Coloquio de Matemáticas Aplicadas

This talk examines a class of linear hyperbolic systems which generalizes the Goldstein–Kac model to an arbitrary finite number of speeds with transition rates. Under the basic assumptions that the transition matrix is symmetric and irreducible, and the speed differences generate all the space, the system exhibits a large-time behavior described by a parabolic advection–diffusion equation. The main contribution is to determine explicit formulas for the asymptotic drift speed and diffusion matrix in term of the kinetic parameters, establishing a complete connection between microscopic and macroscopic coefficients. It is shown that the drift speed is the arithmetic mean of the velocities. The diffusion matrix has a more complicate representation, based on the graph with vertices the velocities and arcs weighted by the transition rates. The approach is based on an exhaustive analysis of the dispersion relation and on the application of a variant of the Kirchhoff’s matrix tree Theorem from graph theory.