# Feynman-Kac formula

1. Apr 30, 2012

### csopi

Hi,

can anyone explain to me, how the Feynman-Kac formula is used to obtain the following expression for the N-particle canonical partition function of a Bose gas (with interaction potential V)?

Z=1/N!Ʃ_{π\in S_N}Ʃ_{x1,...,xn}E[ exp(-∫_0^β Ʃ_{i<j}V(X_i(s), X_j(s) ds*indicator(X_i(β)=xπ(i), i=1...N) ]

where X_i(t) are independent continuous time simple symmetric random walks, X_i(0)=x_i.

As far as I know, the Feynman-Kac formula is a tool for solving parabolic differential equations, and I just don't know why the above is true.