FermiDirac-BoseEinstein-Boltzman derivation

by Morgoth
Tags: derivation
Morgoth is offline
Jul1-12, 09:37 AM
P: 127
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)]

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....
Phys.Org News Partner Physics news on Phys.org
Physicists consider implications of recent revelations about the universe's first light
Vacuum ultraviolet lamp of the future created in Japan
Grasp of SQUIDs dynamics facilitates eavesdropping
Bill_K is online now
Jul1-12, 02:19 PM
Sci Advisor
Bill_K's Avatar
P: 3,856
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 = eβμ.

<ni> = z / (z + eβεi).

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

<ni> = z e-βεi

Register to reply

Related Discussions
boltzman equation Astrophysics 3
maxwell boltzman (very sorry!!) Introductory Physics Homework 1
Steffan-Boltzman Law General Physics 8
How come? Boltzman stat. Classical Physics 1
Boltzman brains Cosmology 13