1549380323-Statistical Mechanics Theory and Molecular Simulation

Numerical evaluation 481

and eigenvectors; 3) From the eigenvectors, construct the orthogonal matrixOijthat
diagonalizes A. The forward and inverse transformations are thengiven by

uk=

1

√

P

∑P

l=1

Oklxl

xk=

√

P

∑P

l=1

OTklul. (12.6.17)

Although the eigenvalues emerge directly from the diagonalization procedure, they can
also be constructed by hand according to

λ 2 k− 1 =λ 2 k− 2 = 2

[

1 −cos

(

2 π(k−1)

P

)]

. (12.6.18)

Note the twofold degeneracy. When evaluated in terms of the normal mode variables,
the harmonic coupling term becomes

∑P

k=1

(xk−xk+1)^2 =

∑P

k=2

λku^2 k. (12.6.19)

As with the staging transformation, the harmonic term is now separable. The trans-
formation also has unit Jacobian. Thus, the transformed partitionfunction is identical
to eqn. (12.6.9) if the massesmkare defined asmk=mλk,m′ 1 =m, andm′k=mk.
With this identification, eqns. (12.6.11), (12.6.12), and (12.6.14) areapplicable to the
normal-mode case exactly as written. The only difference occurs when the chain rule
is used to obtain the forces on the normal mode variables, whence we obtain

1

P

∂U

∂u 1

=

1

P

∑P

l=1

∂U

∂xl

1

P

∂U

∂uk

=

1

√

P

∑P

l=1

∂U

∂xl

OlkT. (12.6.20)

In addition to its utility as a computational scheme, the normal-modeformulation
of the path integral has several other interesting features. First, the variableu 1 can
be shown to be equal to

u 1 =

1

P

∑P

k=1

xk P−→→∞

1

β ̄h

∫βh ̄

0

dτ x(τ). (12.6.21)

That is, the sum on the left becomes the continuous integral on theright whenP→∞.
The variableu 1 is an average over all of the path variables and, therefore, corresponds,
for finiteP, to the center-of-mass of the cyclic polymer. This point is also known as
thepath centroid. Note that the force on this mode is also just the average force