\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? \ \\ \ \\ 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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%