Alessandro Ciallella, Mario Pulvirenti, Sergio Simonella
We present and analyze two simple $N$-particle particle systems for the spread of an infection, respectively with binary and with multi-body interactions. We establish a convergence result, as $N \to \infty$, to a set of kinetic equations, providing a mathematical justification of related numerical schemes. We analyze rigorously the time asymptotics of these equations, and compare the models numerically.
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Individual-Based Foundation of SIR-Type Epidemic Models: mean-field limit and large time behaviour
本文提出基于个体的SIR型传染病模型，通过动能框架和均值场极限分析，揭示了动态的耗散结构与稳定平衡配置的形成机制。
We present a Boltzmann equation for mixtures of three species of particles reducing to the Kermack-McKendrick (SIR) equations for the time-evolution of the density of infected agents in an isolated population. The kinetic model is potentially more detailed and might provide information on space mixing of the agents.
Controllability and kinetic limit of spherical particles immersed in a viscous fluid
本文研究粘性流体中球形颗粒系统的可控性与动力学极限，分别利用几何控制理论和Boltzmann型表述分析少颗粒可控性及颗粒数趋于无穷时的极限行为。
We analyze the continuous time evolution of a $d$-dimensional system of $N$ self propelled particles with a kinematic constraint on the velocities inspired by the original Vicsek's one \cite{VCB-JCS}. Interactions among particles are specified by a pairwise potential in such a way that the velocity of any given particle is updated to the weighted average velocity of all those particles interacting with it. The weights are given in terms of the interaction rate function. When the size of the system is fixed, we show the existence of an invariant manifold in the phase space and prove its exponen…
We discuss the Cucker-Smale's (C-S) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasov-type kinetic model for the C-S particle model and prove it exhibits time-asymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic description of flocking based on the C-S Vlasov-type kinetic model and prove flocking behavior \emph{without} closure of higher moments.
In this paper, we prove the global existence of the weak solution to the mean field kinetic equation derived from the $N$-particle Newtonian system. For $L^1\cap L^\infty$ initial data, the solvability of the mean field kinetic equation can be obtained by using uniform estimates and compactness arguments while the difficulties arising from the non-local non-linear interaction are tackled appropriately using the Aubin-Lions compact embedding theorem.
Propagation of chaos for moderately interacting particle systems related to singular kinetic McKean-Vlasov SDEs
本文研究中等相互作用粒子系统中混沌传播性质，用于近似由alpha稳定过程驱动的奇异动能麦肯-瓦拉斯斯德过程。
We provide a detailed multiscale analysis of a system of particles interacting through a dynamical network of links. Starting from a microscopic model, via the mean field limit, we formally derive coupled kinetic equations for the particle and link densities, following the approach of [Degond et al., M3AS, 2016]. Assuming that the process of remodelling the network is very fast, we simplify the description to a macroscopic model taking the form of single aggregation-diffusion equation for the density of particles. We analyze qualitatively this equation, addressing the stability of a homogeneou…
We study kinetic theories for isotropic, two-dimensional grain boundary networks which evolve by curvature flow. The number densities $f_s(x,t)$ for $s$-sided grains, $s =1,2,\ldots$, of area $x$ at time $t$, are modeled by kinetic equations of the form $\partial_t f_s + v_s \partial_x f_s =j_s$. The velocity $v_s$ is given by the Mullins-von Neumann rule and the flux $j_s$ is determined by the topological transitions caused by the vanishing of grains and their edges. The foundations of such kinetic models are examined through simpler particle models for the evolution of grain size, as well as…
We are interested in a kinetic equation intended to describe the interactions of particles with their environment. We focus on the long time behaviour. We prove that the time derivative of the spatial density goes to 0 and exhibit the omega limit set for the distribution function. We then apply this result to the empirical density associated to a finite number of particles and prove that the speeds of all of them go to 0. It also allows us to prove the convergence of all the positions with an increasing external potential and to get a precise description of their long time behaviour with a non…
In this paper we consider a system of $N$ particles on the real line evolving according to Newton's law, interacting through a singular (repulsive) force deriving from the potential $\frac{|x|^{1-α}}{1-α}$ with $α\in (0,1/8)$ and Brownian force. Thanks to the entropy dissipation along the Liouville equation associated to this particles system and a control of the mechanic energy, we establish a quantitative estimate which enables to conclude to a convergence/consistency the particles system toward the limiting Vlasov-Fokker-Planck equation with singular force.
Combining the results of [14] and [10], the trend to equilibrium in large time is studied for a large particle system associated to a Vlasov-Fokker-Planck equation. Under some conditions (that allow non-convex confining potentials) the convergence rate is proven to be independent from the number of particles. From this are derived uniform in time propagation of chaos estimates and an exponentially fast convergence for the nonlinear equation itself.
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