Poincaré's Recurrence Theorem

Poincaré's Recurrence Theorem states that in any isolated, finite system with a fixed total energy, the system's state will, after a sufficiently long time, return arbitrarily close to its initial state. Essentially, the system's behavior is recurrent; over time, it revisits previous configurations. This doesn't mean it repeats exactly or quickly, but given enough time, the system's microstates will recur in a pattern that approximates earlier conditions. The theorem highlights the idea of long-term predictability in closed systems, emphasizing that in a finite, conserved-energy environment, states are not permanently lost or forever evolving away from their initial arrangements.