Convection

You may not realize it, but convection is an effect that's constantly taking place in the Earth's atmosphere, where hot air from the tropics(the two lines near the equator) and cool air from the poles often interact such that the cool, denser air from the poles falls to the bottom while the hot, less dense air rises to the top.

Convection isn't really defined by any certain equations and is more of an observable phenomenon. In fact, convection is really just buoyancy applied to gases, specifically, and not just any fluid.

Take a look at the house above. The furnace is warming up the room such that the hot air rises. When the hot air reaches the other side of the room, it's no longer being heated by the furnace so it cools down and goes back down.

This is nothing but just buoyancy at work. The hot air is less dense than the cold air, so the cold air will sink until it becomes heated up again. This is analogous to when objects less dense than water float on the surface of water while objects denser than water sink.

So, why is hot air less dense than cold air, specifically?

Well, let's think about this conceptually. If air is hotter, the molecules that air is composed of will have a greater average kinetic energy(and thus speed). This means they're far more likely to be further apart from each other. If they're further apart, the density will be lower because they're not as packed together as colder air. Therefore, hotter air must be less dense than colder air.