Perfect and $\varepsilon$-Perfect Simulation Methods for the One Dimensional Kac Equation

TL;DR

This paper reviews the Kac master equation for particle collisions, introduces a new efficient simulation algorithm leveraging Poisson process properties, and proposes an $oldsymbol{ ext{ extit{ extepsilon}}}$-perfect sampling method for the limiting distribution, with potential for broader applications.

Contribution

It presents a novel, efficient simulation algorithm for the Kac equation and introduces an $ ext{ extepsilon}$-perfect sampling method for the stationary distribution, expanding simulation techniques.

Findings

The new algorithm performs at least as well as existing methods.

Empirical evidence suggests improved tail capturing of velocity distribution.

The $ ext{ extepsilon}$-perfect sampling method is a proof of concept with potential for broader use.

Abstract

We review the derivation of the Kac master equation model for random collisions of particles, its relationship to the Poisson process, and existing algorithms for simulating values from the marginal distribution of velocity for a single particle at any given time. We describe and implement a new algorithm that efficiently and more fully leverages properties of the Poisson process, show that it performs at least as well as existing methods, and give empirical evidence that it may perform better at capturing the tails of the single particle velocity distribution. Finally, we derive and implement a novel "$\varepsilon$-perfect sampling" algorithm for the limiting marginal distribution as time goes to infinity. In this case the importance is a proof of concept that has the potential to be expanded to more interesting (DSMC) direct simulation Monte Carlo applications.