# Eli5 What is fermi gas?

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Eli5 What is fermi gas?

In: Physics

Suppose you have a room with a billion seats. You want to have ten people sit down. How many ways can they sit in the seats?

Well, one way to model it is to say each of the 10 people has a billion choices, so you have (1 billion)^10 = 10^90 possible states. But a more precise model considers the fact that two people can’t sit in the same seat, so you actually have (1 billion) * (1 billion – 1) * … (1 billion – 9) possible seats, or 9.999999 x 10^89 possible states. The room is so big this difference barely matters (it’s about 1 part in a million), and so you can sort of ignore the fact that two people can’t sit in the same seat.

This situation is analogous to an ideal gas (aside from the fact that we can distinguish between people and not between gas particles, which is important to the physics but not necessary for your question). The people represent the particles of gas, and the chairs represent their possible states (not just location, but also energy level). In an ideal gas, we imagine that particles may take any random state they want, and may be pushed into any other state. This isn’t true! Two gas particles can’t be in the same place with the same energy! But molecules are so small relative to the space between them, and the number of available energy states is so large, that it’s vanishingly unlikely any two particles would ever be in the same state anyway. So excluding those states makes almost no difference.

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Now, shrink the room. There are now only 11 seats. People still can’t both sit in the same chair.

This time, the collisions matter. A naive estimate gets you 11^10 = 2.6 x 10^(10) possible configurations. But if you handle the situation properly, with the first person having 11 choices, the second having 10, and so on, you get 11 * 10 * 9 * 8 * … * 2 = 3.99 x 10^(7) possible states. That’s almost a thousand times fewer possible states! This time, the number of possible states changes *dramatically* when you ignore the collisions.

This situation is analogous to where you’d need to model a gas as a fermi gas. The gas is either so dense (so many people) or so cool (so few ‘chairs’, i.e. possible states) that the number of “chairs” (states) isn’t much bigger than the number of “people” (particles). The collisions start to matter, which changes the statistical properties of the whole system.

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In a real setting, the number of “chairs” (states) isn’t actually hard-capped, the chairs just get increasingly “uncomfortable” (higher energy) in a way that makes particles prefer other states. Adding more people, or reducing the number of chairs, forces particles into higher energy states, which requires extra energy input beyond what you would expect if you were modeling the gas as an ideal gas.

This happens, for example, in the core of a white dwarf star: there’s enough pressure that an ideal gas should just continue to compress without limit, but an ideal gas isn’t a good model in that environment. Instead, the particles run out of comfortable chairs (low energy states) and there’s not enough pressure to keep pushing new particles into increasingly uncomfortable chairs (high energy states). You can view this as there being an extra outward pressure of particles trying to escape into lower energy states, and this extra outward pressure stops the white dwarf from collapsing.