Example 9.10: Entropy of free massless scalar fields Show that the entropy S of a free scalar field in

(a) 1+1 dimensions (length L), and in

(b) 1+3 dimensions (volume V ) is given by

(a) S =

1 3

πLT, and

(b) S =

2

π2 Vζ(4)T3, (9.45)

‡‡F. Reif [59], Ex. 7.12.

§§From P.T. Landsberg [38], problem 14.9 on pp.272, 417.

¶¶R.B. Dingle [21], p.121.

134 9. Debye Theory of Specific Heat of Solids where

ζ(4) =

1 6



∞

0 x3dx ex − 1

=

π4 90

. (9.46)

[Hint: Use in

(a) Eq. (6.23) and look at Eq. (9.6). Thus, with  = c = k = 1 and with (Tds)V = dE, E =



ndn en/T − 1

=



2Ldν

c hν

ehν/T − 1

=

1 6

πLT2, ζ(2) =

π2 6

, dS =

Lπ

3 dT ].