“Apple in a Box” Theory

The “Apple in a Box” theory is a popular thought experiment that illustrates the concept of infinity and Poincaré recurrence.

It gained recent popularity from the Netflix documentary A Trip to Infinity, but it is based on principles of thermodynamics and statistical mechanics that have been debated for over a century.

Here is an explanation of the theory, the physics behind it, and why it might (or might not) happen.

1. The Thought Experiment

Imagine you place a fresh apple inside a perfectly sealed box that allows absolutely nothing to enter or leave—no matter, no energy, nothing. It is a closed system. You then leave it there for an infinite amount of time.

Here is the timeline of what theoretically happens:

1. Decay: The apple rots. Bacteria consume it, it turns into mold, and eventually, it decomposes into dust and gas.
2. Equilibrium: The contents of the box eventually reach “thermal equilibrium.” At this point, it is just a soup of particles (protons, neutrons, electrons) and photons (energy) bouncing around in a disordered state.
3. The Long Wait: For an unimaginably long time, these particles will just bounce around randomly.
4. Recombination: Because there is a finite number of particles in the box but an infinite amount of time, the particles will eventually go through every possible arrangement they can physically form.
5. Recurrence: Eventually, purely by chance, the particles will cycle back into the exact configuration they started in. They will reform the apple—fresh, un-rotted, and identical to the original.

2. The Physics: Poincaré Recurrence Theorem

This idea is not magic; it is based on the Poincaré Recurrence Theorem (proven by Henri Poincaré in 1890).

The theorem states that if you have a system with a finite amount of energy and a finite amount of space (like our box), the system will, after a sufficiently long time, return to a state arbitrarily close to its initial state.

• Why? Think of it like a deck of cards. If you shuffle a deck of 52 cards enough times, eventually you will repeat a sequence of cards you have seen before. The “deck” in this box is the particles of the apple. There are a huge number of them (roughly 10²⁵ atoms), so the number of possible combinations is staggering—but it is not infinite. If time is infinite, you will eventually run out of new combinations and have to repeat the old ones.

3. Why it’s terrifying (The Philosophical Implication)

If this theory holds true for an apple in a box, and if our Universe is a finite, closed system (which is a big “if”), then everything must eventually repeat.

• You will be born again.
• You will read this sentence again.
• You will live every variation of your life (e.g., one where you didn’t click this link, one where you became an astronaut) an infinite number of times.

This leads to the concept of Eternal Recurrence (or Eternal Return), a philosophical concept explored by Friedrich Nietzsche, which suggests that existence recurs in an infinite cycle.

4. The “Catch” (Why it might not happen)

While the math works on paper, there are physical reasons why your apple might never return.

• The Box Must Be Perfect: In the real world, no box is perfectly isolated. Energy leaks out (Hawking radiation, quantum tunneling). If energy escapes, the system “cools down” and eventually freezes (Heat Death), stopping the particles from moving and recombining.
• Proton Decay: Some theories in physics suggest that protons (the building blocks of atoms) are not immortal. If protons eventually decay (after 10³⁴ years or more), the matter in the box will vanish before it has time to reform the apple.
• The Expansion of the Universe: If the box is the Universe itself, and the Universe is expanding (Dark Energy), the particles might get too far apart to ever interact again, preventing recurrence.

Summary

The “Apple in the Box” is a demonstration that in a finite world with infinite time, everything that can happen will happen—and will happen an infinite number of times. It turns the chaos of a rotting apple back into order, simply by waiting long enough for the odds to reset.