3.3 Rudiments of thermodynamics 87over antifermions is
nf−nf ̄=gT^3
6β[
1 +
β^2
π^2]
. (3.54)
Substituting the expressions above into (3.31) for the entropy density, we obtain
sf+s ̄f=
7 π^2
90gT^3[
1 +
15 β^2
7 π^2]
. (3.55)
If the chemical potential is much larger than the temperature, the main contribution
to the total energy density comes from the degenerate fermions and is equal to
gμ^4 f/ 8 π^2 .These fermions fill the states with energies smaller than the Fermi energy
εF=μf, which determines the Fermi surface. The temperature correction to the
energy, which to leading order is of ordergT^2 μ^2 f/ 4 ,is due to the particles located
in the shell of widthT near this Fermi surface. One can see from (3.55) that the
only states which contribute to the entropy are those near the Fermi surface. As
the temperature approaches zero, the entropy vanishes. In this limit all fermions
occupy definite states and information about the system is complete. Note that the
antiparticles, for whichμf< 0 ,disappear as the temperature vanishes.
Ifβ 1 ,then
nf−nf ̄gT^3
6β (3.56)and the excess of fermions over antifermions is small compared to the number
density. In this case we can neglect the chemical potential in (3.27) and, to leading
order, the number densities of fermions and antifermions are the same, namely,
nf 3 ζ( 3 )
4 π^2
gT^3. (3.57)The energy density, pressure and entropy density of the fermions are
εf
7 π^2
240gT^4 , pf
εf
3, sf4
3
εf
T, (3.58)
respectively. If the mass is small compared with the temperature but nonzero, there
exist mass corrections, as can be inferred from the formulae derived in the previous
subsection. They are nonanalytic inα≡m/Tand ifβ≡μ/T= 0 ,cross-terms
simultaneously containing mass and chemical potential are also present.
Finally, we note the useful relation between the entropy of ultra-relativistic
fermions with a small chemical potential and the entropy of ultra-relativistic bosons
when the two types of particles have the same number of internal degrees of freedom:
sf=7
8
sb. (3.59)