# How to reduce fermi-dirac to maxwell-boltzmann in a solid?

• kof9595995
In summary, for distinguishable particles like atoms in a solid, we typically use Maxwell-Boltzmann statistics. However, it is possible to treat these particles as indistinguishable and use Fermi-Dirac or Bose-Einstein statistics instead. This is because the overlaps between the particles' wavefunctions are very small, allowing us to approximate the result of Maxwell-Boltzmann statistics. This can be seen mathematically through the relationship between the degeneracy and occupation numbers for each energy level in the three different statistics.
kof9595995
For indistinguishable particles we use fermi-dirac(FD) or bose-einstein(BE), and for distinguishable we use maxwell-boltzmann(MB).For the distinguishable case our prof gave us the example of atoms in solid, because the positions of the atoms are fixed, so they are distinguishable, thus satisfy MB statistics.
But the so called "fixed" i think is just extremely narrow wave packets with very small overlaps, so in the strict sense atoms in solid are still indistinguishable, then how can we reduce FD or BE to MB, from the condition "wave packets are narrow with small overlaps".
Like in the treatment of dilute gas, when we assume the occupation number is much smaller than the number of degeneracies for each energy level, from the math FD and BE reduce to MB nicely. So it puzzled me whether for the atoms in solid we can find a nice way to reduce to MB.

Well, no reply for a long time...Maybe I did not state the question clear enough, let me try again:
For atoms in solid, the overlaps between wavefunctions are so small that we can treat them as distinguishable, hence the MB statistics.
But let's take a alternative point of view: atoms in solid are actually indistinguishable, so it's always correct to use FD statistics (let say they're fermions), but we said we could use MB because overlaps are small, which means that $${\Omega _{MB}} \approx {\Omega _{FD}}$$ when overlaps are small. So how do we get "$${\Omega _{MB}} \approx {\Omega _{FD}}$$" from "overlaps are small"
For reference:
MB:$${\Omega _{MB}} = \prod\limits_{j = 1}^n {\frac{{g_j^{{N_j}}}}{{{N_j}!}}}$$
FD:$${\Omega _{FD}} = \prod\limits_{j = 1}^n {\frac{{{g_j}!}}{{{N_j}!\left( {{g_j} - {N_j}} \right)!}}}$$
Where gj is the degeneracy for different energies and Nj is the occupation number

## 1. What is the difference between Fermi-Dirac and Maxwell-Boltzmann statistics?

Fermi-Dirac and Maxwell-Boltzmann statistics are two different ways of describing the distribution of particles in a solid. Fermi-Dirac statistics apply to particles with half-integer spin, such as electrons, and take into account the Pauli exclusion principle. Maxwell-Boltzmann statistics apply to particles with integer spin, such as atoms, and do not take into account the Pauli exclusion principle.

## 2. Why would one want to reduce Fermi-Dirac to Maxwell-Boltzmann in a solid?

Reducing Fermi-Dirac statistics to Maxwell-Boltzmann statistics can make calculations simpler and more accurate in certain situations. For example, at high temperatures, the effects of the Pauli exclusion principle become less significant, and Maxwell-Boltzmann statistics can be used instead of Fermi-Dirac statistics.

## 3. Is it possible to completely reduce Fermi-Dirac to Maxwell-Boltzmann in a solid?

No, it is not possible to completely reduce Fermi-Dirac to Maxwell-Boltzmann in a solid. The two statistics are fundamentally different and apply to different types of particles. However, in certain situations, such as at high temperatures or low densities, the effects of the Pauli exclusion principle can be neglected and Maxwell-Boltzmann statistics can be used instead of Fermi-Dirac statistics.

## 4. How does one go about reducing Fermi-Dirac to Maxwell-Boltzmann in a solid?

The process of reducing Fermi-Dirac to Maxwell-Boltzmann in a solid involves taking the limit of high temperatures or low densities. In this limit, the effects of the Pauli exclusion principle become less significant and the distribution of particles in a solid can be described by Maxwell-Boltzmann statistics instead of Fermi-Dirac statistics.

## 5. Can Fermi-Dirac and Maxwell-Boltzmann statistics be used interchangeably in all situations?

No, Fermi-Dirac and Maxwell-Boltzmann statistics cannot be used interchangeably in all situations. They are two different ways of describing the distribution of particles in a solid and apply to different types of particles. However, in certain situations, the effects of the Pauli exclusion principle can be neglected and Maxwell-Boltzmann statistics can be used instead of Fermi-Dirac statistics.

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