We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neuron and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.[-]

We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neuron and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.[-]

The momentum transport in a fusion device such as a tokamak has been in a scope of the interest during last decade. Indeed, it is tightly related to the plasma rotation and therefore its stabilization, which in its turn is essential for the confinement improvement. The intrinsic rotation, i.e. the part of the rotation occurring without any external torque is one of the possible sources of plasma stabilization.
The modern gyrokinetic theory [3] is an ubiquitous theoretical framework for lowfrequency fusion plasma description. In this work we are using the field theory formulation of the modern gyrokinetics [1]. The main attention is focussed on derivation of the momentum conservation law via the Noether method, which allows to connect symmetries of the system with conserved quantities by means of the infinitesimal space-time translations and rotations.
Such an approach allows to consistently keep the gyrokinetic dynamical reduction effects into account and therefore leads towards a complete momentum transport equation.
Elucidating the role of the gyrokinetic polarization is one of the main results of this work. We show that the terms resulting from each step of the dynamical reduction (guiding-center and gyrocenter) should be consistently taken into account in order to establish physical meaning of the transported quantity. The present work [2] generalizes previous result obtained in [4] by taking into the account purely geometrical contributions into the radial polarization.[-]

We investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system in a regime studied by F. Golse and L. Saint-Raymond [1, 3]. First we establish the convergence towards the Euler equation under several assumptions on the energy and on the norms of the initial data. Then we analyze the asymptotics for a Vlasov-Poisson system describing the interaction of a bounded density of particles with a moving point charge, characterized by a Dirac mass in the phase-space.[-]

This talk introduces, in a simplified setting, a novel commutator method to obtain averaging lemma estimates. Averaging lemmas are a type regularizing effect on averages in velocity of solutions to kinetic equations. We introduce a new bilinear approach that naturally leads to velocity averages in $L^{2}\left ( \left [ 0,T \right ],H_{x}^{s} \right )$. The new method outperforms classical averaging lemma results when the right-hand side of the kinetic equation has enough integrability. It also allows a perturbative approach to averaging lemmas which provides, for the first time, explicit regularity results for non-homogeneous velocity fluxes.[-]

This talk introduces, in a simplified setting, a novel commutator method to obtain averaging lemma estimates. Averaging lemmas are a type regularizing effect on averages in velocity of solutions to kinetic equations. We introduce a new bilinear approach that naturally leads to velocity averages in $L^{2}\left ( \left [ 0,T \right ],H_{x}^{s} \right )$. The new method outperforms classical averaging lemma results when the right-hand side of the kinetic equation has enough integrability. It also allows a perturbative approach to averaging lemmas which provides, for the first time, explicit regularity results for non-homogeneous velocity fluxes.[-]

This talk is devoted to the quasi linear approximation for solutions of the Vlasov equation a very popular tool in Plasma Physic cf. [4] which proposes, for the quantity:
(1)
$$
q(t,\ v)=\int_{\mathbb{R}_{v}^{d}}f(x,\ v,\ t)dx)\ ,
$$
the solution of a parabolic, linear or non linear evolution equation
(2)
$$
\partial_{t}q(t,\ v)-\nabla_{v}(D(q,\ t;v)\nabla_{v}q)=0
$$
Since the Vlasov equation is an hamiltonian reversible dynamic while (2) is not reversible whenever $D(q,\ t,\ v)\# 0$ the problem is subtle. Hence I did the following things :

1. Give some sufficient conditions, in particular in relation with the Landau damping that would imply $D(q,\ t,\ v)\simeq 0$. a situation where the equation (2) with $D(q,\ t;v)=0$ does not provides a meaning full approximation.

2. Building on contributions of [7] and coworkers show the validity of the approximation (2) for large time and for a family of convenient randomized solutions. This is justified by the fact that the assumed randomness law is in agreement which what is observed by numerical or experimental observations (cf. [1]).

3. In the spirit of a Chapman Enskog approximation formalize the very classical physicist approach (cf. [6] pages 514-532) one can show [3] that under analyticity assumptions this approximation is valid for short time. As in [6] one of the main ingredient of this construction is based on the spectral analysis of the linearized equation and as such it makes a link with a classical analysis of instabilities in plasma physic.

Remarks

In some sense the two approaches are complementary The short time is purely deterministic and the stochastic is based on the intuition that over longer time the randomness will take over of course the transition remains from the first regime to the second remains a challenging open problem. The similarity with the transition to turbulence in fluid mechanic is striking It is underlined by the fact that the tensor
$$
\lim_{\epsilon\rightarrow 0}\mathbb{D}^{\epsilon}(t,\ v)=\lim_{\epsilon\rightarrow 0}\int dx\int_{0}^{\frac{t}{\epsilon^{2}}}d\sigma E^{\epsilon}(t,\ x+\sigma v)\otimes E^{\epsilon}(t-\epsilon^{2}\sigma,\ x)
$$
which involves the electric fields here plays the role of the Reynolds stress tensor.

2 Obtaining, for some macroscopic description, a space homogenous equation for the velocity distribution is a very natural goal. Here the Vlasov equation is used as an intermediate step in the derivation. And more generally it appears as an example of weak turbulence. In particular defining what would be the physical natural probability seems related to the derivation of $\mathrm{e}$ of the Lenard-Balescu equation as done in [5].[-]

The Hamiltonian Mean-Field (HMF) model is a 1D simplified version of the gravitational Vlasov-Poisson system. I will present two recent works in collaboration with Mohammed Lemou and Ana Maria Luz. In the first one, we proved the nonlinear stability of steady states for this model, using a technique of generalized Schwarz rearrangements. To be stable, the steady state has to satisfy a criterion. If this criterion is not satisfied, some instabilities can occur: this is the topic of the second work that I will present.[-]

We are concerned with a mixture of Boltzmann and McKean-Vlasov type equations, this means (in probabilistic terms) equations with coefficients depending on the law of the solution itself, and driven by a Poisson point measure with the intensity depending also on the law of the solution. Both the analytical Boltzmann equation and the probabilistic interpretation initiated by Tanaka (1978) have intensively been discussed in the literature for specific models related to the behavior of gas molecules. In this paper, we consider general abstract coefficients that may include mean field effects and then we discuss the link with specific models as well. In contrast with the usual approach in which integral equations are used in order to state the problem, we employ here a new formulation of the problem in terms of flows of endomorphisms on the space of probability measure endowed with the Wasserstein distance. This point of view already appeared in the framework of rough differential equations. Our results concern existence and uniqueness of the solution, in the formulation of flows, but we also prove that the 'flow solution' is a solution of the classical integral weak equation and admits a probabilistic interpretation. Moreover, we obtain stability results and regularity with respect to the time for such solutions. Finally we prove the convergence of empirical measures based on particle systems to the solution of our problem, and we obtain the rate of convergence. We discuss as examples the homogeneous and the inhomogeneous Boltzmann (Enskog) equation with hard potentials.
Joint work with Aurélien Alfonsi.[-]