Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From pinned billiard balls to partial differential equations

Krzysztof Burdzy, Jeremy G. Hoskins, Stefan Steinerberger

TL;DR

The paper studies energy transport in a line of pinned billiard balls, where fixed centers exchange energy through a two-step, random update while maintaining elastic-collision-like properties. By positing a modulated white-noise local equilibrium, it derives a coupled system of partial-difference equations for the mean velocity and its variance, which are then rescaled to a nonlinear PDE framework. Numerical simulations (large-scale, with many balls) demonstrate strong agreement between the discrete model and the proposed PDEs, supporting the modulated white-noise hypothesis and the transport-type hydrodynamic description. The work also discusses connections to related hydrodynamic limits, offers justification for the modeling choices, and provides rigorous steps for the partial-difference derivations, including detailed estimates and series expansions.

Abstract

We discuss the propagation of kinetic energy through billiard balls fixed in place along a one-dimensional segment. The number of billiard balls is assumed to be large but finite and we assume kinetic energy propagates following the usual collision laws of physics. Assuming an underlying stochastic mean-field for the expectation and the variance of the kinetic energy, we derive a coupled system of nonlinear partial difference equations. Our results are illustrated by numerical simulations.

From pinned billiard balls to partial differential equations

TL;DR

The paper studies energy transport in a line of pinned billiard balls, where fixed centers exchange energy through a two-step, random update while maintaining elastic-collision-like properties. By positing a modulated white-noise local equilibrium, it derives a coupled system of partial-difference equations for the mean velocity and its variance, which are then rescaled to a nonlinear PDE framework. Numerical simulations (large-scale, with many balls) demonstrate strong agreement between the discrete model and the proposed PDEs, supporting the modulated white-noise hypothesis and the transport-type hydrodynamic description. The work also discusses connections to related hydrodynamic limits, offers justification for the modeling choices, and provides rigorous steps for the partial-difference derivations, including detailed estimates and series expansions.

Abstract

We discuss the propagation of kinetic energy through billiard balls fixed in place along a one-dimensional segment. The number of billiard balls is assumed to be large but finite and we assume kinetic energy propagates following the usual collision laws of physics. Assuming an underlying stochastic mean-field for the expectation and the variance of the kinetic energy, we derive a coupled system of nonlinear partial difference equations. Our results are illustrated by numerical simulations.
Paper Structure (13 sections, 3 theorems, 132 equations, 5 figures)

This paper contains 13 sections, 3 theorems, 132 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Evolution of pinned billiard balls model parameters
  3. Modulated white noise
  4. Partial difference equations
  5. Numerical Examples
  6. A System of PDEs
  7. Different coordinates
  8. Related hydrodynamic models
  9. Justification of our model
  10. Proof of partial difference equations
  11. Estimates for $I_1$ and $I_2$
  12. Series expansion
  13. Acknowledgments

Key Result

Theorem 2.3

Suppose functions $\widetilde{\mu}: (0,1)\times[0,\infty) \to \mathbb{R}$ and $\widetilde{\sigma}: (0,1) \times [0,\infty)\to (0,\infty)$ are $C^3_b$ (with bounded third derivative). Assume that $v(x,t)$ satisfies a28.1 at a fixed time $t\geq 0$ and for $1\leq x \leq n$. Then for $2\leq x \leq n-1$,

Figures (5)

  • Figure 1: Billard balls arranged along a one-dimensional line. The balls touch but are fixed for all time.
  • Figure 2: Empirical histogram (blue) of white noise values at the time $0.37T$. The standard normal density is drawn in red.
  • Figure 3: Correlation between white noise values at the distance $k$ at time $0.37 T$, for $k=1,\dots , 50$, for a single run. The values of the empirical white noise were calculated for a spatial position $x$ by subtracting the mean $\mu(x,0.37T)$ and dividing by the standard deviation $\sigma(x,0.37T)$, where the last two functions were evaluated as averages over all runs.
  • Figure 4: Pairs of values of the noise $(W(300,0.37T),W(301,0.37T))$ for 100,000 runs of the simulation. The RGB scheme is $((k/100,000)^5,0, 1-(k/100,000)^5)$ where $k$ is the number of the simulation.
  • Figure 5: The moments $\mu$ and $\sigma$ at times $0.01 T,0.2 T,0.4 T,0.6 T,0.8 T$ and $0.99 T$ (top from left to right, then bottom left to right). The mean $\mu$ (dotted red) and $\sigma$ (dotted orange) were estimated by averaging values over 100,000 repetitions of the pinned balls model. The mean $\mu$ (solid blue) and $\sigma$ (solid green) were numerically computed using the equations \ref{['m29.5']}-\ref{['m29.6']}. The curves were horizontally and vertically rescaled to show agreement.

Theorems & Definitions (8)

  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • proof : Proof of Theorem \ref{['a28.2']}
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof