@DavidZ hey, do you have a second for a question about Fermi-Dirac and Bose-Einstein distirbutions?

Ah, sorry I missed this, but unfortunately I'm busy now.

@JohnRennie hey John

@psa hi :-)

I've got a question from your favourite subject

(not really your favourite subject)

I was thinking because they seem randomly assorted, they should be fermions in the high temperature limit.

A friend of mine argued that it could be either bosonic or fermionic, but I think that since $$\bar{n}_{FD} \to 1/2$$ and $$\bar{n}_{BE} \to \infty$$ as T gets large, a bosonic distribution should have many more particles in the spectrum. My friend said that maybe it's just a system with a really small number of particles, so now I don't know.

Any ideas?

Well they could also be bosons. In the high temperature limit the probability of getting two bosons in the same state is small, so the fact we don't see two particles in the same state could just be random chance.

oh right

i forgot about that !

In fact the Fermi-Dirac and Bose-Einstein distributions both tend towards the Boltzmann distribution in the high temperature limit.

So at high temperatures you can't tell the difference.

@JohnRennie what's the mathematical description of the Boltzmann distribution?

$e^{-\beta(\epsilon-\mu)}$?

in the grand canonical ensemble

I can't remember enough about stat mech to recognise your notation.

I just remember it as $P(E) = e^{-E/kT}$

hmm okay

I'm confused about something from the lecture notes then

so, $\beta = 1/kT$, $\epsilon = E$, and $\mu$ is chemical potential

we had that the average number of particles in the state of energy $\epsilon$ (Fermi-Dirac) was $\bar{n}_{FD} = \frac{1}{e^{\beta(\epsilon-\mu)}+1}$

likewise $\bar{n}_{BE} = \frac{1}{e^{\beta(\epsilon-\mu)}-1}$ for Bose-Einstein

so we get:

notice that it says where $\epsilon-\mu \gg kT$, isn't this the low T limit?

I'm pretty confused about that...

because I think it should be the high T limit as well, but mathematically something seems to be wrong wrong?

the reasoning I guess is that $\beta$ gets large for low $T$ so the +1 or -1 in the denominators for the FD and BE distributions don't really matter, and they tend to $e^{-\beta(\epsilon-\mu)} = \text{Boltzmann Distribution}$

The argument for approaching the Boltzmann distribution is that if $e^{\beta(\epsilon-\mu)} \gg 1$ the $1$ is negligible and can be removed, which gives the Boltzmann distribution.

yes, but if $\epsilon > \mu$ then low $T$ gives high $\beta$ which gives $e^{\beta(\epsilon-\mu)} \gg 1$?

That's the high $E$ limit.

I'll have to think about it. I need to work now for a while.

OK sounds good

no rush really : )