The Poincare Recurrence Theorem states that a dynamical system with a bounded phase space and a measure-preserving evolution will, after a sufficiently long but finite time, return arbitrarily close to almost any given starting state. Named for Henri Poincare, it applies in particular to classical mechanical systems with finite energy and volume, and its consequence that an isolated gas of colliding particles must eventually return near its initial configuration was historically cited as a challenge to the irreversibility built into the second law of thermodynamics.
Facts
Partially Attested
Proof YearPoincare discussed the theorem in 1890; a measure-theoretic proof was given by Caratheodory in 1919. StatementAlmost every point in a set A of a finite-measure, measure-preserving system returns to A; the set of points that never return has measure zero. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Poincare recurrence theorem (Wikipedia)
Proved By
Sources
1. Poincare recurrence theorem (Wikipedia)
Formal statement
Almost every point in A returns to A. More precisely, the set of points in A that never return to A has a measure of exactly zero.
Introduction
The theorem is named after Henri Poincaré, who discussed it in 1890.
In Branch: Dynamical Systems, Lead sentence
currence Theorem is a foundational result in both dynamical systems and statistical mechanics.
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