I
Then, absolute internal energy
A
can be calculated for any
λ
with
A
(
λ
) 
A
(
λ
0
) =
Z
λ
λ
0
∂
U
∂λ
d
λ
.
(40)
I
The real potential function, for which we want to calculate
A
is
U
0
.
I
So, constructed
U
interpolates between
U
0
and a harmonic
lattice
U
(
r
,
λ
) =
U
0
(
r
) +
λ
∑
i
(
r
i

r
0
)
2
⇒
A
(
λ
=
0
) =
A
(
λ
) 
R
λ
0
∂
U
∂λ
d
λ
0
.
(41)
I
Because at large
λ
we now have harmonic lattice, for
which we know the solution, we can integrate the real
solution
A
(
λ
=
0
)
from
h
∂
U
/∂λ
i
.
Notes
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R
F
I
Response functions describe how the system reacts to a
change in some thermodynamic variables.
I
Some of the most important response functions are:
Constant volume heat capacity:
C
V
=
(
∂
E
∂
T
)
V
Constant pressure heat capacity:
C
P
=
(
∂
H
∂
T
)
P
Thermal expansion coefficient:
α
P
=
V

1
(
∂
V
∂
T
)
P
Isothermal compressivity:
β
T
= 
V

1
(
∂
V
∂
P
)
T
Bulk modulus:
B
=
1
/β
T
Thermal pressure coefficient:
γ
V
=
(
∂
P
∂
T
)
V
I
Many of them can be directly calculated with an MD
simulation.
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 Winter '12
 Kotakoski
 Thermodynamics, Energy, Statistical Mechanics, Canonical Ensemble

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