For a bidimensional system of N particles, the hamiltonian of pair-interaction is:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]H(\vec{q}_1,\vec{q}_2;\vec{p}_1,\vec{p}_2)=K(\vec{p}_1,\vec{p}_2)+U(\vec{q}_1,\vec{q}_2;\vec{p}_1,\vec{p}_2)[/tex]

where K is the kinetic (translational) energy and U is the potential energy

i want to solve this multiple integral:

[tex]\int\int\int\int{e^{-\frac{H(\vec{q}_1,\vec{q}_2;\vec{p}_1,\vec{p}_2)}{{k_BT}}}}d\vec{q}_1d\vec{q}_2d\vec{p}_1d\vec{p}_2[/tex]

But the pair-potential depends on positions, and momentums as well:

[tex]U=\frac{k}{\tau}e^{\tau/\tau_0}[/tex]

where τ_{0}and κ are parameters and [tex]τ=τ(\vec{q}_{12};\vec{p}_{12})[/tex]

is a function that depends on relative positions and relative momentums.

how could i solve this horrible integral? i don't need an analytical solution, a numerical solution with any software like SAGE or Mathematica is fine.

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# How can i solve this multiple Integral?

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