There must also be a temperature difference between the gas inside the chimney and the atmosphere outside. Simply erecting a chimney, with no heat source, will not produce a draft.
It's the difference between density of the gas inside the chimney and the density of air outside that causes the draft to flow. Or, you might say, it's the difference between the weight of the column of gas inside the chimney, and the weight of a similar column of ambient atmosphere outside the chimney, that causes a buoyancy effect; this force of buoyancy pushes the column upward, just as the weight of water displaced by a boat's hull produces an upward force that keeps the boat afloat.
Of course, the source of heat must be ongoing in order to generate a constant draft. If the heat source is removed, the density of gas inside the chimney will increase as it cools, and the draft will stop. Under certain conditions, it can actually reverse.
Uh, heat rises.