\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?



