The energy density ϵ follows from the relation ϵ = T s + µn + F : It is interesting to note that the above result, which in terms of the pressure is simply ϵ = pd/ 2, is a consequence of the mechanics of free gases. This shows that this relation continues to hold for interacting gases at a quantum critical point, as pointed out by Ho[5]. Also of interest is the energy per particle ϵ$_{1}$ = ϵ/n : Consider first the limit T /µ → 0, with µ positive. In this limit, if b is a smooth function of T /µ as T /µ → 0 $^{+}$, then the b ′ terms in the density vanish and one simply obtains which is the same result as for a free gas, where for fermions µ is equal to the Fermi energy ϵ$_{F}$ . The above formula is usually not appropriate to the T → 0 limit when µ is negative. For the interacting gas µ = ϵ$_{F}$ , so this leads us to define the scaling functions ξ : As T /µ → 0 $^{+}$, the functions ξ$_{d}$ should become universal constants, and for free fermions ξ$_{d}$ (0 $^{+}$) = 1. The Fermi energy ϵ$_{F}$ can be defined based on its relation to density in the zero temperature free fermion gas. For bosons, one can formally use the same definition. For d = 2, ϵ$_{F}$ = 2 πn/m , whereas for d = 3, ϵ$_{F}$ = (3 π 2 $_{n/}$√$_{2)}$ 2 / $^{3}$/m . This leads to the definitions: ϵ = − d 2 F (30) ϵ$_{1}$ = − d $_{2}$F n (31) lim T/µ → 0 $_{+}$ϵ$_{1}$ = d d + 2 µ (32) ξ$_{d}$ ( T /µ ) = d + 2 d ϵ$_{1}$ ϵ$_{F}$ (33) ξ$_{2}$ = ( mµ 2 πn ) 2 b$_{2}$ ( T /µ ) (34) ξ$_{3}$ = √$_{2}$ 3 π $^{2}$n 5 / 3 ( mµ ) 5 / 2 b$_{3}$ ( T /µ ) 14