Event № 1055
Event № 1055
TE Special Seminar - Amit Einav 11/06/2014, Wednesday, 13:30
Name: Special Seminar
Title: Kac?s Model, Chaoticity and Local Central Limits.
Speaker: Amit Einav
Place: Amado 814, Technion
Abstract: One of the most influential equations in the kinetic theory of gases is the so-called Boltzmann equation, describing the time evolution of the probability density of a particle in dilute gas. The problem of proving its validity is an important, and significant, problem in the field of Kinetic Theory. In 1956 Marc Kac presented an attempt to solve this problem in a particular settings of the spatially homogeneous Boltzmann equation. Kac considered a stochastic linear model of N indistinguishable particles, with one-dimensional velocities, that undergo a random binary collision process. Under the property of 'chaoticity' Kac managed to show that when one takes the number of particles to infinity, the limit of the first marginal of the N-particle distribution function satisfies a caricature of the Boltzmann equation, the so-called Boltzmann-Kac equation. The concept of chaoticity, and that of propagation of chaos, has become a fundamental one in many other models of systems of many objects. As such, there is much interest in identifying chaotic states, as well as the more robust states we call 'entropically chaotic?. In our talk we will shortly present Kac's model, and explain it connection to the Boltzmann equation. We will then discuss a particular, intuitive, type of chaotic, and even entropically chaotic, family - one that is generated by a known one-particle function. We will see how local central limit theorems play an essential role in proving that the states are indeed chaotic, and present a new Levy type local central limit theorem that allowed us to extend previous results.
SubmittedBy: nick crawford , email@example.com
EventLink: Event № 1055