University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Let’s define g(
r
), density – density correlation function that gives us the
probability to find a particle in the volume element d
r
located at
r
if at
r
= 0 there is another particle.
()
∑
−
=
N
j
j
r
r
δ
)
r
ρ
(
r
r
r
Densitydensity correlation function I
Pair correlation function, g(r)
∑∑
−
=
=
N
i
N
j
ij
2
0
r
r
δ
N
V
ρ
)
r
C(
)
r
c(
r
r
r
r
At atomic level the density distribution in a system of N particles can be
described as
Then, by definition, the density – density autocorrelation function is
()(
)
i
i
i
r
r
ρ
r
ρ
)
r
C(
r
r
r
r
+
=
1
r
r
δ
)
r
ρ
(
N
j
j
i
i
=
−
=
∑
r
r
r
where
(
)
∑
∑
−
=
−
+
=
+
N
j
ij
N
j
j
i
i
r
r
δ
r
r
r
δ
)
r
r
ρ
(
r
r
r
r
r
r
r
and
()( )
∑
−
=
−
=
+
=
N
i
N
j
ij
i
N
j
ij
i
i
i
r
r
δ
N
1
r
r
δ
r
r
ρ
r
ρ
)
r
C(
r
r
r
r
r
r
r
r
Therefore
To relate the probability to find a particle at
r
to what is expected for a
uniform random distribution of particles of the same density, we can
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 Fall '11
 Zhigilei
 Statistical Mechanics, Covariance and correlation, correlation function

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