When I open a window to air out the room, how does the smell disperse? Let's say I'm in a room with some kind of noxious stink, possibly of flatulent nature.  The quickest way to right the world that comes to mind is to open a window.  When I open a window, how do the stank particles leave the room?
 A: Gases diffuse from higher concentration to lower concentration because this process of diffusion increases the gas's entropy.
Every thing in the universe, in general, aims for two things : Minimum energy and maximum entropy.
Entropy is the degree of randomness of a substance. A gas in a container has low entropy because the probability of finding that gas in the universe is less but when the container is opened the gas diffuses into the atmosphere and this diffusion increases the probability of finding that gas in the universe and thus, increasing the entropy.
To move a little more into science, The Gibbs energy change of the system must be -ve in order for the process under observation (say diffusion) to be spontaneous or naturally occurring.
∆G = ∆H - T∆S

Here, ∆G, ∆H and ∆S are the change in the gibbs energy, enthalpy and entropy of the system respectively.
(enthalpy is the heat content of the system at constant pressure).
Now, when gas diffuse from higher concentration to lower concentration the heat change is negligible but the entropy change is highly +ve (∆S = +ve).
The net result is the decrease in the gibbs energy of the system.
Therefore, Diffusion of a gas from higher concentration to lower concentration is a spontaneous or naturally occurring process.
A: They leave slowly because of the air exchange via window, depend upon rate of exchange.
On a lighter note, even if you do not open the window, the smell will seem to disappear in a little bit :)
A: That process is called diffusion, the stinky small in the room gradually leaves the room until the stinky smell is evenly distributed. when the air is evenly distributed the room assumes a new state of smell.
