How fast will a small disturbance propagate to affect all of Earth?

As far as I understand it, quantum mechanics requires that a particle's position to be not specifically determined in space, but rather be 'spread' out through space, in the sense that we can only know the probability a particle is at a particular location. This can be visualised through the wavefunction. When we then try to measure the particle's position, (say by firing a high energy photon at it), the particle will turn out to be at some particular location, which corresponds to the wave function collapsing (I'm not too sure if this is the right use of the term). The particle could however be found in a large range of positions. Consider an air particle, which has an initial wavefunction (in black), which we then fire a photon at to determine the position of. The air particle could then be found at position A or found at position B, with roughly equal probability.

The two circumstances however cause a slight disturbance, which 'propagates' through space. What I mean by this is this air particle's position and momentum will affect the air particles near it, which will affect the air particles near those, and so on and so forth. Through chaos theory, a small change in initial conditions will result in a very different outcome, so this single misplaced air molecule has the potential to change everything about Earth.

In the above diagram, air particles are depicted by dashes, with their velocities depicted by the length of the dash. As can be seen, the two situations A and B lead to a 'propagation of disturbance' (the area in which the air particles are different between situation A and situation B) which is depicted by the black circle. What I'm interested in is how quickly the circle grown in size.

At first, I thought that it should propagate at the speed of light. Imagine that our air particle is situated at the North Pole. An air particle situated at the south pole will have a wavefunction that is VERY NEARLY zero at the north pole, but it will still be finite(I think). For this reason, a small disturbance of the air particle at the north pole will result in a disturbance at the South Pole directly, at the speed of light.

Another voice in my head however said that this was rather silly. The disturbance should propagate rather slowly, and should only propagate through the collision of air particles. Through the atmosphere it would travel at roughly the average speed of air particles in the atmosphere, and through the solid ground it would travel very slowly.

Which of the two, if either, is correct?

• What do you mean by "an air particle"? A molecule? Commented Nov 9, 2015 at 10:31
• The premise of the question is false - you assume that, unless we fire a photon at it, the "air particle" isn't localized well. But it is constantly interacting - with other particles, with infalling sunlight, etc. - the assumption it can be in a delocalized state for any significant span of time at all is simply false. Commented Nov 9, 2015 at 13:36
• "..For this reason, a small disturbance of the air particle at the north pole will result in a disturbance at the South Pole directly, at the speed of light. " why? there's no reason Commented Nov 9, 2015 at 20:56

a small disturbance of the air particle at the north pole will result in a disturbance at the South Pole directly, at the speed of light.

Anything that QM tells us has to be consistent with what we observe in the real world and consistent with classical physics in the range of circumstances for which classical physics is valid.

Even quite energetic and focussed disturbances initially affecting vast numbers of molecules in air only propagate at a speed of a few meters a second. They also dissipate quite rapidly.

The highest known wind speeds on Earth are far far lower than the speed of light.

this single misplaced air molecule has the potential to change everything about Earth.

I think this overstates the case. The observable macroscopic effects we notice in air are far too large scale to be affected by the fate of one $N_2$ molecule.

• Isn't there like chaos theory though, like a butterfly flaps its wings and causes a tornado like halfway across the earth? Like it seems like a change in a single molecule wouldn't do anything, but the Earth is so super complex that it would actually change quite a lot given enough time wouldn't it? Commented Nov 13, 2015 at 11:05
• @Joshua: No, the Butterfly Effect is a metaphor, it specifically does not mean the flap of the butterfly wing causes the tornado. Commented Nov 13, 2015 at 11:12
• It's a metaphor, but isn't it correct in stating that for a chaotic system a small change in initial conditions (such as a butterfly wing flapping) can cause a tornado? Commented Nov 13, 2015 at 11:13
• "The phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that may ultimately alter the path of a tornado or delay, accelerate or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado. The term is not intended to imply—as is often misconstrued—that the flap of the butterfly's wings causes the tornado." - wikipedia. Commented Nov 13, 2015 at 11:17
• So the tiny changes in the atmosphere may ultimately create a tornado, wouldn't a change in a single molecule suffice given enough time? Commented Nov 13, 2015 at 11:20