Energy fluctuations 143which measures the width of the energy distribution, i.e. the root-mean-square devia-
tion ofH(x) from its average value. The quantity under the square root can also be
expressed as
〈(H(x)−〈H(x)〉)^2 〉=〈(
H^2 (x)− 2 H(x)〈H(x)〉+〈H(x)〉^2)
〉
=〈H^2 (x)〉− 2 〈H(x)〉〈H(x)〉+〈H(x)〉^2=〈H^2 (x)〉−〈H(x)〉^2. (4.4.2)The first term in the last line of eqn. (4.4.2) is, by definition, given by
〈H^2 (x)〉=CN
∫
dxH^2 (x)e−βH(x)
CN∫
dx e−βH(x)=
1
Q(N,V,β)∂^2
∂β^2Q(N,V,β). (4.4.3)Now, consider the quantity
∂^2
∂β^2lnQ(N,V,β) =∂
∂β[
1
Q(N,V,β)∂Q(N,V,β)
∂β]
=−
1
Q^2 (N,V,β)[
∂Q(N,V,β)
∂β] 2
+
1
Q(N,V,β)∂^2 Q(N,V,β)
∂β^2. (4.4.4)
The first term in this expression is just the square of eqn. (4.3.19) or〈H(x)〉^2 , while
the second term is the average〈H^2 (x)〉. Thus, we see that
∂^2
∂β^2lnQ(N,V,β) =−〈H(x)〉^2 +〈H^2 (x)〉= (∆E)^2. (4.4.5)However, from eqn. (4.3.27),
∂^2
∂β^2lnQ(N,V,β) =kT^2 CV= (∆E)^2. (4.4.6)Thus, the variance in the energy is directly related to the heat capacity at constant
volume. If we now consider the energy fluctuationsrelativeto the total energy, ∆E/E,
we find
∆E
E
=
√
kT^2 CV
E. (4.4.7)
SinceCVis an extensive quantity,CV∼N. The same is true for the energy,E∼N, as
it is also extensive. Therefore, according to eqn. (4.4.7), the relative energy fluctuations
should behave as
∆E
E
∼
√
N
N
∼
1
√
N
. (4.4.8)
In the thermodynamic limit, whenN→ ∞, the relative energy fluctuations tend to
zero. For very large systems, the magnitude of ∆Erelative to the total average energy