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. 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 so-called Boltzmann-Kac equation. Kac hoped that using this mean field approach will lead to new results in the convergence to equilibrium of the limit equation using the simpler, yet dimension dependent, linear ODE . In our talk we will introduce Kac’s model and the concept of Chaoticity. We will then discuss possible trends to equilibrium and review recent results in the matter. Time permitting, we will describe related research that has been done recently in connection to the above.