1549380323-Statistical Mechanics Theory and Molecular Simulation

482 The Feynman path integral

(1/P)

∑P

k=1∂U/∂xk. It can be shown that, for the staging transformation, the force
on the modeu 1 is also the average force; however this mode is not physically the same
as the centroid.
In a seminal paper by Feynman and Kleinert (1986), it was shown that the path
centroid could be used to capture approximate quantum effects in asystem. Consider
eqn. (12.6.9) with the variablesu 1 ,...,uPrepresenting the normal modes. If we inte-
grate over the variablesu 2 ,...,uPand the corresponding momentap 1 ,...,pP, then the
result can, in the spirit of Section 8.10, be written as

QP(L,T)∝

∫

dp 1 du 1 exp

{

−β

[

p^21

2 m 1

+W(u 1 )

]}

, (12.6.22)

whereW(u 1 ) is the potential of mean force on the centroid given by

W(u 1 ) =−kTln

{∫

du 2 ···duP

×exp

[

−β

∑P

k=1

(

1

2

mkω^2 Pu^2 k+

1

P

U(xk(u))

)]}

(12.6.23)

up to an additive constant. Although we cannot determineW(u 1 ) for an arbitrary
potentialU(x), Feynman and Kleinert were able to derive an analytical expressionfor
W(u 1 ) for a harmonic oscillator using the functional integral techniquesdiscussed in
Section 12.4. In particular, they showed how to derive the parameters of a harmonic
oscillator potentialUho(x;u 1 ) that minimize the expectation value〈U(x)−Uho(x;u 1 )〉.
The parameters of the potential depend on the position of the centroid so that the
harmonic potential takes the general formUho(x;u 1 ) = (1/2)Ω^2 (u 1 )(x−u 1 )^2 +L(u 1 ).
That is, the frequency and vertical shift depend on the centroidu 1. The optimization
procedure leads to a simple potential functionW ̃(u 1 ) of the centroid that can then be
used in eqn. (12.6.22) to obtain approximate quantum equilibrium and thermodynamic
properties.
We close this section by showing how the path-integral molecular dynamics pro-
tocol extends toN Boltzmann particles inddimensions. Since most path-integral
calculations fall into this category, we will limit our discussion to these. Excellent
descriptions of path-integral algorithms for bosons and fermionscan be found in the
literature (Ceperley, 1995; Miura and Okazaki, 2000). The discreteN-particle partition
function for Boltzmann particles follows directly from eqn. (12.5.10). After introduc-
ingdNPmomentum integrations as in eqn. (12.6.2), the discrete partition function
becomes

QP(N,V,T) =

∏N

i=1

(

miP

2 πβ ̄h^2

)dP/ 2 ∫ ∏N

i=1

dr(1)i ···dr(iP)dp(1)i ···dp(iP) (12.6.24)

×exp

{

−β

∑P

k=1

[N

∑

i=1

p(k)

2

i

2 m′i

+

∑N

i=1

1

2

miω^2 P

(

r(ik+1)−r(ik)

) 2

+

1

P

U

(

r( 1 k),...,r(Nk)

)

]}