FermiDirac-BoseEinstein-Boltzman derivation

  • Thread starter Morgoth
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    Derivation
In summary, the FermiDirac and BoseEinstein distributions have a limit at high temperatures or low densities, where they approach the Boltzmann's distribution. To reach the classical limit, one must take into account both high temperatures and low densities, which can be represented using the fugacity, z. The Maxwell-Boltzmann distribution can be obtained from the FermiDirac distribution by taking the limit of z approaching 0.
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Morgoth
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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...
 
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  • #2
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
 

What is the Fermi-Dirac distribution and how is it derived?

The Fermi-Dirac distribution is a probability distribution used to describe the distribution of fermions (particles with half-integer spin) in a system. It is derived from the principles of quantum mechanics and statistical mechanics.

What is the Bose-Einstein distribution and how is it derived?

The Bose-Einstein distribution is a probability distribution used to describe the distribution of bosons (particles with integer spin) in a system. It is derived from the principles of quantum mechanics and statistical mechanics.

What is the Boltzmann distribution and how is it related to the Fermi-Dirac and Bose-Einstein distributions?

The Boltzmann distribution is a probability distribution used to describe the distribution of particles in a classical, non-quantum system. It is derived from the principles of classical mechanics and statistical mechanics. The Fermi-Dirac and Bose-Einstein distributions are special cases of the Boltzmann distribution, with additional quantum effects taken into account.

What are the key differences between the Fermi-Dirac and Bose-Einstein distributions?

The key difference between the Fermi-Dirac and Bose-Einstein distributions lies in their statistics. Fermi-Dirac statistics follow the Pauli exclusion principle, meaning that no two fermions can occupy the same quantum state. Bose-Einstein statistics do not follow this principle, allowing multiple bosons to occupy the same state. This leads to different distributions and behaviors for fermions and bosons.

How are the Fermi-Dirac and Bose-Einstein distributions used in real-world applications?

The Fermi-Dirac and Bose-Einstein distributions are used in many areas of physics, including condensed matter physics, quantum mechanics, and astrophysics. They are particularly useful in understanding the behavior of particles in systems at low temperatures, such as in superconductivity and Bose-Einstein condensates.

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