\sect{Condesation for general dispersion}

An ideal Bose gas whose single particle spectrum is given by ${\epsilon =C\cdot|p|^{s}, s>0}$ is contained in volume $V$ of dimention d. The gas is in uniform temperature $T$.

\Dn

(1)  Calculate the one-particle density of states.\\

(2) Find the condition on $s$ and $d$ for the existence of Bose-Einstein condensation.  In particular relate to relativistic (s=1)
and nonrelativistic (s=2) particles in two dimensions (d=2).\\

(3) Find the dependence of the number of particles $N$ on the chemical potential $\mu$.\\

(4) Find the dependence of the total energy $E$ on the chemical potential, and show how the pressure $P$ is obtained from this result.\\

(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..\\  

(6) Repeat item (1) for relativistic gas whose particles have finite mass such that their dispersion relation is $\epsilon =\sqrt{m^2c^4+c^2p^2}$.\\

(7) Consider a relativistic gas in $2D$. Find expressions for $N$ and $E$ and $P$. Should one expect Bose-Einstein condensation?




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OLD: 

\sect{Baruch's  B03.}

Consider an ideal Bose gas in d dimensions whose single particle
spectrum is given by ${\epsilon =\alpha |{\bf p}|^{s}, s>0}$.

\begin{itemize}
\item [(a)]
Find the condition on ${s, d}$ for the existence of Bose-Einstein
condensation. In particular show that for nonrelativistic particles
in two dimensions ${\left(s=d=2\right)}$ the system does not exhibit
Bose-Einstein condensation.

\item [(b)]
Show that
\[\ P= \frac{s}{d}\frac{E}{V}\,\,\,\,\,\,
\mbox{and}\,\,\,\,\,\,
C_{V}(T\rightarrow \infty) = \frac{d}{s}
Nk_{B}\]\\
\end{itemize}

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