Bose–Einstein Condensate
en.wikipedia.org
en.wikipedia.org
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
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...
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.
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.]
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"
https://johncarlosbaez.wordpress.com/2020/01/31/superfluid-q...
Apart from the weird quantum stuff, another interesting part of that perspective is that neurons are made up of countless microtubules and it might be the microtubules that are the basic computation unit of the brain. So then you can think of each neuron as a processor in its own right, and the computational complexity of the brain might be an order of magnitude higher than commonly supposed.
One can detect magnetic fields from a shielded laptop, even many meters away, and figure out what's happening inside (e.g. leaking secrets etc).
I think, not a terribly difficult experiment, but nobody has performed it: need ultracool physicists and hackers in one lab room
Anyone working with BECs?