Why diffusion happens?

Question

The reason we smell fragrance from a far distance is due to the diffusion of its molecules. But which force causes this diffusion?
Which forces cause the material to propagate from higher density to lower density regions?

J Kusin · Accepted Answer

Fundamentally why densities seem to spontaneously go from higher to lower is the same reason why entropy tends to increase, systems go from non-equilibrium to equilibrium, and why we have an arrow of time.
It's simply the laws of physics plus initial conditons and letting the system evolve. The laws of physics alone have no direction. Small assemblages of particles have no arrow of time/entropy
and thus no "spontaneous" equilibration of systems like we see in our macroscopic view. It's only with initial conditions that the laws "pick out a particular direction". And in our universe we started with incredibly low entropy (and large enough) initial conditions. And as such the laws were basically forced to picked out this arrow of time toward increasing entropy.
Entropy inceases because there are simply more ways to arrange a system (by the laws of physics) in one direction than another once it's "large" enough. This gets at the other answer dealing with probability. There are exponentially fewer movements for particles in a corner of a box than at t+1 where they are a little more spread out. Why right now is a little more "disordered" than yesterday is because yesterday was a little more disordered than two days ago. We follow this logic until the big bang.
Yes at any point any particle can "go the other direction" toward lower entropy. The laws allow it and it happens with decreasing odds as the systems grows/evolves. In fact whole (open) systems can lower their entropy and go against the arrow temporarily. But the universe as a whole trends toward higher entropy with higher and higher odds at every timeslice, going back to the big bang.
So things diffuse because of their corresponding dynamical laws and their lower entropy and large enough prior conditions essentially. I hope this doesn't sound like a tautology because it isn't. There is a direction to systems because of the laws and initial conditions combined.

CGS · Answer

The basic phenomenon of diffusion can be studied by simple kinetic theory.  The approach is similar to that of studying heat transport or charge transport.  A gradient of a physical property will exist in a substance - in this case the number of particles of a particular species (your fragrance molecules) in a gas is not uniform.  From a thermodynamics point of view, this is not an equilibrium situation.  The attainment of equilibrium will require an increase in entropy - which will mean in this case a uniform distribution of the fragrance molecules.
Collisions between molecules are going to be responsible for bringing about equilibrium.  In kinetic theory, this is usually examined by looking at a plane surface and calculating the number of molecules crossing this plane in both directions. The next flux across this plane turns out to be proportional to the fragrance particle gradient, which will be non-zero in a non-equilibrium situation.
Kinetic theory allows one to come up with simple formulas for transport coefficients such as diffusion or thermal conductivity.

Gert · Answer

In a nutshell: probability!
Imagine a partitioned container. To the left of the partition all (gas) molecules are of the substance "$A$", to the right all (gas) molecules are of the substance "$B$". As long as the partition remains in place that's a stable arrangement.
Now we carefully remove the partition. The separation between $A$ and $B$ is now highly ordered and of very low probability. It is comparable to a checkers board with all white pieces to the left and all black pieces to the right: if you were to shake that board up, you'd end up with a random configuration because the random configuration is far more probable than the separated, ordered one.
Pieces on a board board checker can of course not move themselves but gas molecules are always moving in random direction caused by random collisions with other molecules. This way, the 'molecular checkers board' will over time rearrange itself from a high degree of order (improbable) to a high degree of disorder (probable)
This is reflected in Fick's First Law of Diffusion (here in $text{1D}$ only):
$$J=-Dfrac{text{d}varphi}{text{d}x}$$
where $J$ is the diffusion flux, which measures the amount of substance that will flow through a unit area during a unit time interval.
$frac{text{d}varphi}{text{d}x}$ is the concentration gradient: the higher it is the faster diffusion takes place.

Why diffusion happens?

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