462 The Feynman path integral
= lim
P→∞〈PP(x 1 ,...,xP)〉f, (12.3.23)where
PP(x 1 ,...,xP) =P
βL−
1
L
∑P
k=1[
mP
β^2 ̄h^2(xk+1−xk)^2 +1
P
xk∂U
∂xk]
. (12.3.24)
Thus,PP(x 1 ,...,xP) is an estimator for the pressure (Martynaet al., 1999) so that
we can calculate the average pressure fromP= limP→∞〈PP(x 1 ,...,xP)〉f. As we dis-
cussed in Section 4.6.3, if the potentialUhas an explicit length (volume) dependence,
then the estimator becomes
PP(x 1 ,...,xP) =P
βL−
1
L
∑P
k=1[
mP
β^2 ̄h^2(xk+1−xk)^2 +1
P
xk∂U
∂xk−
L
P
∂U(xk,L)
∂L]
. (12.3.25)
The basic thermodynamic relations of the canonical ensemble can beused to derive
estimators for other thermodynamic quantities such as the constant-volume heat ca-
pacity (Glaesemann and Fried, 2002) (see Problem 12.2). We will explore the utility of
expressions like eqns. (12.3.20) and (12.3.24) in practical calculations in Section 12.6.1.
12.4 The continuous limit: Functional integrals
Before we discuss the numerical implementation of path integrals, let us examine the
physical content of the path integral in greater detail by formallyanalyzing theP→∞
limit. This limit gives rise to a mathematical construct known as afunctional integral.
Because the physical picture associated with the functional integral is clearer for real-
time quantum mechanics, we will begin the discussion by analyzing the propagator of
eqn. (12.2.23) and then perform the Wick rotation to imaginary time to obtain the
canonical density matrix and partition function. For this analysis, itis convenient to
introduce a parameterǫ=t/P, so thatP→ ∞impliesǫ→0. In terms ofǫ, eqn.
(12.2.23) can be written as
U(x,x′;t) = lim
P→∞
ǫ→ 0( m
2 πiǫ ̄h)P/ 2 ∫
dx 2 ···dxP×exp{
iǫ
̄h∑P
k=1[
m
2(
xk+1−xk
ǫ) 2
−
1
2
(U(xk+1) +U(xk))]}∣∣
∣
∣
∣
xP+1=x′x 1 =x. (12.4.1)
In the limitP→∞andǫ→0, the time interval between the pointsx 1 ,x 2 ,...,xP,xP+1
become infinitely small, while the number of points becomes infinite. Thus, in this