what makes the particles move from a high concentration to a lower concentration?
Consider two adjacent regions A,B,(of water, for example) which has different concentrations of a solvent (for example, salt).
Let the boundary between these two regions be permeable for the solvent.
(Think of A and B as the two halves of a tank)
Now, every molecule of the solvent moves in a random fashion.
Hence, the fraction of molecules in A that passes over into B (that is, the percentege of the molecules contained in A) should equal the percentage of molecules originally in B that passes into A.
But then it follows that the actual number of molecules passing from the region of higher concentration must be higher than the number of molecules passing over from the region of lower concentration.
Hence, you have a net drift of molecules from the region of higher concentration into the region of lower concentration.
It is important to realize that the molecules themselves couldn't care less about which region they belong to; diffusion is the net effect of random motion.
"Perfectly" random motion.
To maximize entropy. It's not that they specifically seek out regions of low concentration, though it could be viewed that way. They just get randomly shoved around (as arildno pointed out) till they are pretty much spread out uniformly. Even though I use the word "uniformly", the system is now more disordered, and it's harder for me to guess where a certain type of particle can be found. In the initial state of the system, I would have better odds of correctly guessing the location of this type of particle - since I know they are all concentrated in a certain region.
Isn't this more of an answer to the question:
"Why do the particles exhibit (practically perfect) random motion?"
It seems to me that you chose to explain this fundamental phenomenon in your reply (which I merely stated as a fact)
Perhaps, but I was also setting up a framework for analysing other such problems - the concept of maximizing entropy is a useful tool and allows you to not have to go down to first priciples each time.
Is this random motion caused by the bombardment of air molecules? Like Brownian Motion?
Not necessarily. You can have random motion with a pure species that is not one of the species found in air. The random nature is due to our inability to monitor the particles individually (since there can be billions of billions of billions of them). The temperature and pressure of a gas are statistical properties. It would be another thing entirely to specify the state of a system by the positions and momenta of every particle in the system. Since we cannot do that (not practically nor even theoretically AFAIK), we use temperature, pressure, volume, entropy, and such to specify the state of the system. Since these properties are statistical wrt what each particle in the system is doing, then the system can be in a well defined statistical state (i.e. T=300K, P=1atm) while any given particle undergoes completely unspecified motion.
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