Analyze the behavior of the atoms’ total energy near the onset of condensation, in the limit of arbitrarily large N (i.e., keeping only the leading order in our 1/N^1/3 expansion and approximating the condensation as turning on discontinuously at T = T_c⁰). More specifically, do the following.

(a) Show that the total energy of the atoms in the magnetic trap is

(b) Show that the specific heat C = (∂E_total/∂T)_N is discontinuous across the critical temperature T_c⁰:

Note that for gas contained within the walls of a box, there are two specific heats: C_V = (∂E/∂T)_N,V when the box volume is held fixed, and C_P = (∂E/∂T)_N,P when the pressure exerted by the box walls is held fixed. For our trapped atoms, there are no physical walls; the quantity held fixed in place of V or P is the trapping potential V(r).

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Etotal E total = 35 (4) NKBT (3) T ()* 3Li4(el/(KBT)) (3) when T < To T NkBT NL₂Tc (7)¹ T when T> TO, (4.55a) where (at T > T) e/(KBT) is a function of N and T determined by N = (kgT/(ħw))³ Liz(eH/(KBT)), so μ = 0 at T = To (see Ex. 4.12). Because Lin (1) = (n), this energy is continuous across the critical temperature Te