Question: Consider two glass bottles filled with humid air, connected with a tube. One of the glasses is hot (heated), the other is cold (cooled) and the situation is steady state. How will humidity be distributed among the two bottles? Will the relative humidity be the same between them? Or will it be the absolute humidity? Or something inbetween?
EDIT: Suppose that these is no condensed water anywhere in the system, because the temperatures are high and the total amount of water is low.
Commentary 1: The question is inspired by humidity in cellars. If the cellar (cold bottle) and the outside air (hot bottle) are in steady state, what will be the relation between their relative humidities? (The steady state assumption reduces pretty much to the air circulation being faster than weather changes.) If the relative humidity is the same, then the cellar will be dry if there's sunshine outside and wet if it rains. If the absolute humidity is the same, then the cellar will be dry if it's cold outside and wet if it's hot outside.
Commentary 2: Ultimately, I'd like to expand this question to: I know the temperatures and relative humidities both in the cellar and outside, will opening the window result in a drop of relative humidity in the cellar? (By opening the window I cause a transition from two isolated bottles to two connected bottles in steady state. To make the calculations, however, I need to know whether it's the relative or absolute humidities which move toward equilibrium.)
Commentary 3: Physics explanation based on molecule movement will be greatly appreciated. While you are at it - I have a hypothesis why humidity of the atmosphere is seldom 100 %. Water molecules are lighter than nitrogen (or average air) molecules and thus water vapour rises upward due to buoyancy. Depending on the temperature and vicinity of open water surfaces, the rate at which water molecules float upward may be faster than the rate of evaporation, resulting in a temporary steady state where relative humidity is below 100 %. The water vapour in the atmosphere is not lost to the space, however. At some point it cools down so much that it condenses. Thus clouds are made, which float until they can't support their own weight (by whatever means, up-drafts, buoyancy...) at which point the liquid water obeys gravity and falls down again. This greatly enhances evaporation rates while cooling the atmosphere down, and thus relative humidity rises to near 100 % while it's raining. Once the rain is done and the excess water is either absorbed or evaporates again, the cycle starts up again. Water molecules are lost to the upper layers of the atmosphere faster than evaporation can supply them, and thus relative humidity falls below 100 % again. How far below depends on the temperature, major air currents, open bodies of water, and so on.