Multitime fields and hard rod scaling limits

Abstract: A Poisson line process is a random set of straight lines contained in the plane, as the image of the map $(x,v)\mapsto (x+vt)_{t\in\mathbb{R}}$, for each point $(x,v)$ of a Poisson process in the space-velocity plane. By associating a step with each line of the process, a random surface called multitime walk field is obtained. The diffusive rescaling of the surface converges to the multitime Brownian motion, a classical Gaussian field also called L\'evy-Chentsov field. A cut of the multitime fields with a perpendicular plane, reveals a one dimensional continuous time random walk and a Brownian motion, respectively. A hard rod is an interval contained in $\mathbb{R}$ that travels ballistically until it collides with another hard rod, at which point they interchange positions. By associating each line with the ballistic displacement of a hard rod and associating surface steps with hard rod jumps, we obtain the hydrodynamic limits of the hard rods in the Euler and diffusive scalings. The main tools are law of large numbers and central limit theorems for Poisson processes. When rod sizes are zero we have an ideal gas dynamics. We describe the relation between ideal gas and hard-rod invariant measures.