FermiDirac-BoseEinstein-Boltzman derivation

[Morgoth]

I am having hard times, trying to find out how the FermiDirac and BoseEinstein distributions give you at a limit the Boltzmann's one.


Let's see the FermiDirac one:

<n_i>= 1/ { 1+ e^{[β(ε_i-μ)]} }

where β=1/kT, where T:Temperature and k the Boltzmann's constant.

As we know the limits from quantum to classical physics for these are either at high Temperatures (so T→∞ So β→0) or low densities (n<<n_Q=(2πmkT/h²)^3/2).

So I am trying to put on Fermi-Dirac's distribution the limit β→0.


I just want you to reconfirm my work:
I multiplied on numerator and denominator with e^{[-β(ε_i-μ)]}
getting:

e^{[-β(ε_i-μ)]} / (e^{[-β(ε_i-μ)]} +1)

Now again for β→0 I get
e^{[-β(ε_i-μ)]} /2

which is half what I want to get...

 

[Bill_K]

Morgoth, If you only let T → ∞, then β → 0 and <n> → 1/2 = const, a uniform distribution which is what you got, and of course is not the classical limit. To get the classical limit you must take high temperature and low density. This is most easily done in terms of the fugacity, z = e^βμ.

<n_i> = z / (z + e^βε_i).

Let z approach 0, and then you get the Maxwell-Boltzmann distribution,

<n_i> = z e^-βε_i