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The energy density ϵ follows from the relation ϵ = T s + µn + F : 

<img src='content_image/352221.jpg'>

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 : 

<img src='content_image/352222.jpg'>

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 

<img src='content_image/352223.jpg'>

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 : 

<img src='content_image/352224.jpg'>

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: 

<img src='content_image/352225.jpg'>

