# Witnessing a Poincar\'e recurrence with Mathematica

## TL;DR

This paper demonstrates how to observe Poincaré recurrence in a separable system using Mathematica, leveraging Diophantine approximation and the LLL algorithm to compute large recurrence times accurately.

## Contribution

It introduces a method to witness Poincaré recurrence in separable systems through Diophantine approximation and Mathematica's LatticeReduce, providing exact large recurrence times.

## Key findings

- Recurrence times follow the expected scaling law
- The method accurately computes very large recurrence times
- Demonstration with a harmonic chain system

## Abstract

The often elusive Poincar\'e recurrence can be witnessed in a completely separable system. For such systems, the problem of recurrence reduces to the classic mathematical problem of simultaneous Diophantine approximation of multiple numbers. The latter problem then can be somewhat satisfactorily solved by using the famous Lenstra-Lenstra-Lov\'{a}sz (LLL) algorithm, which is implemented in the Mathematica built-in function \verb"LatticeReduce". The procedure is illustrated with a harmonic chain. The incredibly large recurrence times are obtained exactly. They follow the expected scaling law very well.