Specialized to N=2, D=2
specialize to N=2
Z
=
∫
∫
exp
-
V
β
𝑑
r
1
𝑑
r
2
replace potential with N=2
Z
=
∫
∫
exp
-
β
V
r
1
r
2
𝑑
r
1
𝑑
r
2
Switch variables
Z
=
∫
∫
exp
-
β
V
r
1
r
2
𝑑
r
12
𝑑
r
cm
Specialize to a potential that depends only on interparticle distance
Z
=
∫
∫
exp
-
β
V
r
12
𝑑
r
12
𝑑
r
cm
Depend only on the magnitude of the distance
Z
=
∫
∫
exp
-
β
V
Abs
r
12
𝑑
r
12
𝑑
r
cm
Integrate out r_cm (this step is still a hack)
Z
=
Ω
∫
exp
-
β
V
Abs
r
12
𝑑
r
12
Decompose into vector components
Z
=
Ω
∫
∫
exp
-
β
V
r
12
x
2
+
r
12
y
2
𝑑
r
12
x
𝑑
r
12
y
Add integration limits
Z
=
Ω
∫
1.00000000000000e-5
L
∫
1.00000000000000e-5
L
exp
-
β
V
r
12
x
2
+
r
12
y
2
𝑑
r
12
x
𝑑
r
12
y
Parent:
partition
Children:
lj_n2