Freezing in Space-time: A functional equation linked with a PDE system

TL;DR

This paper explores a functional equation and its connection to a PDE system modeling the freezing behavior of a line of billiard balls, revealing how solutions link boundary conditions to the entire space-time evolution.

Contribution

It establishes a novel link between a specific functional equation and a PDE system describing freezing dynamics in a billiard ball model.

Findings

Functional equation solutions determine PDE boundary conditions.

The PDE models the freezing process of billiard balls with fixed terminal velocities.

The approach connects boundary data to the entire space-time solution.

Abstract

We analyze the functional equation $$F(x+F(x))=-F(x)$$ and reveal its relationship with a system of partial differential equations arising as the hydrodynamic limit of a system of pinned billiard balls on the line. The system of balls must freeze at some time, i.e., no velocity may change after the freezing time. The terminal velocity and the freezing time profiles play the role of boundary conditions for the PDEs (qua terminal conditions, despite being initial conditions from the wave equation perspective pursued here). Solutions to the functional equation provide the link between the freezing time and terminal velocity profiles on the one hand, and the solution to the PDE in the entirety of the space-time domain on the other.