Statistical Physics, Second Revised and Enlarged Edition

90 Fermi–Diracgases

Pressure P. The pressureisalso readily deducedfromU.Infactit remains true that
for a gas of massive particlesP=^23 U/Vindependent of statistics. HencePVandU
are proportional.Thisrelationfollowsfrombasic statisticalideas

P=−(∂U/∂V)S basic thermodynamics

=−(∂U/∂V){ni}fixed Sdependson{ni}only

=−

∑

ni(∂ε/∂V) compare section 2.3.2 – onlythe energylevels

depend onV

=

2

3

∑

niεiV see below

=

2

3

U/V as stated!

Thevolumedependence ofthe energy levelsdependsonthedispersion relation alone.
For our gas of particles of massM,we know thatε(k)∝k^2 .Butk,a dimension in
k-space,is proportionalto the reciprocalofthebox sizea,i.e.k∝V−^1 /^3 .Hence
ε∝V−^2 /^3 .Eachenergy leveldependsonVwiththis samepowerlaw, so that we
can replace∂ε/∂Vby(−^23 )ε/V.
ThereforePfollows the energydensityU/V.The Fermigas thereforeis seen to
exert a verylarge zero-point pressureP( 0 )=^23 U( 0 )/V. This plays an important role
in considerations ofstabilityinwhite-dwarfstars, where the pressure (oforder 10^17
atm) inhibits further collapse under the influence of gravitational forces. In the case
ofmetals, electrostaticbindingforces arelarge andtheelectrons are containedinthe
metalbythe so-calledworkfunction ofthe metal, an energywhichmustbegreater
thanμfor the electrons not to leak out.

Entropy S. The temperature dependence ofSis readily obtained fromCV.We obtain
alineardependence onT(just asforC)inthedegenerate region,Stendingto zero as
it should at the absolute zero. Thispasses towards the lnTvariation in the classical

T T

TTTF TTTF

CV

3 (a) (b)

UU= 2 NkkkBT

PP=NkkkBT

3

2 NkkkB

UUU, P

UUU(0),P( 0 )

Fig. 8. 3 The variation with temperature of the thermodynamic functions for an ideal FDgas. Note the low-
temperature (quantum) and high-temperature (classical) limits. (a) Internal energyand pressure. (b) Heat
capacity.