# How can i solve this multiple Integral?

1. Feb 15, 2016

### Korbid

For a bidimensional system of N particles, the hamiltonian of pair-interaction is:
$$H(\vec{q}_1,\vec{q}_2;\vec{p}_1,\vec{p}_2)=K(\vec{p}_1,\vec{p}_2)+U(\vec{q}_1,\vec{q}_2;\vec{p}_1,\vec{p}_2)$$
where K is the kinetic (translational) energy and U is the potential energy
i want to solve this multiple integral:
$$\int\int\int\int{e^{-\frac{H(\vec{q}_1,\vec{q}_2;\vec{p}_1,\vec{p}_2)}{{k_BT}}}}d\vec{q}_1d\vec{q}_2d\vec{p}_1d\vec{p}_2$$
But the pair-potential depends on positions, and momentums as well:
$$U=\frac{k}{\tau}e^{\tau/\tau_0}$$
where τ0 and κ are parameters and $$τ=τ(\vec{q}_{12};\vec{p}_{12})$$
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.

2. Feb 16, 2016