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Chemistry LibreTexts

General distribution functions and correlation functions

We begin by considering a general N-particle system with Hamiltonian

\[ H = \sum _{i=1}^{3N} {p^2_i \over 2m} + U (r_1, \cdots, r_N)\]

For simplicity, we consider the case that all the particles are of the same type. Having established the equivalence of the ensembles in the thermodynamic limit, we are free to choose the ensemble that is the most convenient on in which to work. Thus, we choose to work in the canonical ensemble, for which the partition function is

\[ Q (N, V, T ) = {1 \over N!h^{3N}} \int d^{3N}pd^{3N} re^{-\beta \sum_{i=1}^{3N} \frac {P^2_i}{2m}} e^{-\beta U(r_1, \cdots, r_N)} \]

The 3N integrations over momentum variables can be done straighforwardly, giving

\[Q(N, V, T) = {1 \over N!\lambda^{3N}} \int dr_1 \cdots dr_N e^{-\beta U(r_1, \cdots, r_N)} \]

\[ = {Z_N \over N!\lambda^{3N}} \]

where \(\lambda = \sqrt {\frac {\beta h^2}{2\pi m}} \) is the thermal wavelength and the quantity \(Z_N\) is known as the configurational partition function

\[Z_N = \int dr_1 \cdots dr_N e^{-\beta U(r_1, \cdots, r_N)} \]

The quantity

\[ {e^{-\beta U(r_1, \cdots, r_N)} \over Z_N} dr_1 \cdots dr_N \equiv P^{(N)} (r_1, \cdots, r_N ) dr_1 \cdots dr_N \]

represents the probability that particle 1 will be found in a volume element \(dr_1\) at the point \(r_1\) , particle 2 will be found in a volume element \(dr_2\) at the point \(r_2\) ,..., particle N will be found in a volume element \(dr_N\) at the point \(r_N\). To obtain the probability associated with some number n;SPMlt;N of the particles, irrespective of the locations of the remaining n+1,...,N particles, we simply integrate this expression over the particles with indices n+1,...,N:

\[ P^{(n)} (r_1, \cdots , r_n) dr_1 \cdots dr_n = {1 \over Z_N} \left [ \int dr_{n+1} \cdots dr_N e^{-\beta U(r_1, \cdots, r_N)} \right ] dr_1 \cdots dr_n \]

 The probability that any particle will be found in the volume element \(dr_1\) at the point \(r_1\) and any particle will be found in the volume element \(dr_2\) at the point \(r_2\) ,...,any particle will be found in the volume element \(dr_n\) at the point \(r_n\) is defined to be

\[P^{(n)} (r_1, \cdots , r_n) dr_1 \cdots dr_n = {N! \over (N - n)!} P^{(n)} (r_1, \cdots , r_n) dr_1 \cdots dr_n \]

which comes about since the first particle can be chosen in N ways, the second chosen in N-1 ways, etc.

Consider the specal case of n=1. Then, by the above formula,

\[P^{(1)} (r_1) = {1 \over Z_N}{N! \over (N - 1)!} \int dr_2 \cdots dr_N e^{-\beta U(r_1, \cdots, r_N)} \]

\[ = {N \over Z_N} \int dr_2 \cdots dr_N e^{-\beta U(r_1, \cdots, r_N)} \]

Thus, if we integrate over all \(r_1\) , we find that

\[ {1 \over V } \int dr_1 p^{(1)} (r_1) = {N \over V} = p \]

Thus, \(P^{(1)} \) actually counts the number of particles likely to be found, on average, in the volume element \(dr_1\) at the point \(r_1\). Thus, integrating over the available volume, one finds, not surprisingly, all the particles in the system.