11.3.3: Extensive and Intensive Quantities

This partition model leads to an important property that physical quantities can have. Some physical quantities will be called “extensive” and others “intensive.”

Whether the system is isolated from the environment or is interacting with it, and whatever the probability distributions \(p_{s,i}\) of the system, \(p_{e,j}\) of the environment, and \(p_{t,i,j}\) of the combination, the energies of the system state and of the environment state add up to form the energy of the corresponding total state (subscripts \(s\), \(e\), and \(t\) mean system, environment, and total):

\(E_{t,i,j} = E_{s,i} + E_{e,j} \tag{11.37}\)

The probability of occupancy of total state \(k\) is the product of the two probabilities of the two associated states \(i\) and \(j\):

\(p_{t,i,j} = p_{s,i}p_{e,j} \tag{11.38}\)

With this background it is easy to show that the expected value of the total energy is the sum of the expected values of the system and environment energies:

\(\begin{align*}
E_{t} &=\sum_{i, j} E_{t, i, j} p_{t, i, j} \\
&=\sum_{i, j}\left[E_{s, i}+E_{e, j}\right] p_{s, i} p_{e, j} \\
&=\sum_{i} \sum_{j}\left[E_{s, i}+E_{e, j}\right] p_{s, i} p_{e, j} \\
&=\sum_{i} p_{s, i} \sum_{j} E_{e, j} p_{e, j}+\sum_{j} p_{e, j} \sum_{i} E_{s, i} p_{s, i} \\
&=\sum_{j} E_{e, j} p_{e, j}+\sum_{i} E_{s, i} p_{s, i} \\
&=E_{e}+E_{s} \tag{11.39}
\end{align*}\)

This result holds whether the system and environment are isolated or interacting. It states that the energy of the system and the energy of the environment add up to make the total energy. It is a consequence of the fact that the energy associated with each total state is the sum of the energies associated with the corresponding system and environment states.

A quantity with the property that its total value is the sum of the values for the two (or more) parts is known as an extensive quantity. Energy has that property, as was just demonstrated. Entropy is also extensive. That is,

\(\begin{align*}
S_{t} &=\sum_{i, j} p_{t, i, j} \ln \left(\frac{1}{p_{t, i, j}}\right) \\
&=\sum_{i, j} p_{s, i} p_{e, j}\left[\ln \left(\frac{1}{p_{s, i}}\right)+\ln \left(\frac{1}{p_{e, j}}\right)\right] \\
&=\sum_{i} \sum_{j} p_{s, i} p_{e, j}\left[\ln \left(\frac{1}{p_{s, i}}\right)+\ln \left(\frac{1}{p_{e, j}}\right)\right] \\
&=\sum_{i} p_{s, i} \sum_{j} p_{e, j} \ln \left(\frac{1}{p_{e, j}}\right)+\sum_{j} p_{e, j} \sum_{i} p_{s, i} \ln \left(\frac{1}{p_{s, i}}\right) \\
&=\sum_{j} p_{e, j} \ln \left(\frac{1}{p_{e, j}}\right)+\sum_{i} p_{s, i} \ln \left(\frac{1}{p_{s, i}}\right) \\
&=S_{e}+S_{s} \tag{11.40}
\end{align*}\)

Again this result holds whether or not the system and environment are isolated or interacting.

Not all quantities of interest are extensive. In particular, \(\alpha\) and \(\beta\) are not. Consider \(\beta\). This is an example of a quantity for which the values associated with the system, the environment, and the total configuration may or may not be related. If the system and environment are isolated, so that a separate application of the Principle of Maximum Entropy is made to each, then there is no reason why \(\beta _s\) and \(\beta _e\) would be related. On the other hand, if the system and environment are interacting so that they are exchanging energy, the distribution of energy between the system and the environment may not be known and therefore the Principle of Maximum Entropy can be applied only to the combination, not to the system and environment separately. Then, the same value of \(\beta\) would apply throughout.

Quantities like \(\beta\) that are the same throughout a system when analyzed as a whole are called intensive.