Bose-Einstein Condensation Condition

Bose-Einstein condensation (BEC) is a state of matter that occurs in a gas of bosons at very low temperatures, where a large fraction of bosons occupy the lowest quantum state (the ground state).

For a gas of non-interacting bosons, the condition for BEC to occur at a finite critical temperature $T_c$ is related to the density of states $g(E)$ at low energies. The dispersion relation given is $E(k) = ak^s$ in $d$ dimensions, where $a$ and $s$ are positive constants.

Density of States in d Dimensions

In $d$ dimensions, the number of states in phase space with wavevector magnitude between $k$ and $k+dk$ is proportional to $k^{d-1} dk$. The density of states $g(k)$ in k-space is proportional to $k^{d-1}$.

To find the density of states $g(E)$ in energy space, we use the dispersion relation $E = ak^s$. From this, we have $k = (E/a)^{1/s}$.

Differentiating $E$ with respect to $k$, we get $\frac{dE}{dk} = sak^{s-1}$. Thus, $dk = \frac{dE}{sak^{s-1}}$.

Substituting $k$ in terms of $E$: $k^{s-1} = ((E/a)^{1/s})^{s-1} = (E/a)^{(s-1)/s}$. So, $dk \propto \frac{dE}{(E/a)^{(s-1)/s}} \propto E^{-(s-1)/s} dE$.

The density of states $g(E) dE$ is proportional to the phase space volume element $k^{d-1} dk$.

So, $g(E) dE \propto k^{d-1} dk \propto (E/a)^{(d-1)/s} E^{-(s-1)/s} dE \propto E^{(d-1)/s - (s-1)/s} dE \propto E^{(d-1-s+1)/s} dE \propto E^{(d-s)/s} dE$.

So, the density of states for low energy $E$ is proportional to $E^{d/s - 1}$, i.e., $g(E) \propto E^{d/s - 1}$.

Condition for Bose-Einstein Condensation

Bose-Einstein condensation occurs at a finite critical temperature $T_c$ if the total number of particles in the excited states at chemical potential $\mu=0$ and temperature $T$ is finite, allowing a macroscopic fraction of particles to occupy the ground state ($E=0$) when the total number of particles $N$ exceeds this finite value.

The total number of particles in the excited states is given by:

\begin{equation*} N_{ex} = \int_0^\infty \frac{g(E)}{e^{\beta E} - 1} dE \end{equation*}

where $\beta = 1/(k_B T)$. For BEC to occur at a finite $T_c$, the integral for $N_{ex}$ must converge at the lower limit $E \to 0$ when $\mu=0$.

Near $E=0$, $e^{\beta E} - 1 \approx \beta E$. Using the low-energy behavior of $g(E)$, the integrand behaves as:

\begin{equation*} \frac{g(E)}{e^{\beta E} - 1} \propto \frac{E^{d/s - 1}}{\beta E} \propto E^{d/s - 2} \end{equation*}

The integral $\int_0^\epsilon E^\alpha dE$ converges at the lower limit $E=0$ if $\alpha > -1$. In our case, $\alpha = d/s - 2$.

For the integral for $N_{ex}$ to converge at $E=0$, we require:

\begin{equation*} d/s - 2 > -1 \\ d/s > 1 \\ d > s \end{equation*}

If $d > s$, the number of particles in excited states $N_{ex}(\mu=0, T)$ is finite for $T > 0$. If the total particle number $N$ is greater than the maximum capacity of the excited states at $\mu=0$ (which occurs at $T_c$), the excess particles must occupy the ground state, leading to BEC.

If $d \le s$, the integral for $N_{ex}$ diverges at $E=0$. This means the excited states can accommodate an infinite number of particles even at $\mu=0$, implying that no finite fraction of particles is forced into the ground state at finite temperature. BEC at a finite temperature does not occur in this case (or $T_c=0$).

Therefore, Bose-Einstein condensation will occur for all values of $d$ and $s$ such that $d > s$.