- #1
Morgoth
- 126
- 0
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:
<ni>= 1/ { 1+ e[β(εi-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<<nQ=(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(εi-μ)]
getting:
e[-β(εi-μ)] / (e[-β(εi-μ)] +1)
Now again for β→0 I get
e[-β(εi-μ)] /2
which is half what I want to get...
Let's see the FermiDirac one:
<ni>= 1/ { 1+ e[β(εi-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<<nQ=(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(εi-μ)]
getting:
e[-β(εi-μ)] / (e[-β(εi-μ)] +1)
Now again for β→0 I get
e[-β(εi-μ)] /2
which is half what I want to get...