1549380323-Statistical Mechanics Theory and Molecular Simulation

Path integral derivation 455

x

Imaginary time

0 ℏ/2 ℏ

x

x’

x

Real time

0 t/2 t

x

x’

β β

Fig. 12.6 Representative paths in the path sums of eqns. (12.2.22) and(12.2.23).

x 1 =x, we may rename the integration variable in eqn. (12.2.24)x 1 and perform a
P-dimensional integration

Q(L,T) = lim

P→∞

∫

dx 1 ···dxP

×exp

{

−

1

̄h

∑P

k=1

[

mP

2 β ̄h

(xk+1−xk)^2 +

β ̄h

2 P

(U(xk+1) +U(xk))

]}∣∣

∣

∣

∣

xP+1=x 1

,(12.2.25)

which is subject to the conditionxP+1=x 1. This condition restricts the integration to
paths that begin and end at the same point. All of the coordinate integrations in eqn.
(12.2.25), must be restricted to the spatial domainx∈[0,L], which we will denote
asD(L). Finally, note that

∑P

k=1(1/2)[U(xk) +U(xk+1)] = (1/2)[U(x^1 ) +U(x^2 ) +
U(x 2 ) +U(x 3 ) +···+U(xP− 1 ) +U(xP) +U(xP) +U(x 1 )] =

∑P

k=1U(xk), where the
conditionx 1 =xP+1has been used. Thus, eqn. (12.2.25) simplifies to

Q(L,T) = lim

P→∞

(

mP

2 πβ ̄h^2

)P/ 2 ∫

D(L)

dx 1 ···dxP

×exp

{

−

1

̄h

∑P

k=1

[

mP

2 β ̄h

(xk+1−xk)^2 +

β ̄h

P

U(xk)

]}∣∣

∣

∣

∣

xP+1=x 1

. (12.2.26)

The integration over cyclic paths implied by eqn. (12.2.26) is illustratedin Fig. 12.7.
Interestingly, as the temperatureT→ ∞andβ→0, the harmonic spring constant
connecting neighboring points along the paths becomes infinite, which causes the cyclic
paths in the partition function to collapse onto a single point corresponding to a classi-
cal point particle. Thus, the path-integral formalism shows that the high-temperature
limit is equivalent to the classical limit. Finally, note that the partition function can