# Is there a simple explanation for Fermi-Dirac statistics?

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• YoungPhysicist
In summary: The half-integer spin particles are fermions because their wavefunction is antisymmetric, while the integer spin particles are bosons because their wavefunction is symmetric.

#### YoungPhysicist

Is there a simple explanation for fermi-dirac statistics? I can't understand how a particle can "follow" a statistic.

That is not a question of a single particle, it is a question of how a collection of identical particles behave statistically. Ultimately, this is settled by experiment, but the underlying reason half-integer spin particles follow FD statistics is laid out in the spin-statistics theorem.

So what fundamentally makes fermions and bosons different?

Fermions follow FD statistics and bosons BE statistics. But that is not the question I believe you really want to ask. I believe that the question you want to ask is "Why are half-integer spin particles fermions and integer spin particles bosons?" This, again, is laid out in the derivation of the spin-statistics theorem, see, e.g., https://en.wikipedia.org/wiki/Spin–statistics_theorem , and is ultimately connected to the behaviour under the exchange of particles.

mfb and YoungPhysicist
Thanks man. I'll check it out.

Young physicist said:
Is there a simple explanation for fermi-dirac statistics? I can't understand how a particle can "follow" a statistic.

I think your question here exhibits a deeper issue than simply the Fermi-Dirac statistics.

The world that you see around you arises out of the collective behavior of a gazillion particles. You never deal with just one, or a few particles or interactions. Now, it is impossible to know the dynamics of every single one of these particles. So instead, we find a description of their collective behavior, and we find that in many instances, this collective description works quite adequately. This is the "statistics" that we are talking about.

You also do not need to delve into quantum physics to talk about statistics of particles. The basic thermodynamics laws are based on the classical description of particles via the Maxwell-Boltzmann statistics. So even before QM, we are already well-aware of statistical mechanics.

Zz.

mfb and YoungPhysicist
I think the simple thing you seek without a course in QM is included in the spin statistics page that @Orodruin already linked.

https://en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem said:
Bosons are particles whose wavefunction is symmetric under such an exchange or permutation, so if we swap the particles the wavefunction does not change. Fermions are particles whose wavefunction is antisymmetric, so under such a swap the wavefunction gets a minus sign, meaning that the amplitude for two identical fermions to occupy the same state must be zero. This is the Pauli exclusion principle: two identical fermions cannot occupy the same state. This rule does not hold for bosons.

From that "simple" difference, the statistics arise.

## 1. What are Fermi-Dirac statistics?

Fermi-Dirac statistics, also known as Fermi statistics, is a branch of quantum statistics that describes the behavior of a large number of identical particles with half-integer spin, such as electrons, protons, and neutrons.

## 2. How are Fermi-Dirac statistics different from other statistical models?

Fermi-Dirac statistics differ from other statistical models, such as Bose-Einstein statistics, in that they take into account the exclusion principle, which states that two identical fermions cannot occupy the same quantum state simultaneously. This leads to a unique distribution of particles and their energies.

## 3. Is there a simple explanation for Fermi-Dirac statistics?

While the mathematics behind Fermi-Dirac statistics can be complex, the concept itself is fairly straightforward. It is based on the idea that particles with half-integer spin follow different rules than particles with integer spin, and their behavior is influenced by the exclusion principle.

## 4. How are Fermi-Dirac statistics applied in real-world scenarios?

Fermi-Dirac statistics are used to understand and predict the behavior of fermionic systems, such as electrons in a solid material. This has important applications in fields such as condensed matter physics, materials science, and electronics.

## 5. Are there any limitations to Fermi-Dirac statistics?

While Fermi-Dirac statistics accurately describe the behavior of fermionic systems in most cases, they do have some limitations. For example, they do not take into account the effects of interactions between particles, which can become significant at high densities or temperatures.