80 The hot universe
particle having energy within the interval fromto+. Let us first consider a
particle with no internal degrees of freedom in one dimension. At any moment of
time its state can be specified completely by the coordinatexand the momentum
p.In classical mechanics two infinitesimally different coordinates or momenta
correspond to microscopically different states. Therefore, the number of microstates
is infinite and the entropy can be defined only up to an infinite additive factor.
However, in quantum mechanics, two states within a cell of volume 2πin phase
space are not distinguishable because of the uncertainty relation. Hence, there is
only one possible microstate per corresponding phase volume. The generalization
for the case of a particle withginternal degrees of freedom in three-dimensional
space is straightforward:
g=g∫+
d^3 xd^3 p
( 2 π)^3=
gV
( 2 π)^3∫+
d^3 p, (3.25)where we have assumed homogeneity and integrated over the volumeV. Hence-
forth, we use natural units wherec==kB=G= 1 .The energydepends on
the momentum|p|and in the isotropic case we have
g=
gV
2 π^2∫+
|p|^2 d|p|
gV
2 π^2√
(^2 −m^2 ), (3.26)where the relativistic relation,
^2 =|p|^2 +m^2 ,has been used. Note that the state with the minimal possible energy,=m,drops
out when the approximate expression in (3.26) is used. This state becomes very
important when the chemical potential of the bosons approaches the mass of the
particles. In this case any new particles we add to the system occupy the minimal
energy state and form a Bose condensate.
Taking the limit→0 and considering a unit volume(V= 1 ),we obtain the
following expression for theparticle number density:
n=∑
ng=
g
2 π^2∫∞
m√
(^2 −m^2 )
exp((−μ)/T)∓ 1d, (3.27)