Why are things NOT scale-invariant?

[nonequilibrium]

(posting here as to avoid any discussion about QM or relativity)

I think everyone at first sight wouldn't expect that if you'd enlarge an ant to the size of a house, it would collapse. Well, I'd intuitively except scale-invariance. Yet it would collapse under its own weight.

Another example: we see in the newspaper, every once in a while pictures of a great cathedral made with little matchsticks...If we imagine that this wooden cathedral were actually built on the scale of a real cathedral, we see where the trouble is; it would not last -- the whole things would collapse because of the fact that scaled-up matchsticks are just not strong enough. ...
Today, of course, we understand the fact that phenomena depend on the scale on the grounds that matter is atomic in nature, and certainly if we built an apparatus that was so small that there were only five atoms in it, it would clearly be something we could not scale up and down arbitrarily. The scale of an individual atom is not at all arbitrary -- it is quite definite.
- Feynman (lectures, vol 1, 52-3)

So is the fact our world exists out of atoms the explanation?
Shouldn't we also scale up the atoms if we want to be honest about our scaling-up?

And even if it were the atoms: why didn't people use this as evidence of the atomic theory sooner?

Apparently Galileo already discovered this non-scale-invariance in nature and wrote about it. If it's only explainable by the concept of atoms, then it's explanation was simply a mystery for 100's of years? (although we never read about it having been a problem)

 

[Sakha]

I can't remember right now where I saw or read an article that stated that we as humans are the perfect size. It argued that if you scaled up, the weight increased to the cubic power, while the strength of the muscles increased as the square, because one is volumetric and the other is the surface area.

 

[Klockan3]

Technically you could have scale variance even if matter did not have discrete constituents, it is just a fact that volume and thus weight scales faster that area and thus weight capacity. However there are a lot of phenomena which depends on the fact that things are made up of particles, such as how the air is extremely chaotic if you go down to the micrometer scale.

Shouldn't we also scale up the atoms if we want to be honest about our scaling-up?

If you could build larger particles, be my guest. By scale invariance we mean that we just take something and build a new copy of it just that we make everything larger aka adds more particles. We don't have magical machines which scales things.

And even if it were the atoms: why didn't people use this as evidence of the atomic theory sooner?

Because people did not draw that parallel until Einstein came and invented discrete physics. Einstein saw how chaotic air was for really small objects and he argued that the only way this could happen was if air was not a homogeneous soup but rather a huge set of really small objects bouncing around.

 

[cjl]

It has nothing to do with matter's inherent granularity due to atoms. Instead, it would happen even with a perfectly homogeneous material. It's simply due to the way that various parameters scale.

The ability of a column (such as, for example, a leg) to support weight is dependent primarily on its cross sectional area. As you scale a column up, the cross sectional area will increase proportional to the scale factor squared, since it's an area (and therefore scales as length²). However, the mass the column has to support scales with the volume of the creature or item it is supporting, since the density isn't changing in this example. So, the strength of the thing to support itself is scaling with its size squared, but the amount of stuff to support is scaling with size cubed. As a result, the strength to weight ratio is actually going down (specifically, it's proportional to 1/size). This is the reason why scaling something directly doesn't work. I'm somewhat surprised that Feynman said what is in your quote - although it is true that atoms are the basis for certain types of scale dependence, they certainly are not the root cause of all scale dependent quantities.

 

[nonequilibrium]

Imagine a perfectly empty universe. Now let there be only one big massive shiny featureless orb-like planet and one ant on that planet: even though you have no way of comparing it with anything, just by looking at the ant and the fact it's not being squished to the ground by its own size, you can be certain that the ant is not bigger (expressed in meter) than a certain critical size? Am I the only one who finds this extremely counterintuitive?

 

[Pythagorean]

So is the fact our world exists out of atoms the explanation?
Shouldn't we also scale up the atoms if we want to be honest about our scaling-up?

It's precisely because atoms don't scale that the problem lies. For example, if you miniaturized yourself (Honey, I Shrunk The Kids), there would be technical difficulties in both cases:

If your atoms shrunk, then you wouldn't be able to breathe oxygen because oxygen atoms aren't going to shrink... so how are you hemoglobins going to bond to them now that your hemoglobins have scaled down.

If your atoms don't shrink... well, hopefully you can imagine the difficulty of cramming all those atoms into a tighter space (or using less atoms).

 

[Pythagorean]

For a more classical, straight forward point of view. Imagine scaling a simple geometric object (like a sphere or a cube)

As you increase it's size, the volume goes up by d^3, while the surface area increases by d^2, so any classical force depending on volume (i.e. gravity, assuming constant density) is going to to have a stronger effect on the object than any force depending on surface area as you scale the object up (which is why the fullsize matchstick cathedral collapses)

Unfortunately, the surface area as an indicator of structural integrity is ultimately a particle question, but it's quite intuitive if you imagine something like the scaling of a water balloon, isn't it?