IS26: Invariant measures and scaling limits of integrable systems

date: 7/14/2025, time: 16:00-17:30, room: IM B

Organizer: Davide Gabrielli (University of L’Aquila)

Chair: Davide Gabrielli (University of L’Aquila)

Scaling limits of a tagged soliton in the randomized box-ball system

Hayate Suda (Institute of Science Tokyo)

The box-ball system (BBS) is a cellular automaton that exhibits the solitonic behavior. In recent years, with the rapid progress in the study of the hydrodynamics of integrable systems, there has been a growing interest in BBS with random initial distribution. In this talk, we consider the scaling limits for a tagged soliton in the BBS starting from certain stationary distribution. Furthermore, we show that solitons of the same type are completely correlated at the diffusive scale. This talk is based on a joint work with Stefano Olla and Makiko Sasada.

Bibliography

\([1]\) Stefano Olla, Makiko Sasada and Hayate Suda : Scaling limits of solitons in the Box-Ball system. arXiv:2411.14818

Simple nonlinear PDEs inspired by billiards

Krzysztof Burdzy (University of Washington)

“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

Continuous Box Ball System

Ines Armendariz (Universidad de Buenos Aires and CONICET)

The Box Ball System was introduced by Takahashi and Satsuma in 1990 as a cellular automaton that exhibits solitons (travelling waves). We study a continuous version, where blocks of consecutive occupied boxes are replaced by occupied intervals of the real numbers, separated by empty intervals. The walk representation of a configuration is given by a zig-zag function, and the dynamics by Pitman’s transformation. We describe how to identify solitons, and show that they are conserved under the dynamics. We also show that the soliton decomposition of some random zig-zag walks can be mapped to a bidimensional Poisson process, a representation that linearizes the dynamics. This extends discrete space results by Ferrari, Gabrielli, Nguyen, Rolla and Wang \([1]\), \([2]\). This is joint work with Pablo Blanc, Pablo Ferrari and Davide Gabrielli.

Bibliography

\([1]\) Ferrari, P.A. and Gabrielli, D., BBS invariant measures with independent soliton components. 2020.

\([2]\) Ferrari P.A., Nguyen C., Rolla L.T., Wang M., Soliton Decomposition of the Box-Ball System. Forum of Mathematics. Sigma, 9. Cambridge University Press, Cambridge, 2021.