- #1
csopi
- 82
- 2
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.
Thank you for your help!
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.
Thank you for your help!