General Correlation Functions

A general correlation function can be defined in terms of the probability distribution function $$p^{(n)} (r_1, \cdots , r_n)$$ according to

\begin{align*} g^{(n)}(r_1, \cdots , r_n) &= {1 \over p^n} p^{(n)} (r_1, \cdots , r_n) \\[4pt] &= {V^n N! \over Z_N N^n (N - n)!} \int dr_{n+1} \cdots dr_N e^{-\beta U(r_1, \cdots , r_N)} \end{align*}

Another useful way to write the correlation function is

\begin{align*} g^{(n)} (r_1, \cdots , r_n) &= {V^n N! \over Z_N N^n (N - n)!} \int dr'_1 \cdots dr'_N e^{-\beta U(r_1, \cdots , r_N)} \delta (r_1 - r'_1) \cdots \delta (r_n - r'_n) \\[4pt] &= {V^n N! \over Z_N N^n (N - n)!} \left < \Pi _{i=1}^n \delta (r_i - r'_i \right >_{r'_1, \cdots , r'_N} \end{align*}

i.e., the general n-particle correlation function can be expressed as an ensemble average of the product of $$\delta$$ -functions, with the integration being taken over the variables $$r'_1, \cdots , r'_N$$.