[eindiran]

At the beginning of the article, it mentions the phrase "the 5th state of matter", which I've heard before; personally, I find it quite misleading. Even if you exclude all of the intermediary states and all of the non-classical states (things like glasses and liquid crystals), there are a lot of exotic states of matter. The Wikipedia page for "States of matter" includes a lot of them[0]. Though perhaps the idea is that there is something fundamental enough about the Bode-Einstein condensate that it should count as the fifth fundamental state of matter?

On an unrelated note, its interesting to see the Riemann zeta function turn up here. Does anyone know why it cropped up here, in the critical temperature equation? The page on Bose-Einstein statistics[1] doesn't seem to include it at all.

[0] https://en.wikipedia.org/wiki/State_of_matter

[1] https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statisti...

EDIT:

This paper seems to provide some insight into the latter question - https://arxiv.org/abs/1101.3116

[contravariant]

>On an unrelated note, its interesting to see the Riemann zeta function turn up here. Does anyone know why it cropped up here, in the critical temperature equation? The page on Bose-Einstein statistics[1] doesn't seem to include it at all.

A complete derivation is beyond me at the moment but if you look at the equation in the section on the non-interacting Bose Einstein's gas [2] you'll note that they provide an equation for the critical point, the right hand side of which is quite similar to the integral definition of the Riemann Zeta function [3]. This is the likely origin of the Riemann Zeta function.

[2]:https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensa... [3]: https://en.wikipedia.org/wiki/Riemann_zeta_function#Definiti...

[sdenton4]

Qua? I believe the zeta function goes back to 1859, and arose in number theory... https://en.m.wikipedia.org/wiki/On_the_Number_of_Primes_Less...

[contravariant]

I meant that it's the reason the Zeta function appears in the formula, not that it is the original reason for the Zeta function to exist.

[sdenton4]

Ah, sorry!

[ssivark]

> interesting to see the Riemann zeta function turn up here. Does anyone know why it cropped up here, in the critical temperature equation?

I don’t know if this is satisfactory, but here’s a technical answer: A lot of field theory calculations (when including 1-loop quantum effects) involve summing over all the modes/states in the field. In thermal equilibrium we can think of the time axis as a circle, and the wave frequencies get quantized into integers (upto a multiple of the temperature) aka Matsubara frequencies. So now, the calculation involves summing over all natural numbers (raised to some power), which is basically the zeta function. A more careful explanation can even justify the 3/2 (from the fact that we have 3 spatial dimensions, and a boson has action scaling as momentum squared) but it’s been a while since I’ve thought about this.

[jpmattia]

> The page on Bose-Einstein statistics[1] doesn't seem to include it at all.

Unless I'm mistaken, it [e: meaning the page on condensates] does: Look at the "Derivation" section, and you'll see that g_3/2(f) is eventually manipulated into zeta.

[Or maybe I'm misunderstanding your comment.]

[eindiran]

I see that on the page for "Bose-Einstein condensate", in the section "Derivation". I don't see it in the "Derivation" section on the "Bose-Einstein statistics" page, which is where I was hoping to see an explanation (of what it means that it showed up here).

But this page has some of the desired info: https://en.wikipedia.org/wiki/Polylogarithm#Integral_represe...

"For ''z'' = 1 the polylogarithm reduces to the Riemann zeta function:

\operatorname{Li}_s(1) = \zeta(s) \qquad (\operatorname{Re}(s)>1)"

And

"The polylogarithm can be expressed in term of the integral of the Bose-Einstein distribution:

\operatorname{Li}_{s}(z) = {1 \over \Gamma(s)} \int_0^\infty {t^{s-1} \over e^t/z-1} \,dt"

[jpmattia]

FWIW, I think the reason it does not show up on the "Bose-Einstein statistics" page: The zeta function is not so relevant to describing how the bosons populate states at particular energies, but zeta is relevant once you start summing/integrating over those populations (eg calculating expectation values.)