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A system of N bosons in two dimensions has an energy-momentum relation
$\epsilon_p = c p^f$
and density n = N/A (A = area). c and f are constants.
(a) Show that at low temperatures the system will Bose condense, for some values of f. Find these values of f. (In what follows, assume that f indeed takes these values.) Show that the Bose condensation temperature $T_c \sim n^{\alpha}$. Find $\alpha$ in terms of f.
(b) Show that the entropy below Tc goes as $S \sim T^{\beta}$, and the "pressure" below Tc (i.e. its equivalent in two dimensions) goes as $P \sim T^{\gamma}$. Find $\beta$ and $\gamma$ (again in terms of f).
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Answer
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