Do gases mix faster when of unequal temperature? Or, 

Is my room aired quicker during winter?

I always feel that in winter, after opening the window, my study room becomes breathable quicker than in summer. Now, this is of course highly subjective, but even though I see no rational explanation supporting such a phenomenon, I neither can intuitively outrule it.
Surprisingly, Google was no help here.
 A: There are a few things that would be going on here.
1. Convection
There's a common expression "heat rises".  That is because things are heated, their density often lowers.  In fluids (such as air and water), if there is a density difference in an open space, you will create a flow from high to low density (going from high to low pressure, and the diffusion will both contribute).
The cooler it is outside compared to inside; the greater this effect.
On the same note, once you get air moving, it not only cools the room quicker; but it feels cooler if it moves more.
2. Humidity
Warm air can pick up more moisture, and humid air feels warmer.  It's harder to cool yourself off when it is humid.  
I imagine this is a large part of why it becomes breathable quicker.  The humidity in the air is a big part of comfort.  When you open the window and it's humid outside,there is less diffusion of moisture (on really bad days I've had it bring moisture in).  In the winter the cool air doesn't have as much moisture, so the humidity is quick to start mixing with the dry air outside.
A: Probably yes.
The warmer air inside your room is less dense than the colder air from outside, so the latter can easily flow over the windowsill into the room, dislodging the warmer air, which escapes like smoke through the window.
If there are two openings allowing a strong draft to form, there shouldn't be much difference between summer and winter, but you'll probably still find it "becoming breathable quicker" in winter, since the additional temperature difference makes the fresh air more noticeable.
