Q:

Why doesn't the solute settle out of a solution, such as brine or sugar water?

If I have a solution of salt or sugar dissolved in water, the solute will never settle out or "fall to the bottom", no matter low long I wait. Why is that?
Since salt and sugar (in their pure, solid forms) are denser than water, I would intuitively expect that gravity would eventually pull all the salt and sugar to the bottom. Why doesn't this happen?
Now, I'm guessing that salt and sugar solutions have lower energy than mixtures of solid salt or solid sugar with water. So that may explain why salt and sugar don't spontaneously settle out; reactions tend to proceed from a high-energy state to a lower-energy state, not vice versa. (In "explain like I'm 5" terminology, maybe sugar sticks to water better than sugar sticks to sugar.) Is that accurate?
I've heard that the favorability of a reaction is determined by such things as energy, entropy and enthalpy, but I don't know the details. I'm guessing the answer has something to do with those.
This question is inspired by this question over at the Earth Science site: Why does the salt in the oceans not sink to the bottom?
(I'm surprised that I wasn't able to find a previous question about why solutions don't settle. All I found was a question about why colloids don't settle, which is also an interesting question, but it's not the same thing.)

A:

You discuss two different phenomena, solubility and concentration gradients due to gravity. 
Solubility can be explained using the concept of entropy. Solubility requires the the free energy of a solid be higher than the corresponding substance dispersed in solution. Surprinsingly perhaps, additional entropy associated with dispersion of the solute in the solvent suffices to drive the process (interactions can certainly be important, but are not at minimum necessary, although much depends on how much you are attempting to solubilize, and at what temperature).
Gravity can (and does) cause concentration gradients, and can therefore under the right circumstances cause precipitation. The problem is that gravity is a very weak force, so it's effects on light molecules are, well, weak. Gravitational effects require large scales or masses to be significant. Density differences are argued to be critical driving force behind water cycling in the oceans. The air atop Mt Everest is so thin people suffocate without access to an auxiliary supply of oxygen. So these effects do happen, and matter. And you can use a centrifuge to separate components in a mixture, which is analogous to applying gravitational forces, just scaled up in strength.

A:

Supplemental answer for additional background/insight:
Biologists generate salt gradients in test tubes all the time using high-speed centrifugation, which allows them to produce much higher potential energy gradients than gravity can, and in an environment free of ocean-scale mixing and temperature gradients.
If the concentration is close to saturation and the gradient is high enough, this can certainly lead to precipitation. Otherwise you just end up with a static gradient in concentration, being the balance between buoyancy forces downward and thermally driven diffusion upward.
See for example Wikipedia's Buoyant density centrifugation 
Heavy salts like $\ce{CsCl}$, $\ce{Cs2SO4}$, and $\ce{KBr}$ have been used historically.
However, you can't use this equation which applies to centrifugation because it is based on "gravity" increasing with distance from the center of the rotor or "downward" whereas in the ocean gravity is fairly constant, but can decrease or increase slightly with depth depending on the specific location and how close you are to dense rock at the bottom.

