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C. Tang (唐晨宇) and Y. Wang (王延颋)
5.7.3.2 Structure Factor
The RDF cannot be directly measured by experiment. However, we can introduce the Fourier transform of it in k-space where
k =
2π
L
(k x , k y , k z ), which is the
experimentally measurable structure factor, that is
S(k) = 1 + 4πρ
∞
0
r
2 sin kr
kr
g(r)dr
(5.7.11)
5.7.4 Diffusion Coefficient
Apart from the structural properties, to understand the system dynamics, it is also
important to investigate the diffusion coefficients of particles. The mean square
displacement (MSD) marks the square of the average distance moved in time duration
t. The MSD is a time autocorrelation function, based on which the diffusion coefficient
D can be calculated from the Einstein Relation:
r
2
(t)
≡
r
2
(t)
=
1
N
N
i=1
r
2
i (t) = 2dDt
(5.7.12)
where the parameter d labels the dimension of the system. Equation (7.12) is strictly
correct only when time t → ∞.
The diffusion coefficient can also be represented by time autocorrelation function
of velocity. The MSD can be rewritten as the integral of velocity autocorrelation
function:
r
2
(t)
=
⎛
⎝
t
0
dt
v(t
)
⎞
⎠
2
=
t
0
t
0
dt
dt
v(t
) v(t
)
= 2
t
0
t
0
dt
dt
v(t
) v(t
)
(5.7.13)
Also, we have:
v(t
) · ·
v(t
)
=
v(t
− t
) · ·
v(0)
(5.7.14)