration of the typical dependence of the precession frequencies (induced by the cluster and relativistic corrections) as a function of the distance to the central BH

We derive the kinetic equation that describes the secular evolution of a large set of particles orbiting a dominant massive object, such as stars bound to a supermassive black hole or a proto-planetary debris disc encircling a star. This degenerate Balescu-Lenard equation describes self-consistently the long-term evolution of the distribution of quasi-Keplerian orbits around the central object: it is the master equation that describes the secular effects of resonant relaxation.

Expression of the resonant dressed dynamical friction as given by the inhomogeneous Balescu-Lenard equation

We derive general self-consistent expressions for the coefficients of diffusion and dynamical friction in a stable, bound, multicomponent, self-gravitating, and inhomogeneous system.

Norm of the collisional diffusion flux given by the thickened WKB limit of the Balescu-Lenard equation. The presence of an enhanced diffusion flux in the inner region of the disc is compatible with the self-induced formation of vertical resonant ridges.

We investigate the secular thickening of a self-gravitating stellar galactic disc using the dressed collisionless Fokker-Planck equation and the inhomogeneous multi-component Balescu-Lenard equation, by relying on a generalised thickened WKB approximation. When applied to a tepid stable tapered disc perturbed by shot noise, these two frameworks predict the formation of ridges of resonant orbits towards larger vertical actions, as found in direct numerical simulations, but over-estimates the timescale involved in their appearance. Swing amplification is likely needed to resolve this discrepancy, as demonstrated in the case of razor-thin discs.

Response of the Galactic's DF to a bar perturbation, in the region of trapping.

Distribution functions for resonantly trapped orbits in the Galactic disc

We show how to compute the response of the DF of a galactic disc to a bar-like perturbation in the region of orbit-trapping, where the traditional perturbation theory fails.

Illustration in physical space of the mass segregation of a two-component quasi-Keplerian disc.

Relying on Gauss' method, we compute the drift and diffusion coefficients characterising the properties of the resonant relaxation of a razor-thin quasi-Keplerian disc.

Growth rate of the two main modes of an isochrone disc as a function of the softening length. Full lines trace the linear stability analysis, while bullets are data from N-body simulations.

How gravitational softening affects galaxy stability. I. Linear mode analysis of disc galaxies

We use linear pertubation theory to investigate the effect of graviational softening on the spiral eigenmodes of razor-thin stellar discs.

Diffusion coefficient in a self-interacting HMF bath, obtained through different methods.

Relaxation in self-gravitating systems

We show how the stochastic η-formalism allows for the recovery of the Balescu-Lenard and Landau equations. This approach provides a new view of the resonant diffusion processes associated with long-term orbital distortions.

Main collaborators

Christophe Pichon (IAP), James Binney (Oxford), Pierre-Henri Chavanis (Toulouse), John Magorrian (Oxford), Simon Prunet (Hawaii), Giacomo Monari (Postdam), Benoit Famaey (Strasbourg), Sven De Rijcke (Ghent), Walter Dehnen (Leicester), Ben Bar-Or (IAS).