Date of Award
Bachelor of Science (BS)
College of Science
The Poincaré recurrence theorem is one of the first and most fundamental theorems of ergodic theory. When applied to a dynamical system satisfying the theorem's hypothesis, it roughly states that the system will, within a finite amount of time, return to a state arbitrarily close to its initial state. This result is intriguing and controversial, providing a contradiction with the Second Law of Thermodynamics known as the recurrence paradox. Here, we treat a set of pool balls on a billiard table as a dynamical system that satisfies the hypotheses of the Poincaré recurrence theorem. We prove that time is a volume-preserving transformation from the state space onto itself. After showing that the hypotheses for the recurrence theorem are met, we discuss the theorem's implications, including possible resolutions of the paradox that arises.
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Smith, Aaron, "Applying the Poincaré Recurrence Theorem to Billiards" (2014). Honors Theses. 23.