IS26: Invariant Measures and Scaling Limits of Integrable Systems

Organizer: Davide Gabrielli (University of L’Aquila)

Simple nonlinear PDEs inspired by billiards

Krzysztof Burdzy

“Pinned billiard balls” do not move but have pseudo-velocities. Pairs of balls are chosen one by one, in either a deterministic or random way. When a pair is chosen, it “collides” in the sense that the pseudo-velocities change according to the usual laws of totally elastic collisions. In other words, the total energy and total momentum are conserved.

A one-dimensional pinned billiard balls model with local energy redistribution was considered in \([1]\). The results suggest that the model has a hydrodynamic limit and the parameters for the limit, the local variance \(\sigma\) and mean \(\mu\), satisfy the coupled partial differential equations with freezing \[\begin{aligned} \sigma_t(x,t) & = - \mu_x(x,t) {\bf 1}_{ \sigma(x,t) >0},\\ \mu_t(x,t) &= - \sigma_x(x,t) {\bf 1}_{ \sigma(x,t) >0},\\ \sigma(x,t) & \geq 0. \end{aligned}\] If the initial conditions are continuous then there exist continuous solutions for all time. This and many quantitative and qualitative results were obtained in \([2]\).

Bibliography

\([1]\) Krzysztof Burdzy, Jeremy G. Hoskins and Stefan Steinerberger (2024) From pinned billiard balls to partial differential equations, Math Arxiv 2209.01503

\([2]\) Krzysztof Burdzy and John Sylvester (2024) Coupled transport equations with freezing, Math Arxiv 2406.02707