The heat-overpowers-gravity size-scale

When going down in the micro and nanoscale thermal motion becomes more and more relevant. At the atomic scale gravity is pretty much irrelevant not because it's not present but because its very much overpowered at room temperature (~300K).

A natural question to ask to strengthen ones intuition is:

• at which size does gravity start to play a notable role? Or:
• Is there a special size-scale where one could say that the "strength" of gravity overtakes "strength" of thermal motion?

As it turns out there is.

Explanation

Thermal motion is characterized by the fact that every independent degree of freedom (DOF) gets on average a defined quantity of energy (a statistical packet - not a quantum) determined by the temperature. This is called the "equipartitioning theorem".

A rigid particle has six DOFs (not considering inner vibrations) three angular rotation DOFs and three linear translation DOFs. We'll focus on the translation DOFs only.

Assuming by sheer chance the particle has exactly the average energy and its present in form of kinetic motion that points upwards exactly. Now with constant temperature the thermal upward moving energy of the nanoparticle stays the same no matter which size the particle has but the mass of the particle changes drastically with size. Halving the size shrinks the mass to an eighth, doubling the size blows the mass up eightfold (a cubic scaling law). With rising size (and consequently much faster rising mass) but constant kinetic energy the "launch speed" goes down.

With rising size of the particle (cube side-length) the "rising hight" of its throwing parabola (motion upward till full stop) shrinks since the rising height of a throwing parabola of a particle is only given by the vertical starting speed.

At some point (specific size-scale) the "rising height" of the throwing parabola will become smaller than the particles actual dimensions.

This could be seen as the size-scale at which gravity overpowers heat (or vice versa depending on which direction one goes, shrinking or blowup).

Discussion

While not too important technically (nothing notable happens when crossing this size-scale) its very useful for an intuitive understanding.

Since this size-scale depends on temperature and gravity it is different on other planets, moons, asteroids, ... .

• It can go way down in deeply cooled systems (quantum effects can kick in though.
• It can go up a bit in very hot systems.

At earth normal conditions particles at the size threshold still lie in the (upper) nanoscale. (wiki-TODO: note exact size) Note that an actual experiment would be difficult because particles of this size scale like to stick to from wherever one wants to launch them from. (One does see brownian motion when free floating in a gas but the throwing parabolas are suppressed / dampened out exactly by this gas so the point is to conduct the experiment in a very good vacuum).

The smallest possible particles are atoms and molecules. Since they are so light their throwing parabolas are really big. Surprisingly this rising height can be easily observed. Its the very crudely equal to the thickness of the atmosphere in therms of the height where the pressure falls to halve.

Note that except for extremely thin atmosphere (that may not even deserve to be called atmosphere) molecules collide a lot before completing a full throwing parabola (the mean free path is shorter than the rising height).

Notes

(wiki-TODO: add math and graph - and the critical size for a cube of diamond in earth conditions !!)
Very simple math: (E_kin=3/2k_BT; v=sqrt(2E/m); m=rho*V; ...)

Related

• Prefixes of size-scales (macro-, micro-, nano-, pico-, femto-)