1549380323-Statistical Mechanics Theory and Molecular Simulation

Green–Kubo relations 511

0 1 2 3 4 5

t (ps)

0

1

2

3

Dr

2 (

t) (Å

2 )

0 1 2 3 4 5

t (ps)

0

10

20

30

40

50

C

vv

(t

) (Å

2 /ps

2 )

(a) (b)

Fig. 13.4 (a) Mean-square displacement for a particular model of heavy water (Lee and
Tuckerman, 2007) at 300 K. (b) Total velocity autocorrelation function for the same water
model.

that is ubiquitous in the velocity autocorrelation functions of diffusing particles and is
of hydrodynamic origin.^1 In practice, convergence of the long-time tail of the velocity
autocorrelation function is slow and is influenced by finite-size effects as well. The noise
in this part of the correlation function makes the calculation of the diffusion constant
via eqn. (13.3.33) numerically difficult. Note, however, that the diffusion constant can
also be computed using the Einstein relation

D=

1

6

lim

t→∞

d

dt

1

N

∑N

i=1

〈|ri(t)−ri(0)|^2 〉, (13.3.34)

where the derivative of the averagemean-square displacementof particles is taken.
The mean-square displacement in eqn. (13.3.34) is a time correlation function, as can
be seen by writing

〈|ri(0)−ri(t)|^2 〉=〈r^2 i(0)〉+〈r^2 i(t)〉− 2 〈ri(0)·ri(t)〉. (13.3.35)

By Liouville’s theorem,〈r^2 i(0)〉=〈r^2 i(t)〉; the last term is the position autocorrelation
function. In general, the mean-square displacement becomes a linear function of time
in the long-time limit so that the diffusion constant is simply related to the slope of
this linear regime. The mean-square displacement for the same heavy water model
used in Fig. 13.4(b) is shown in Fig. 13.4(a). In this case, the model yields a diffusion
constant of 0.055 ̊A^2 /ps, which is smaller than the experimental value of 0.186 ̊A^2 /ps
at 298 K. Eqn. (13.3.34) and eqn. (13.3.33) are statistically equivalent; however, in

(^1) As an example, Alder and Wainwright showed that the decay of the velocity autocorrelation
function of a hard-sphere system decays ast−d/^2 , wheredis the number of spatial dimensions, in
a moderately dense system (Alder and Wainwright, 1967). Theasymptotic behavior of the velocity
autocorrelation function can be analyzed in detail theoretically using an approach known asmode-
coupling theory, a discussion of which, however, is beyond the scope of this book (see, for example,
Ernstet al.(1971)).