Partition function and the canonical distribution

The constant in () can be determined by normalizing the probability distribution i.e. requiring that

$$\int p(E)dE=1$$

Let us define the canonical partition function ( $\alpha =microstate)$

$$Z_c=\sum \limits_\alpha e^{-\beta E(\alpha )} =\int dE g(E) e^{-\beta E}$$

We find that the probability $p(\alpha )$ that a state is in a given microstate

$$p(\alpha )={{1}\over{Z_c}}e^{-\beta E}$$ (17)

Eq. () is the canonical distribution. If $x(\alpha )$ is the value of some physical property in microstate $\alpha ,$ and $E(\alpha )$ the energy of this state then the canonical ensemble average is given by

$$\langle x\rangle ={{1}\over{Z_c}}\sum \limits_\alpha x(\alpha )e^{-\beta E(\alpha )}$$ (18)

Equation () is a very useful formula, and we will give many examples of its use.