# Energy of a Free electron gas in D dimensions

I am trying to calculate the internal energy of a free electron bas in a box in $$D$$ dimensions. To calculate the density of states, I used the following formula:

$$g(E) = \int \frac{d^Dk}{(2\pi)^D} \delta(E(k)- E(k'))$$

From this, I found that the density of states is $$g(E) = \gamma E^{\frac{D}{2}-1}$$, where $$\gamma$$ is a constant in terms of $$D$$, $$\pi$$, $$\hbar$$, etc. Using this, I plan to calculate the energy for finite temperatures. Hence, I used to the following equation:

$$U = \int Eg(E) f(E) dE$$

In the three dimensional case, I know for sure that $$f(E)$$ is the Fermi-Dirac distribution, which enables us to use the Somerfield expansion formula. In $$D$$ dimensions, does the following hold? $$f(E) = \frac{1}{e^{\beta(E - \mu)} + 1}$$

In other words, is there a $$D$$-dimensional version of the Fermi-Dirac distribution, or is the Fermi-Dirac distribution always of the form $$f(E) = \frac{1}{e^{\beta(E - \mu)} + 1}$$ regardless of dimension?

• It's always of that form. Oct 13 '21 at 0:32
• In your formula for the density of states, the Dirac distribution should be $\delta(E(\vec k)-E)$. Oct 13 '21 at 11:11

Fermi-Dirac distribution depends only on the energy - it is a consequence of the grand-canonical ensemble applied to particles, obeying Pauli exclusion principle; a state of energy $$E$$ can be either occupied, with weight $$e^{-\beta(E-\mu)}$$ or empty with weight $$1$$ so that the partition function for this state is $$Z=e^{-\beta(E-\mu)}+1,$$ whereas the probabilities of the occupied and empty states are $$P_1=\frac{1}{Z}e^{-\beta(E-\mu)}=\frac{1}{e^{\beta(E-\mu)+1}}=f(E),\\ P_0=\frac{1}{Z}=\frac{e^{\beta(E-\mu)}}{e^{\beta(E-\mu)+1}}=1-f(E).$$
The statistical physics arguments leading to the grand canonical distribution are independent on the number of dimensions - the only place where the dimensions appear is when neglecting the energy contributions due to the surface encolising the system, scaling as $$R^{D-1}$$, in comparison to the volume that scales as $$R^D$$. This works for arbitrary integer $$D$$ (although some peculiarities may appear in systems with $$D=1$$).