Physics of (household) dust? There are plenty of papers dealing with the interstellar variety, but none dealing with plain old household dust afaict.


*

*Are there any references for the physics of dust?

*Are there for instance any mathematical models for the dust density
as a function of time?

*Is there any treatment from the viewpoint of
thermodynamics/statistical mechanics?

 A: Household dust is not a homogenous material, nor is it well-defined in terms its distribution of particle sizes. According to wikipedia :

Dust in homes, offices, and other human environments contains small amounts of plant pollen, human and animal hairs, textile fibers, paper fibers, minerals from outdoor soil, human skin cells, burnt meteorite particles, and many other materials which may be found in the local environment.

Typical particle sizes range from about $0.1\mu m$ to $1mm$. Because the composition is so variable, its physical properties (even statistical averages) are difficult to predict. There are no useful questions about the physics of household dust, so it is unlikely that anyone has undertaken a serious study. By contrast, studies on the influence of dust on human health are easy to find. 
The kind of issue you are asking about comes under sedimentation physics. Particles suspended in a fluid gradually settle because of gravity. This process is opposed by the thermal processes of diffusion and convection, and also cross-currents (draughts). As you suggest,  a dynamic equilibrium is always reached. Quite obviously, the larger particles eventually fall to the floor, the smaller particles remain suspended in the air. In still air, the density of each size of particle decreases exponentially away from the floor.   
A: Well, till now, I have not seen any references that might interest you. You have to wait for it. I will update my answer as soon as I get some.
Now, as for the thermodynamic treatment that you asked for, even if not done before, it wont be hard to derive one. The motion of a dust particle in air can be modelled as a 3D random walk, taking into consideration- the temperature, wind and humidity of the air. And the interaction of dust particles with the surface of any substance essentially depends on the nature of the substance. 
Another important factor is the number of dust particles in air. The higher the number, the less randomness of their motion.
