University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Monte Carlo evaluation of statisticalmechanics integrals
Why are we interested in integration?
N
N
N
N
N
N
N
N
p
d
r
d
kT
p
r
E
p
r
A
Z
p
r
A
r
r
r
r
r
r
r
r
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
∫
)
,
(
exp
)
,
(
1
)
,
(
Ensemble average of quantity A(
r
,
p
) can be calculated for a given distribution function.
For NVT ensemble we have distribution function
ρ
(
r
,
p
),
N
N
N
N
N
N
p
d
r
d
kT
)
p
,
r
(
E
exp
Z
1
)
p
,
r
(
r
r
r
r
r
r
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
ρ
Energy can always be
expressed as a sum of kinetic and potential contributions. The
contribution of the kinetic part is trivial and we can consider integral in only
configurational 3N dimensional space, where Z is configurational integral.
N
N
N
N
r
d
kT
r
U
r
A
Z
r
A
r
r
r
r
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
∫
)
(
exp
)
(
1
)
(
∫
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
N
N
r
d
kT
)
r
(
U
exp
Z
r
r
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View Full DocumentUniversity of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Monte Carlo evaluation of statisticalmechanics integrals
Statisticalmechanics integrals typically have significant contributions only from very small
fractions of the 3N space.
For example for hard spheres contributions are coming from the areas
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 Fall '11
 Zhigilei
 Statistical Mechanics, Monte Carlo method, University of Virginia, Leonid Zhigilei

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