# LOG#167. D-dimensional laws(II).

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, or if the dispersion relationship is in d-dimensional SPACE. The Bose-Einstein integral reads

(1) We define and the energy density

(2) 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 and where and  for the solid angle . Moreover, a trivial calculation  For a dispersion relationship in d-dimensional space, we have and where The pressure gives us the Stefan-Boltzmann law in higher dimensions

(3) since 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

(4) Define  where with the Bose integral diverges. Then, the non-relativistic bosonic BEC temperature reads

(5) In 3d we get the known result where we have used .

In the case of the ultrarelativistic (massless, almost massless) case, we obtain

(6) In the 3d case, we get the very well result and where is the zeta value of 3.

Remark:

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 where .

Remark (II):

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 and then 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

(7) 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:    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 Furthermore 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 with   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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