Averages

We have discussed three partition functions, and their connections to thermodynamic quantities. How can these partition functions be used to calculate averages of other physical quantities? The basic formula for computing an average of a quantity $F$ is simply $<F> = \sum F_i p_i$. Our main focus in this book is on systems for which classical mechanics is a good approximation (for the motions of the nuclei). Therefore we discuss averages in the classical limit. In this limit, the system energy consists of the kinetic energy plus the total potential energy for the interaction of all particles. In the partition function, the kinetic energy part can be integrated out, since it is simply a product of many gaussian integrals (Eq. 1.43). We are interested in quantities which depend on the configuration of all the atoms in the system. We denote such a configuration with the shorthand ${\bf x}^{\vec N}$. Then the average of a physical quantity $F$ which depends only on coordinates is simply

$$<F>_C = \frac{\int d{\bf x}^{\vec N} F({\bf x}^{\vec N}) e^{-U({\bf x}^{\vec N})/k_BT}} {Z_{\vec N}(T,V)}$$ (1.47)

All of the factors involving the $N_i!$ and kinetic energy terms in Eq. 1.43 cancel out since they occur in the numerator and denominator.

In the grand canonical case, the averaging process for the quantity $F({\bf x}^{\vec N})$ is slightly more complicated, and can be expressed as

$$<F>_{GC} = \frac{1}{\Xi} \sum_{\vec N} \frac{{\vec z}^{\vec N}} {\prod N_i!}... ...}^{\vec N} F({\bf x}^{\vec N}) e^{-U({\bf x}^{\vec N})/k_BT}} {Z_{\vec N}(T,V)}$$ (1.48)

$$= \frac{1}{\Xi} \sum_{\vec N} \frac{{\vec z}^{\vec N}} {\prod N_i!} Z_{\vec N}(T,V) <F>_C({\vec N})$$ (1.49)

Thus the grand canonical average is a weighted sum of canonical averages.