Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation

Discrete-velocity approximations represent a popular way for computing the Boltzmann collision operator. The direct numerical evaluation of such methods involve a prohibitive cost, typically $O(N^{2d+1})$ where $d$ is the dimension of the velocity space. In this paper, following the ideas introduced in [27,28], we derive fast summation techniques for the evaluation of discrete-velocity schemes which permits to reduce the computational cost from $O(N^{2d+1})$ to $O(\bar{N}^d N^d\log_2 N)$, $\bar{N} << N$, with almost no loss of accuracy.Comment: v2: 22 pages, improvement of the presentation and more details given in some proofs. arXiv admin note: text overlap with arXiv:1106.1020 by other author

Numerical schemes for a multi-species quantum BGK model

We consider a kinetic model of an N-species gas mixture modeled with quantum Bhatnagar-Gross-Krook (BGK) collision operators. The collision operators consist of a relaxation to a Maxwell distribution in the classical case, a Fermi distribution for fermions and a Bose-Einstein distribution for bosons. In this paper we present a numerical method for simulating this model, which uses an Implicit-Explicit (IMEX) scheme to minimize a certain potential function. This is motivated by theoretical considerations coming from entropy minimization. We show that theoretical properties such as conservation of mass, total momentum and total energy as well as positivity of the distribution functions are preserved by the numerical method presented in this paper, and illustrate its usefulness and effectiveness with numerical examplesComment: arXiv admin note: text overlap with arXiv:2202.0565

Numerical study of Bose-Einstein condensation in the Kaniadakis-Quarati model for bosons

Kaniadakis and Quarati (1994) proposed a Fokker--Planck equation with quadratic drift as a PDE model for the dynamics of bosons in the spatially homogeneous setting. It is an open question whether this equation has solutions exhibiting condensates in finite time. The main analytical challenge lies in the continuation of exploding solutions beyond their first blow-up time while having a linear diffusion term. We present a thoroughly validated time-implicit numerical scheme capable of simulating solutions for arbitrarily large time, and thus enabling a numerical study of the condensation process in the Kaniadakis--Quarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the Kaniadakis--Quarati model in 3D form a condensate in finite time and converge in entropy to the unique minimiser of the natural entropy functional at an exponential rate. Our simulations further indicate that the spatial blow-up profile near the origin follows a universal power law and that transient condensates can occur for sufficiently concentrated initial data.Comment: To appear in Kinet. Relat. Model

The grazing collision limit of Kac caricature of Bose-Einstein particles

We discuss the grazing collision limit of certain kinetic models of Bose-Einstein particles obtained from a suitable modification of the one- dimensional Kac caricature of a Maxwellian gas without cut-off. We recover in the limit a nonlinear Fokker-Planck equation which presents many similarities with the one introduced by Kaniadakis and Quarati in [13 ]. In order to do so, we perform a study of the moments of the solution. Moreover, as is typical in Maxwell models, we make an essential use of the Fourier version of the equation.Comment: 28 page

Classical and Quantum Mechanical Models of Many-Particle Systems

This workshop was dedicated to the presentation of recent results in the field of the mathematical study of kinetic theory and its naturalextensions (statistical physics and fluid mechanics). The main models are the Vlasov(-Poisson) equation and the Boltzmann equation, which are obtainedas limits of many-body equations (Newton’s equations in the classical case and Schrödinger’s equation in the quantum case) thanks to the mean-field and Boltzmann-Grad scalings. Numerical aspects and applications to mechanics, physics, engineering and biology were also discussed

Lower Mass Bounds on FIMP Dark Matter Produced via Freeze-In

Feebly Interacting Massive Particles (FIMPs) are dark matter candidates that never thermalize in the early universe and whose production takes place via decays and/or scatterings of thermal bath particles. If FIMPs interactions with the thermal bath are renormalizable, a scenario which is known as freeze-in, production is most efficient at temperatures around the mass of the bath particles and insensitive to unknown physics at high temperatures. Working in a model-independent fashion, we consider three different production mechanisms: two-body decays, three-body decays, and binary collisions. We compute the FIMP phase space distribution and matter power spectrum, and we investigate the suppression of cosmological structures at small scales. Our results are lower bounds on the FIMP mass. Finally, we study how to relax these constraints in scenarios where FIMPs provide a sub-dominant dark matter component.Comment: 50 pages, 14 figures, 3 tables; version published in JCA