Collective Behavior: Macroscopic versus Kinetic Descriptions

On Chaotic States in Kac’s Model

Amit Einav

Cambridge University


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. While widely used, and intuitive, the Boltzmann equation has no formal validation from Newtonian laws, in macroscopic time scales. 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 concept of chaoticity, and that of propagation of chaos, has become a fundamental one in many other models of systems of many particles. 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 present Kac’s model, and explain it connection to the Boltzmann equation. We will then discuss a particular 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 will present a new Levy type local central limit theorem that allowed us to extend previous results.