Fall 2006 17 Sm

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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