Thermodynamics 459〈Aˆ〉=
1
Q(L,T)
lim
P→∞(
mP
2 πβ ̄h^2)P/ 2 ∫
dx 1 ···dxP[
1
P
∑P
k=1a(xk)]
×exp{
−
1
̄h∑P
k=1[
mP
2 β ̄h
(xk+1−xk)^2 +β ̄h
P
U(xk)]}∣∣
∣
∣
∣
xP+1=x 1, (12.3.8)
which treats thePcoordinatesx 1 ,...,xPon an equal footing.
Eqn. (12.3.8) can be put into a compact form as follows: First, we define a proba-
bility distribution functionf(x 1 ,...,xP) by
f(x 1 ,...,xP) =1
QP(L,T)
(
mP
2 πβ ̄h^2)P/ 2
×exp{
−
1
̄h∑P
k=1[
mP
2 β ̄h(xk+1−xk)^2 +β ̄h
PU(xk)]}∣∣
∣
∣
∣
xP+1=x 1, (12.3.9)
whereQP(L,T) is the partition function for finiteP, which is obtained by removing
the limit asP→∞from eqn. (12.2.26):
QP(L,T) =
(
mP
2 πβ ̄h^2)P/ 2 ∫
dx 1 ···dxP×exp{
−
1
̄h∑P
k=1[
mP
2 β ̄h(xk+1−xk)^2 +β ̄h
PU(xk)]}∣∣
∣
∣
∣
xP+1=x 1. (12.3.10)
Clearly,Q(L,T) = limP→∞QP(L,T). The functionf(x 1 ,...,xP) satisfies the condi-
tions of a probability distribution:f(x 1 ,...,xP)≥0 for allx 1 ,...,xPand
∫
dx 1 ···dxPf(x 1 ,...,xP) = 1. (12.3.11)
In Section 7.2, we introduced the concept of anestimatorfor a multi-dimensional inte-
gral. In path integral calculations, equilibrium expectation values can be approximated
using estimator functions that depend on thePcoordinatesx 1 ,...,xP. Thus, for eqn.
(12.3.8), an appropriate estimator for〈Aˆ〉is the functionaP(x 1 ,...,xP) defined to be
aP(x 1 ,...,xP) =1
P
∑P
k=1a(xk). (12.3.12)The expectation value〈Aˆ〉can be approximated for finitePas an average of the estima-
tor in eqn. (12.3.12) with respect to the probability distribution functionf(x 1 ,...,xP).
We write this approximation as
〈Aˆ〉P= lim
P→∞
〈aP(x 1 ,...,xP)〉f, (12.3.13)where〈···〉findicates an average over the probability distribution functionf(x 1 ,...,xP).
It follows that〈Aˆ〉= limP→∞〈Aˆ〉P.