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?

  • 3
    $\begingroup$ It's always of that form. $\endgroup$ Oct 13 '21 at 0:32
  • $\begingroup$ In your formula for the density of states, the Dirac distribution should be $\delta(E(\vec k)-E)$. $\endgroup$
    – Christophe
    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 $D=1$).


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