Problems
191
Problems
16.1 Verify equation (16.9).
16.2 Verify equation (16.10).
16.3 Show that the momentum space version of (16.18) is (16.19).
16.4 Use (16.31) in (16.33) to verify (16.34).
16.5 Verify (16.68) and (16.70).
16.6 In a modified one-dimensional Ising model, spin variables s n at sites
labelled by n = 1, 2, 3, . . . N take the values s n = ±1, and the energy of each
spin configuration is
N ∑ −1
E = −
J n s n s n+1 ,
n=1
where all the constants J n are positive. Show that the partition function Z N
is given by
N ∏ −1
Z N = 2
(2 cosh K n ) ,
n=1
where K n = J n /k B T . Hence calculate the entropy for the particular case in
which all the J n ’s are equal to J and N ≫ 1, and discuss the behaviour of
the entropy in the limits T → ∞ and T → 0.
Let ‘p’ denote a particular site such that 1 ≪ p ≪ N . Show that the
average value ‹s p s p+1 › of the product s p s p+1 is given by
1 ∂Z
‹
N
s p s p+1 › =
.
Z N ∂K p
Show further that
1
∂
j Z
‹
N
s p s p+j › =
.
Z N ∂K p ∂K p+1 . . . ∂K p+j−1
Hence show that in the case J 1 = J 2 = . . . = J N = J,
‹s p s
ξ
+j › = e
−ja/
p
,
where
ξ = −a/[ln(tanh K)] ,
and K = J/k B T . Discuss the physical meaning of ξ, considering the T → ∞
and T → 0 limits explicitly.