General version of fermi-dirac distribution?

[timmy1234]

general version of fermi-dirac distribution??

merry x-mas everyone!

in the Boltzmann distribution every state with energy Ei can be occupied by an arbitrarily large number of molecules. In contrast, if each state can be occupied by only one particle then one needs to use the fermi dirac distribution.
here comes the question:
does a general solution exist for the case that only Ni particles can occupy energy state Ei? For instance, state E0 can be occupied by <= 5 molecules, energy state E1 can be occupied by <= 20 molecules etc.

I have looked in a couple of books but none talks about this situation.

Many thanks!

Tim

 

[olgranpappy]

merry x-mas everyone!

in the Boltzmann distribution every state with energy Ei can be occupied by an arbitrarily large number of molecules.

boltzmann? you mean Bose-Einstein?

In the classical limit, for either BE or FD statistics, the mean occupation number reduces to

e^{-\beta(E_i-\mu)}

where \mu is very large in magnitude and negative so that the mean occupation number is always tiny.

In contrast, if each state can be occupied by only one particle then one needs to use the fermi dirac distribution.

If incontact with a thermal bath one always starts from the same place. always. One starts from the fact that the probability that a state 'x' is occpied depends on the energy
of the state E_x and is given by


e^{-\beta(E_x)}


in the independant particle approximation the energy of the state is the sum of the individual particle energies, etc, one ends up with either the BE or FD mean occupation number. In general, for independant particles, regardless of the statistics, the energy is a state x is the sum of independant particle energies \epsilon_i and depends only on occupation number for identical particles. And one just performs the sum


Z=\sum_{n1,n2,...}e^{-\beta(\sum_{i}n_i \epsilon_i)}.


For BE n1,n2,etc run from 0 to infinity

For FD n1,n2,etc run from 0 to 1

But you are free to choose n1, n2, etc, to be occupied however you like. Unfortunately you won't be describing any real physics. Cheers.

here comes the question:
does a general solution exist for the case that only Ni particles can occupy energy state Ei? For instance, state E0 can be occupied by <= 5 molecules, energy state E1 can be occupied by <= 20 molecules etc.

I have looked in a couple of books but none talks about this situation.

Many thanks!

Tim

 

[IITian]

Please have a look at this:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/disfd.html

 

[timmy1234]

First off, thanks for the replies!

but i think my question didn't get across too well.

the point i tried to make is the follwoing:

if the pauli exclusion principle doesn't apply then one can use the Boltzmann distribution to determine how many partiles will be sitting on energy state Ei with equation:

Ni = N * exp(Ei/kT) / Z.

where N is the total number of particles and Ni is the number of particles with energy Ei.

If the pauli excultion principle must be obeyed (that is Ni is einther 0 or 1) then one needs to apply the Fermi-Dirac distribution.

The question is: if Ni is restricted to some number Ai (not necessarily 1 but any number)what is the statistics that one is supposed to use?

 

[olgranpappy]

First off, thanks for the replies!

but i think my question didn't get across too well.

the point i tried to make is the follwoing:

if the pauli exclusion principle doesn't apply then one can use the Boltzmann distribution to determine how many partiles will be sitting on energy state Ei with equation:

Ni = N * exp(Ei/kT) / Z.

where N is the total number of particles and Ni is the number of particles with energy Ei.

If the pauli excultion principle must be obeyed (that is Ni is einther 0 or 1) then one needs to apply the Fermi-Dirac distribution.

you need to work through the derivation of the single-particle occupation number <N_a> starting from where I suggested in my previous post

<N_a>=\frac{-1}{E_a-\mu}\frac{\partial}{\partial \beta}\log(Z_a)

The question is: if Ni is restricted to some number Ai (not necessarily 1 but any number)what is the statistics that one is supposed to use?

 

[timmy1234]

$$<N_a>=\frac{-1}{E_a-\mu}\frac{\partial}{\partial \beta}\log(Z_a)$$