Poincaré recurrence theorem asserts that certain systems will return arbitrarily close to their initial state after a sufficiently long time.A deck of cards is a mechanical system with a finite number of states. Shuffling changes the order, but the total number of arrangements is fixed. Sooner or later any sequence will repeat, and the deck will return to its initial state. In 1890, Henri Poincaré transferred this idea to physical systems. He proved that in a closed conservative system, where energy is bounded and the phase space (an imaginary multidimensional hall where each point represents all coordinates and velocities of particles) has finite volume, the trajectory sooner or later comes arbitrarily close to the starting point. The return time is monstrously large — for a mole of gas in a shoebox, it is 10 to the power of 10²³ seconds. This number is impossible to write down in the visible Universe. Poincaré's theorem resolved the paradox of how reversible Newtonian laws can lead to irreversible entropy growth. The answer lies in the timescales.
Henri Poincaré proved the theorem in 1890 in his treatise 'On the Problem of Three Bodies and the Equations of Dynamics'. At that time he was participating in a competition by King Oscar II of Sweden and Norway to solve the problem of the stability of the Solar System. While studying the restricted three-body problem, Poincaré encountered chaotic behavior of trajectories and the need to describe their long-term fate. He succeeded in showing that the phase trajectory of a bounded system sooner or later returns near the starting point. Later it turned out that similar ideas had been expressed by James Clerk Maxwell, reasoning about molecular dynamics, and also by Lars Onsager in the context of statistical mechanics. In the 20th century, the theorem acquired fundamental importance for understanding the connection between reversible microscopic dynamics and irreversible thermodynamics. Ilya Prigogine's works on nonequilibrium processes also rely on the concept of recurrence.
The theorem finds applications in statistical mechanics, ergodic theory, and dynamical systems theory. In practice, it sets the limits of applicability of the second law of thermodynamics: the entropy of an isolated system can increase over intervals short compared to the Poincaré time, but eventually it must return to low values. In astrophysics, recurrent processes are considered for chaotic orbits of celestial bodies, but the timescales make them unobservable. In quantum information science, the memory of the initial state in a closed system does not disappear completely, which is used in error correction protocols.
The theorem explains why in an isolated room the scent of perfume does not disappear without a trace but only disperses — and could spontaneously gather back into the bottle, though the wait would be unimaginably long. In practice, this limits the applicability of the second law of thermodynamics for closed systems.
Ernst Zermelo tried to refute Boltzmann with this theorem: if everything returns, entropy cannot grow forever. Boltzmann retorted: 'You would have to wait so long that by the time of return any living being would have been reborn an infinite number of times.'