Feynman-Kac Formula & N-particle Canonical Partition Function

[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.

Thank you for your help!

 

[azntoon]

The Feynman-Kac formula is an integral representation of solutions of certain kinds of partial differential equations, in particular the heat equation and the Schrödinger equation. It states that the solution of such an equation can be expressed as the expected value of a stochastic process, which in this case is a continuous time random walk. In the above expression, the Feynman-Kac formula is used to calculate the partition function of a Bose gas, which is a measure of the probability of finding a certain number of particles in a given state. The expected value of the stochastic process is then multiplied by an indicator function (which is equal to one if the condition in the indicator is met, and zero otherwise), which ensures that the particles are found in the correct state at the end of the random walk. This is then integrated over the duration of the random walk (from 0 to β) to get the partition function.