Special Session 128: Recent Advances in Kinetic Theory and Related Applications

Derivation of the acoustic system for fermionic condensates from the Boltzmann-Fermi-Dirac equation

Benjamin Anwasia New York University Abu Dhabi United Arab Emirates Co-Author(s):

Abstract: In quantum kinetic theory, the Boltzmann-Fermi-Dirac equation is the model that governs the evolution of a gas of fermions. At extremely low temperatures, fermions transition into a superfluid phase called a fermionic condensate. This superfluid phase is characterized by an equilibrium state of the Boltzmann-Fermi-Dirac equation, where all particles occupy their lowest energy levels while obeying the Pauli exclusion principle. Our objective is to demonstrate how to derive the acoustic equations for fermionic condensates as a hydrodynamic limit of the Boltzmann-Fermi-Dirac equation. This is a joint work with Diogo Ars\`enio.

Asymptotic analysis of kinetic models : Case of general inflow data and Poisson coupling

Samia Ben Ali Faculty of science of tunis Tunisia Co-Author(s):    Samia Ben Ali, Mohamed Lazhar Tayeb

Abstract: We study an approximation by diffusion of a Boltzmann equation. A linear time relaxation model and an inflow boundary data with a general profile are considered. A corrected Hilbert expansion and the contraction property of the collision operator are used to establish a uniform L$^1-$estimate. A correction of the boundary layer at the first order is introduced in order to prove a strong convergence and to exhibit a rate of convergence. The limit fluid model is a Drift-Diffusion model associated with effective boundary data obtained as a decay at infinity of a Half-Space problem. The analysis is performed, in the first step, for the linear case (prescribed potential). In the second step, the analysis is extended to the case of a self-consistent potential (Poisson coupling) in one dimension by carefully combining the relative entropy method and a perturbation of the Hilbert expansion; giving the convergence and rate of convergence.

Optimal transport of measures via autonomous vector fields

Nicola N De Nitti EPFL Switzerland Co-Author(s):

Abstract: We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to construct a suitable autonomous vector field that realizes the (unique) monotone transport map as the time-1 map of its flow. Generically, this vector field can be chosen to be Lipschitz continuous. We then use Sudakov's disintegration approach to deal with the multi-dimensional case by reducing it to a family of one-dimensional problems. This talk is based on a joint work with Xavier Fernández-Real.

Fractional diffusion for the kinetic Fokker-Planck equation: A spectral method in any dimension

Dahmane Dechicha NYUAD United Arab Emirates Co-Author(s):    Marjolaine Puel

Abstract: In recent years, numerous studies have focused on approximating the solution of kinetic equations by an equilibrium whose density satisfies a simpler fractional diffusion equation. This is particularly the case for the linear Boltzmann and Fokker-Planck equations when the equilibria exhibit heavy tails (i.e., polynomial decay). These results have been obtained through various methods. In this presentation, we introduce a method for constructing an eigenpair, which is the solution of the spectral problem associated with the Fokker-Planck operator. This result, first established in dimension 1 and later generalized to higher dimensions, directly leads to the diffusion limit for the Fokker-Planck equation. The eigenvalue provides the correct time scaling and the diffusion coefficient, while the eigenfunction is used as a test function in the moment method.

Small inertia limit for coupled kinetic swarming models

Simone Fagioli University of L`Aquila Italy Co-Author(s):    Y-P. Choi, V. Iorio

Abstract: We investigate various versions of multi-dimensional systems involving many species, modeling aggregation phenomena through nonlocal interaction terms. We establish a rigorous connection between kinetic and macroscopic descriptions by considering the small-inertia limit at the kinetic level. The results are proven either under smoothness assumptions on all interaction kernels or under singular assumptions for \emph{self-interaction} potentials. Utilizing different techniques in the two cases, we demonstrate the existence of a solution to the kinetic system, provide uniform estimates with respect to the inertia parameter, and show convergence towards the corresponding macroscopic system as the inertia approaches zero.

The mean-field Limit of sparse networks of integrate and fire neurons

Pierre-Emmanuel Jabin Pennsylvania State University USA Co-Author(s):    D. Zhou

Abstract: We study the mean-field limit of a model of biological neuron networks based on the so-called stochastic integrate-and-fire (IF) dynamics. Our approach allows to derive a continuous limit for the macroscopic behavior of the system, the 1-particle distribution, for a large number of neurons with no structural assumptions on the connection map outside of a generalized mean-field scaling. We propose a novel notion of observables that naturally extends the notion of marginals to systems with non-identical or non-exchangeable agents. Our new observables satisfy a complex approximate hierarchy, essentially a tree-indexed extension of the classical BBGKY hierarchy. We are able to pass to the limit in this hierarchy as the number of neurons increases through novel quantitative stability estimates in some adapted weak norm. While we require non-vanishing diffusion, this approach notably addresses the challenges of sparse interacting graphs/matrices and singular interactions from Poisson jumps, and requires no additional regularity on the initial distribution.

Compactness and existence theory for a general class of stationary radiative transfer equations

Jin Woo Jang POSTECH Korea Co-Author(s):    Elena Dematte, Juan J. L. Velazquez

Abstract: In this talk, I will introduce a recent proof for the existence of the steady-states of a large class of stationary radiative transfer equations in a $C^1$ convex bounded domain. The main difficulty in proving existence of solutions is to obtain compactness of the sequence of integrals along lines that appear in several exponential terms. Currently available averaging lemmas do not seem to provide sufficient compactness that we require, and I will introduce our new compactness result suitable to deal with such a non-local operator containing integrals on a line segment.

Kinetic and hydrodynamic flocking models with nonlocal velocity alignment

Changhui Tan University of South Carolina USA Co-Author(s):    McKenzie Black

Abstract: The Euler-alignment system describes the collective behaviors of animal swarms. In this talk, we introduce a new type of alignment interaction that depends nonlinearly on velocity. We explore the asymptotic flocking and alignment behaviors. Notably, the introduction of nonlinearity yields a spectrum of distinctive asymptotic behaviors. Moreover, we present a rigorous derivation of our system from a kinetic flocking model.

Non-uniqueness for continuous solutions to 1D hyperbolic systems

Cheng Yu University of Florida USA Co-Author(s):    Ming Chen, Alexis Vasseur

Abstract: In this talk, I will discuss a geometrical condition on $2\times 2$ systems of conservation laws leads to non-uniqueness in the class of 1D continuous functions. This demonstrates that the Liu Entropy Condition alone is insufficient to guarantee uniqueness, even within the mono-dimensional setting. This is a joint work with M.Chen and A. Vasseur.

Landau damping, collisionless limit, and stability threshold for the Vlasov-Poisson equation with nonlinear Fokker-Planck collisions

Weiren Zhao New York University Abu Dhabi United Arab Emirates Co-Author(s):

Abstract: In this talk, I will present a recent work about the asymptotic stability of the global Maxwellian for the Vlasov-Poisson-Fokker-Planck (VPFP) equation with a small collision frequency. Our main result establishes the Landau damping and enhanced dissipation phenomena under the condition that the perturbation of the global Maxwellian falls within the Gevrey-1/s class and obtains that the stability threshold for the Gevrey-1/s class with $s>s_k$ can not be larger than $\gamma=\frac{1-3s_{k}}{3-3s_{k}}$ for $s_{k}\in [0,1/3]$.