1549380323-Statistical Mechanics Theory and Molecular Simulation

Approximations 561

novel Monte-Carlo schemes to be devised for computing this function (Jadhao and
Makri, 2008).
The second alternate time correlation function is the Kubo-transformed correlation
function (Kuboet al., 1985) defined by

KAB(t) =

1

βQ(N,V,T)

∫β

0

dλTr

[

e−(β−λ)

Hˆˆ

Ae−λ

Hˆ

ei

Hˆt/ ̄hˆ

Be−i

Hˆt/ ̄h]

. (14.6.11)

LikeGAB(t),KAB(t) is also purely real. In addition,KAB(t) reduces to its classical
counterpart both in the classical (β→0) and harmonic limits. Consequently,KAB(t)
can be more readily compared to corresponding classical and harmonic time correlation
functions, which can be computed straightforwardly. As withGAB(t), there is a simple
relationship between the Fourier transforms ofKAB(t) andCAB(t):

C ̃AB(ω) =

[

β ̄hω

1 −e−βhω ̄

]

K ̃AB(ω). (14.6.12)

Finally, we note that purely imaginary time correlation functions of the form

GAB(τ) =

1

Q(N,V,T)

Tr

[

e−β

Hˆ ˆ

Ae−τ

Hˆˆ

Beτ

Hˆ]

(14.6.13)

can be computed straightforwardly using the numerical techniques for imaginary-time
path integrals. Eqn. (14.6.13) results from eqn. (14.6.1) when the Wick rotation from
real to imaginary time is applied (see Section 12.2 and Fig. 12.5). An example of such
a correlation function is the imaginary-time mean square displacement given by

R^2 (τ) =〈|xˆ(τ)−ˆx(0)|^2 〉, (14.6.14)

which forNparticles in three dimensions becomes

R^2 (τ) =

1

N

∑N

i=1

〈

[ˆri(τ)−ˆri(0)]^2

〉

. (14.6.15)

Here,τ∈[0,β ̄h/2] (R^2 (τ) is symmetric aboutτ=β ̄h/2) and for a free particle, its
shape is anR^2 (τ)∝τ(β ̄h/ 2 −τ), where the constant of proportionality depends on
the number of dimensions (see Problem 14.11). This important quantity is related
to the real-time velocity autocorrelation functionCvv(t) (more precisely, its Fourier
transformC ̃vv(ω)) via a two-sided Laplace transform

R^2 (τ) =

1

π

∫∞

−∞

dω

e−β ̄hω/^2

ω^2

C ̃vv(ω)

×

{

cosh

[

ω

(

̄hβ

2

−τ

)]

−cosh

(

β ̄hω

2

)}

. (14.6.16)

Eqn. (14.6.16) suggests that performing the Wick rotation from real to imaginary
time is a well-posed problem that requires a Fourier transform followed by a Laplace