What is the microscopic picture for warm air rising? The usual explanation for warm fluids rising past cooler ones is that the warmer fluid has a lower density.  I'm trying to understand what this looks like at a molecular scale.  The density seems to be a large-scale phenomenon, and I don't understand how it can affect whether a particular molecule rises or falls.
Consider a cylinder of fluid that is being heated at the bottom.  The molecules at the bottom have a higher average energy.  How does this result in the tendency of the warmer molecules at the bottom to move upwards past the cooler ones?
 A: Semi-macroscopic view:
The key word for understanding this problem is buoyancy.  Buoyancy is the result of different pressures.  Since the warm air is less dense than the cold one, there are less hits to the (imaginary) balloon of warmer particles from inside than from outside, so there is net pressure toward inside.
However, since this pressure difference is larger at the bottom than at the top of the balloon, net force to the balloon is upwards.  This is simply due to the fact that pressure change is proportional only to density of the matter with the same height difference and gravitational acceleration.
Microscopic view:
The primary question is: why are there any molecules at the top of the container, if gravity pulls them all down?  It is because density rises as you go downwards, so there are more "kicks" up than "kicks" down at certain height, balancing gravitational force and keeping particles at the same height.  A warmer particle makes more space around itself, it is "less dense", so it must travel up.
A: You say that "density seems to be a large-scale phenomenon," and I think you're quite right. Temperature is also a large-scale phenomenon. There's arguably no such thing as a "warmer" or a "cooler" molecule, at least in an ideal gas. There are faster molecules, and slower molecules, and at any given moment in a gas, fast molecules are colliding into slow molecules, transferring momentum in the process. This means that a given molecule may go from being a faster molecule to being a slower molecule many times in a very short duration. 
Indeed, it could be meaningful to describe temperature as a measure of "energy density." Consider the equation relating the average kinetic energy of a particle to the temperature of the system in an ideal gas:
$$\overline{E}_{k} = \frac 1 2 kT$$
This says, very simply, that the average kinetic energy of a single particle in a system is proportional to the temperature of that system. Since that's the average kinetic energy of a single particle, we could get the total quantity of kinetic energy in the system by multiplying the above by the number of particles (or really, three times the number of particles, given these particular proportionality constants, since a single particle has three degrees of freedom in three dimensional space). Temperature, then, really is analogous to density; density is mass per volume, and temperature is kinetic energy per particle. 
So in a very real sense, there is no microscopic picture of the phenomenon you describe. Instead, there are regions in a pocket of gas in which the average velocity of molecules is higher or lower, corresponding to regions of higher or lower temperature. In those regions in which the average velocity of molecules is higher, the gas will tend to push out against its surroundings, becoming less dense if it is allowed to. In those regions in which the average velocity of molecules is lower, the gas will tend to compress, becoming more dense. 
From there, the phenomenon is easily explained by the basic concept of buoyancy. In a dense region of gas, a less-dense pocket of gas will be pushed up by the dense gas, which is itself pushed down by gravity; hence warm air rises. 
A: The molecules are all moving, quite rapidly, all the time, and constantly colliding against each other.
The warmer ones are moving even more rapidly, thus "winning out" in their collisions with the cooler ones, pushing them away. (That's what lower density is.)
Then if there's some gravity field pulling all of them downwards against a surface (they're not in free fall) the cooler ones have less velocity to "get away" from the ones underneath or the surface, therefore they congregate below.
Even in something as dramatic as a rocket engine, the thermal velocity of the molecules is much higher than the exhaust velocity.
This is seen in videos of rocket engines in space, where the exhaust plume is very wide, rather than narrow as near the ground.
