That’s a really interesting question. Here’s what I think the answer is.
The air will move around to minimize the total free energy of the air and the balloons, the same way a fluid will flow downhill as far as it can, minimizing gravitational energy. Let’s make some approximations to see whether this weird behavior follows from simple ingredients or requires some special properties of the balloons.
Let’s say that the total volume of air is pretty much fixed. Then the free energy changes come from the stretching of the rubber in the balloons. Let’s say that, like most springy things, the excess free energy is proportional to the square of how far they’re stretched. Now the volume goes as radius cubed, but the free energy goes as radius squared. So the excess free energy is proportional to the balloon volume to the 2/3 power. The rate of change of free energy with respect to volume increase (the slope or derivative) then is inversely proportional to volume to the 1/3 power. So the more-inflated balloon increases free energy less for a given volume of added air. So free energy goes down when air flows from the less inflated to the more inflated balloon. The process will stop when the less inflated balloon isn’t stretched any more.
(published on 12/08/07)