# What is Fermi-dirac statistics: Definition and 12 Discussions

In quantum statistics, a branch of physics, the Fermi–Dirac distribution is a probability distribution of particles over energy states in systems consisting of many identical particles that obey the Pauli exclusion principle. It is named after Enrico Fermi and Paul Dirac, each of whom discovered the method independently (although Fermi defined the statistics earlier than Dirac).Fermi–Dirac (F–D) statistics apply to identical and non-distinguishable particles with half-integer spin in a system with thermodynamic equilibrium. Additionally, the particles in this system are assumed to have negligible mutual interaction. That allows the multi-particle system to be described in terms of single-particle energy states. The result is the F–D distribution of particles over these states which includes the condition that no two particles can occupy the same state; this has a considerable effect on the properties of the system. F–D statistics apply to particles that are called fermions. It is most commonly applied to electrons, a type of fermion with spin 1/2. Fermi–Dirac statistics are a part of the more general field of statistical mechanics and use the principles of quantum mechanics.
A counterpart to F–D statistics is Bose–Einstein statistics, which apply to identical and non-distinguishable particles with an integer spin (0, 1, 2, etc.). These particles, such as photons (spin 1) and the Higgs bosons (spin 0), are called bosons. Contrary to fermions, bosons do not follow the Pauli exclusion principle, meaning that more than one boson can simultaneously be in the same quantum configuration.
In classical physics, Maxwell–Boltzmann statistics is used to describe particles that are identical and distinguishable.

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1. ### Average speed of molecules in a Fermi gas

My first most obvious attempt was to use the relation ##<\epsilon> = \frac{3}{5}\epsilon_F## and the formula for kinetic energy, but this doesn't give the right answer and I'm frankly not sure why that's the case. My other idea was to use the Fermi statistic ##f(\epsilon)## which in this case...
2. ### Quantum Gases - Mixing of 3He and 4He

Homework Statement Consider a solution in which 99% of the atoms are 4He and 1% are 3He. Assuming that the 3He atoms behave as an ideal gas of spin-1/2 particles determine the Fermi energy of the 3He atoms. You may assume that one mole of 4He occupies a volume of 28 cm3.Homework Equations EF =...
3. ### A Fermi-Dirac statistics, finding all electron configurations

Hello everyone. I'm having trouble understanding this example: https://ecee.colorado.edu/~bart/book/book/chapter2/ch2_5.htm#2_5_2 In this system of 20 electrons with equidistant energy levels, how is it known that there are only 24 possible configurations, and how are those configurations found?
4. ### I Charge carrier density: Hall experiment vs Fermi-Dirac statistics

Many times, the charge carrier density of a material is determined from a Hall effect experiment, from ##R_H=1/(ne)## (SI units). Where ##R_H## is determined from a measured voltage and other controllable parameters. As far as I know, this simple formula comes from the obsolete Drude's model...
5. ### I Is there a simple explanation for Fermi-Dirac statistics?

Is there a simple explanation for fermi-dirac statistics? I can't understand how a particle can "follow" a statistic.
6. ### I A question on Bose enhancement & Pauli blocking

Say I have ##n_{a}## bosons in some state ##a##, then the transition rate from some state ##b## to state ##a##, ##W^{boson}_{b\rightarrow a}##, is enhanced by a factor of ##n_{a}+1## compared to the corresponding transition probability for distinguishable particles, ##W_{b\rightarrow a}##, i.e...
7. ### I Solving Fermi-Dirac Integral: Step-by-Step Guide

i am completely lost. there is an integral in my textbook in fermi dirac statistics whose result is written directly and am not able to understand . it is ∫⌽(u) du /exp.((u-uf)/kt) + 1 from 0 to ∞ expanded by tayor's series to give...
8. ### Entropy and Internal Energy in Fermi-Dirac statistics

What are the formulas for entropy and internal energy in Fermi-Dirac statistics, and how do I derive them?
9. ### Fermi-Dirac statistics at the Fermi level

Hi all, I've search for my question and found no answer. I think it should be pretty simple... Fermi energy corresponds to the last occupied energy, as I understand it. So, energy levels in the Fermi gas are all filled with two electron of opposite spins, up to the Fermi energy. Saying it...
10. ### Fermi-Dirac statistics valid for electron gas in metals?

Hello! In my course of solid states physics we use the fermi-dirac statistics for a free electron gas in metals. The fermi wave length of the electrons is about 1 Angström. Now, the wavelength may be intepreted as something as a coherence range - the electron should forget about the state of...
11. ### Fermi-dirac statistics, Griffiths 5.28

Homework Statement Evaluate the integrals (eqns 5.108 and 5.109) for the case of identical fermions at absolute zero. Homework Equations 5.108 N=\frac{V}{2\pi^{2}}\int_{0}^{\infty}\frac{k^2}{e^{[(\hbar^{2}k^{2}/2m)-\mu]/kT}+1}dk 5.109...
12. ### Exploring Fermi-Dirac Statistics at T=0K

Hey kids, The question I'm having trouble with (this time) is as follows: Show that the Fermi-Dirac distribution function, f_{FD}(E)=\frac{1}{e^{(\frac{E-E_f}{kT})}+1} Has the following functional form at T= 0K (see attachment) Now, the first thing that screamed at me was...