Specialized to N=2, D=2

Z = ∫ exp ⁡ - V ⁢ β 𝑑 R

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