172 Canonical ensemble
The expressions forω 1 ,ω 2 , andω 3 are known as the first, second, and thirdcumulants
ofU 1 (r 1 ,...,rN), respectively. The expansion in eqn. (4.7.12) is, therefore, known as a
cumulant expansion, generally given by
A(1)=
∑∞
k=1(−β)k−^1
k!〈U 1 k〉c, 0 , (4.7.19)where〈U 1 k〉c, 0 denotes thekth cumulant ofU 1 with respect to the unperturbed system.
In general, suppose a random variableyhas a probability distribution function
P(y). The cumulants ofycan all be obtained by the use of acumulant generating
function. Letλbe an arbitrary parameter. Then, the cumulant generating function
R(λ) is defined to be
R(λ) = ln
〈
eλy〉
. (4.7.20)
Thenth cumulant ofy, denoted〈y〉cis then obtained from
〈yn〉c=dn
dλnR(λ)∣
∣
∣
∣
λ=0. (4.7.21)
Eqn. (4.7.21) can be generalized toNrandom variablesy 1 ,...,yNwith a probability
distribution functionP(y 1 ,...,yN). The cumulant generating function now depends on
Nparametersλ 1 ,...,λNand is defined to be
R(λ 1 ,...,λN) = ln〈
exp(N
∑
i=1λiyi)〉
. (4.7.22)
The definition of the cumulants is now generalized as
〈y 1 ν^1 yν 22 ···yNνN〉c=[
∂ν^1
∂λν 11∂ν^2
∂λν 22···
∂νN
∂λνNN]
R(λ 1 ,...,λN)∣
∣
∣
∣
λ 1 =···=λN=0. (4.7.23)
More detailed discussions about cumulants and their application in quantum chem-
istry and quantum dynamics are provided by Kladko and Fulde (1998)and Causoet
al.(2006), respectively, for the interested reader.
Substituting eqns. (4.7.16), (4.7.17), and (4.7.18) into eqn. (4.7.19)and addingA(0)
gives the free energy up to third order inU 1 :
A=A(0)+ω 1 −β
2ω 2 +β^2
6ω 3 ···=−
1
βln(
Z(0)(N,V,T)
N!λ^3 N)
+〈U 1 〉 0
−
β
2(
〈U 12 〉 0 −〈U 1 〉^20
)
+
β^2
6(
〈U 13 〉 0 − 3 〈U 1 〉 0 〈U 12 〉 0 + 2〈U 1 〉^30
)
+···. (4.7.24)
It is evident that each term in eqn. (4.7.24) involves increasingly higher powers ofU 1
and its averages.