Why are particles distributed differently in statistical physics?

[TheDestroyer]

Hi guyz, My question is very easy, I just don't convince by the distribution laws, and the professor of this subject isn't good in mathematics so he just written them on the board + i don't have right now a book for number theory...

Please answer these questions with complete details:
1-Why are the distinct particles distributed in this law in ALL LAYERS (Maxwell-Boltzmann) (more than one particle can take 1 cell)?

W=N! [Pi Product over i] ((gi^Ni)/Ni!)

i is the layer number, We have N distinct particles in the whole system, Ni Particles in the layer i, gi cells in the layer i

2-Why are the fermions distributed in this law in one layer (Fermi-Dirac) (I know that every fermion can take one cell)?

W=g!/(N!(g-N)!)
where we have g cells, N fermions

3-Why are the bosons distributed in this law in one layer (Boze-Einstein) (I know that all bosons can take one cell or more)?

W=(N+g-1)!/(N!(g-1)!)

I can't understand the distribution theories, anyone can help?
Thanks

 

[TheDestroyer]

Very Funny! The Whole Forum Doesn't Have An Answer For My Silly Question! Lol

 

[DonnerJack]

Maybe this will help...

1-Why are the distinct particles distributed in this law in ALL LAYERS (Maxwell-Boltzmann) (more than one particle can take 1 cell)?

W=N! [Pi Product over i] ((gi^Ni)/Ni!)

i is the layer number, We have N distinct particles in the whole system, Ni Particles in the layer i, gi cells in the layer i

it might help to think of it like this:

1. gi^Ni => you have for each particle in the layer, an option of gi cells to populate.
2. dividing by Ni! => you don't care about the order of placement you take care of that later.
3. Pi Product => you multiply each "option" for each layer by the other "options" (for the rest of the layer)
4. Finally, you multiply by N! for permutations between all of the particles because they are distinguishable.

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2-Why are the fermions distributed in this law in one layer (Fermi-Dirac) (I know that every fermion can take one cell)?

W=g!/(N!(g-N)!)
where we have g cells, N fermions

they are not distinct, and you can populate each cell with only one fermion.

it's like selecting N cells out of g cells without caring for the order of selection => g over N !

--------------------------------------------
3-Why are the bosons distributed in this law in one layer (Boze-Einstein) (I know that all bosons can take one cell or more)?

W=(N+g-1)!/(N!(g-1)!)

similar to the fermions but with the difference you noted above.

so it turns out to be a question of lining all of the bosons in a line and deciding where to put the dividers (between cells).

you have N particles + g-1 dividers => consider it as a line of N+g-1 objects.
Now permute all of them => (N+g-1)!
But you need to take into consideration that you have N identicle particles ang (g-1) identicle dividers => divide by N! and (g-1)!

 

[nrqed]

Very Funny! The Whole Forum Doesn't Have An Answer For My Silly Question! Lol

DonnerJack gave an excellent explanation, please see his/her post.

But just to letyou know: I just saw your post for the first time a few seconds ago when logging in. It takes some time for people to get around to seeing the posts (we don't all live in the same time zone! And we have other thinsg to do as well). It does not encourage people to spend time typing a detailed reply if you show this type attitude.

Again, DonnerJack's reply is excellent.