The often elusive Poincare 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 verbLatticeReduce. The procedure is illustrated with a harmonic chain. The incredibly large recurrence times are obtained exactly. They follow the expected scaling law very well.
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