Calculation of Mean Values

Consider a system in contact with a heat reservoir, or with a specified mean energy. The systems in the representative ensemble are distributed over their accessible states in accordance with the canonical distribution. Thus, the probability of occurrence of some state $r$ with energy $E_r$ is given by

$$P_r = \frac{\exp(-\beta E_r)}{\sum_r \exp(-\beta E_r)}.$$ (7.31)

The mean energy is written

$$\overline{E} = \frac{\sum_r \exp(-\beta E_r) E_r}{\sum_r \exp(-\beta E_r)},$$ (7.32)

where the sum is taken over all states of the system, irrespective of their energy. Note that

$$\sum_r \exp(-\beta E_r) E_r = -\sum_r \frac{\partial}{\partial \beta} \exp(-\beta E_r) = -\frac{\partial Z}{\partial \beta},$$ (7.33)

where

$$Z = \sum_r \exp(-\beta E_r).$$ (7.34)

It follows that

$$\overline{E} = - \frac{1}{Z} \frac{\partial Z}{\partial \beta} = - \frac{ \partial \ln Z}{\partial \beta}.$$ (7.35)

The quantity $Z$ , which is defined as the sum of the Boltzmann factor over all states, irrespective of their energy, is called the partition function. (Incidentally, the partition function is represented by the symbol $Z$ because in German it is known as the ``zustandssumme,'' which means the ``sum over states.'') We have just demonstrated that it is fairly easy to work out the mean energy of a system using its partition function. In fact, as we shall discover, it is straightforward to calculate virtually any piece of statistical information from the partition function.

Let us evaluate the variance of the energy. We know that

$$\overline{(\Delta E)^{ 2}} = \overline{E^{ 2}} - \overline{E}^{ 2}.$$ (7.36)

(See Chapter 2.) Now, according to the canonical distribution,

$$\overline{E^{ 2}} = \frac{\sum_r \exp(-\beta E_r) E_r^{ 2}}{\sum_r \exp(-\beta E_r)}.$$ (7.37)

However,

$$\sum_r \exp(-\beta E_r) E_r^{ 2} = -\frac{\partial} {\partial... ...rac{\partial}{\partial \beta}\right)^2 \left[ \sum_r \exp(-\beta E_r)\right].$$ (7.38)

Hence,

$$\overline{ E^{ 2}} = \frac{1}{Z} \frac{\partial^{ 2} Z}{\partial \beta^{ 2}}.$$ (7.39)

We can also write

$$\overline{ E^{ 2}} = \frac{\partial}{\partial \beta}\!\left( \fr... ...\right)^2 = -\frac{\partial \overline{E}}{\partial \beta} + \overline{E}^{ 2},$$ (7.40)

where use has been made of Equation (7.35). It follows from Equation (7.36) that

$$\overline{(\Delta E)^{ 2}} = -\frac{\partial \overline{E}}{\partial \beta} = \frac{\partial^{ 2} \ln Z}{\partial \beta^{ 2}}.$$ (7.41)

Thus, the variance of the energy can be worked out from the partition function almost as easily as the mean energy. Because, by definition, a variance can never be negative, it follows that $\partial \overline{E}/\partial \beta \leq 0$ , or, equivalently, $\partial \overline{E}/\partial T \geq 0$ . Hence, the mean energy of a system governed by the canonical distribution always increases with increasing temperature.

Suppose that the system is characterized by a single external parameter $x$ (such as its volume). The generalization to the case where there are several external parameters is straightforward. Consider a quasi-static change of the external parameter from $x$ to $x+dx$ . In this process, the energy of the system in state $r$ changes by

$$\delta E_r = \frac{\partial E_r}{\partial x} dx.$$ (7.42)

The macroscopic work ${\mathchar'26\mkern-11mud}W$ done by the system due to this parameter change is

$${\mathchar'26\mkern-11mud}W = \frac{ \sum_r \exp(-\beta E_r) (-\partial E_r/\partial x dx)} {\sum_r \exp(-\beta E_r)}.$$ (7.43)

In other words, the work done is minus the average change in internal energy of the system, where the average is calculated using the canonical distribution. We can write

$$\sum_r \exp(-\beta E_r) \frac{\partial E_r}{\partial x} = -\fr... ...m_r \exp(-\beta E_r)\right] = -\frac{1}{\beta} \frac{\partial Z}{\partial x},$$ (7.44)

which gives

$${\mathchar'26\mkern-11mud}W = \frac{1}{\beta Z}\frac{\partial Z}{\partial x} dx = \frac{1}{\beta} \frac{\partial \ln Z}{\partial x} dx.$$ (7.45)

We also have the following general expression for the work done by the system

$${\mathchar'26\mkern-11mud}W = \overline{X} dx,$$ (7.46)

where

$$\overline{X} = - \overline{\frac{\partial E_r}{\partial x}}$$ (7.47)

is the mean generalized force conjugate to $x$ . (See Chapter 4.) It follows that

$$\overline{X} = \frac{1}{\beta} \frac{\partial \ln Z}{\partial x}.$$ (7.48)

Suppose that the external parameter is the volume, so $x=V$ . It follows that

$${\mathchar'26\mkern-11mud}W = \overline{p} dV = \frac{1}{\beta} \frac{\partial \ln Z}{\partial V} dV$$ (7.49)

and

$$\overline{p} = \frac{1}{\beta} \frac{\partial \ln Z}{\partial V},$$ (7.50)

where $p$ is the pressure. Because the partition function is a function of $\beta$ and $V$ (the energies $E_r$ depend on $V$ ), it follows that the previous equation relates the mean pressure, $\overline{p}$ , to $T$ (via $\beta = 1/k T$ ) and $V$ . In other words, the previous expression is the equation of state. Hence, we can work out the pressure, and even the equation of state, using the partition function.