\sect{Condesation for general dispersion}
An ideal Bose gas consists of particles that ahve the
dispersion relation ${\epsilon =c |p|^s}$ with $s>0$.
The gas is contained in a box that has volume $V$ in $d$ dimentions.
The gas is maintained in a uniform temperature~$T$.
\Dn
(1) Calculate the single particle density of states.
\Dn
(2) Find a condition involving $s$ and $d$ for the existence of Bose-Einstein condensation.
In particular relate to relativistic $(s=1)$ and nonrelativistic $(s=2)$ particles in two dimensions.
\Dn
(3) Find the dependence of the number of particles $N$ on the chemical potential $\mu$.
\Dn
(4) Find the dependence of the total energy $E$ on the chemical potential,
and show how the pressure $P$ is obtained from this result.
\Dn
(5) Find an expression for the heat capacity $C_v$.
Show how this result can be expressed using $N$ in the limit of infinite temperature.
\Dn
(6) Repeat item1 for relativistic gas whose particles have finite mass
such that their dispersion relation is $\epsilon =\sqrt{m^2c^4+c^2p^2}$.
\Dn
(7) Consider a relativistic gas in $2D$.
Find expressions for $N$ and $E$ and $P$.
Should one expect Bose-Einstein condensation?