1549380323-Statistical Mechanics Theory and Molecular Simulation

510 Classical time-dependent statistical mechanics

D=

kT

f

Jx=

kT

f

lim

t→∞

〈ux〉t. (13.3.26)

The average〈ux〉t can now be evaluated using linear response theory. From eqns.
(13.3.17), we see thatFe(t) = 1,Di(r,p) =fˆex,Ci(r,p) = 0, so that the dissipative
flux becomes

j(r,p) =−f

∑N

i=1

pxi

mi

=−f

∑N

i=1

x ̇i=−Nfux. (13.3.27)

Therefore, substituting eqn. (13.3.27) into eqn. (13.2.27), we obtain

〈ux〉t=〈ux〉+βNf

∫t

0

ds〈ux(0)ux(t−s)〉. (13.3.28)

Once again, lettingτ=t−sgives

〈ux〉t=〈ux〉+βNf

∫t

0

dτ〈ux(0)ux(τ)〉. (13.3.29)

The average ofuxover an equilibrium canonical ensemble vanishes, since the average
of any component of the velocity is zero. Letting the upper limit of the time integral
go to infinity, and changing the integration variable fromτtot, the diffusion constant
expression becomes

D=N

∫∞

0

dt〈ux(0)ux(t)〉. (13.3.30)

Substituting eqn. (13.3.22) into eqn. (13.3.30) gives

D=

∫∞

0

dt

1

N

〈(N

∑

i=1

x ̇i(0)

)(N

∑

i=1

x ̇i(t)

)〉

. (13.3.31)

Recall that in equilibrium, the velocity (momentum) distribution is a product of inde-
pendent Gaussian distributions. Hence,〈x ̇ix ̇j〉is 0, and moreover, all cross correlations
〈x ̇i(0) ̇xj(t)〉vanish wheni 6 =j. Thus, the Green–Kubo relation for the diffusion con-
stant becomes

D=

∫∞

0

dt

1

N

∑N

i=1

〈x ̇i(0) ̇xi(t)〉. (13.3.32)

The correlation function in eqn. (13.3.32) is known as thevelocity autocorrelation
function. Since we could have chosen any direction for the external force,we can
computeDby averaging over the three spatial directions and obtain

D=

1

3

∫∞

0

dt

1

N

∑N

i=1

〈r ̇i(0)·r ̇i(t)〉. (13.3.33)

An example of a velocity autocorrelation function for a particular model of heavy
water (D 2 O) at 300 K (Lee and Tuckerman, 2007) is shown in Fig. 13.4(b). We note
that the velocity autocorrelation function exhibits a long-time algebraic decay in time