Let me begin this article in D=d+1 spacetime. We are going to study quantum gases and their statistics in multidimensional space. Usual notation:
In D=d+1 spacetime, the massless free ideal relativistic gas satisfies, as we will show, certain relations between thermodinamical variables. For instance,
if the dispersion relationship is in d-dimensional SPACE. The Bose-Einstein integral reads
We define and the energy density
Generally, we will work with natural units (they will be reintroduced if necessary) and with zero chemical potentical and massless particles. We will explore the issue of massive particle statistics though.
Spherical coordinates (in d-dimensional euclidean space they have d-1 angles ) introduce a (d-1)dimensional solid angle
for the solid angle . Moreover, a trivial calculation
For a dispersion relationship in d-dimensional space, we have
The pressure gives us the Stefan-Boltzmann law in higher dimensions
We are ready to study the phenomenon of Bose-Einstein condensation (BEC) for ideal massive (relativistic and non-relativistic) bosonic gases. Take the dispersion relation to be now
and where NR denotes non-relativistic, UR denotes ultrarelativistic (massless relativistic or almost massless relativistic). Now, but the ideal bosonic gas in a box of size L, with
The critical temperature of the BEC, is approache when and . The number density will be then
where with the Bose integral diverges. Then, the non-relativistic bosonic BEC temperature reads
In 3d we get the known result
where we have used .
In the case of the ultrarelativistic (massless, almost massless) case, we obtain
In the 3d case, we get the very well result
and where is the zeta value of 3.
The 2d UR case HAS a critical temperature (unlike the 2d non-relativistic case, where BEC does NOT exist) . Indeed, you can easily check that
BEC does depend not only on the number of dimensions but also on the density of states, i.e., it is highly dependent on the dispersion relationship we use!
An additional important issue is the following. If we allow , then pairs boson-antiboson can be created. In particular, we have that
for each particle. Moreover, you have
with + for bosons and – for antibosons, so
with . Thus, we get
Case 1. Low T, with . Then , as we would expect.
Case 2. High T, with . Then, the critical temperature DOES change to take into account the boson-antiboson pair creation. It yields
Note that the UR boson-antiboson massive case is not equal to the UR boson (massless) case, even in 3d! Indeed, you find that
In fact, with no antiboson (massless), the case provides the following thermodynamical quantities:
When antibosons are present, these integrals become nastier and more complicated:
Challenge for eager readers:
Take the density of states
and take the UR limit to get and prove
and calculate/check the critical temperature
obtaining the value of the “constant” above.
By the other hand, we can also study the fermionic gas in D-dimensional spacetime, d-dimensional space. In order to simplify the discussion, we are going to study only the non-relativistic (NR) ideal gas. I will study the relativistic Fermi gas in a future post because it is important in extremely degenerate systems, as some particular kind of stars. The ideal non-relativistic Fermi gas in d-dimensional space has the following interesting features:
Moreover, you have
and the Fermi energy reads ():
The dimensional Fermi weights are, for D=d+1 dimensional spacetime and the NR and UR case
The non-relativistic (NR) fermionic (or fermi) gas has a dispersion relationship
and the energy in terms of Fermi quantities reads
and the density of states
Thus, the energy density will be
with the thermal wavelength
where is the fugacity. The Fermi function reads
and it uses the polylogarithm as well!!! Wonderful, isn’t it? The average energy per fermion in d-space is
You get if , if and you also have
and the number density
The massless spinless bosonic particles in D=d+1 dimensions have a free energy
The Casimir energy of such a bosonic field requires
and the regularized energy in vacuum has to be
and it shows that the Riemann zeta functional equation holds iff
This striking consequence and relationship between the vacuum structure and pure mathematics is fascinating and not yet completely understood. But this will be a topic for a future discussion here.
See you in my next blog post!
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