11.2.1: General Properties

Because \(\beta\) plays a central role, it is helpful to understand intuitively how different values it may assume affect things.

First, if \(\beta\) = 0, all probabilities are equal. This can only happen if the number of states is finite.

Second, if \(\beta\) > 0, then states with lower energy have a higher probability of being occupied. Similarly, if \(\beta\) < 0, then states with higher energy have a higher probability of being occupied. Because of the exponential dependence on energy, unless | \(\beta\) | is small, the only states with much probability of being occupied are those with energy close to the minimum possible (\(\beta\) positive) or maximum possible (\(\beta\) negative).

Third, we can multiply the equation above for \(\ln (1/p_i)\) by \(p_i\) and sum over \(i\) to obtain

\(S = k_B(\alpha + \beta E) \tag{11.7}\)

This equation is valid and useful even if it is not possible to find \(\beta\) in terms of \(E\) or to compute the many values of \(p_i\).

Fourth, in Section 11.2.2 we will look at a small change \(dE\) in \(E\) and inquire how the other variables change. Such first-order relationships, or “differential forms,” provide intuition which helps when the formulas are interpreted.

Fifth, in Section 11.2.3 we will consider the dependence of energy on an external parameter, using the magnetic dipole system with its external parameter \(H\) as an example.

The critical equations above are listed here for convenience

\(\begin{align*}
1 &=\sum_{i} p_{i} \tag{11.8} \\
E &=\sum_{i} p_{i} E_{i} \tag{11.9} \\
S &=k_{B} \sum_{i} p_{i} \ln \Big(\frac{1}{p_{i}}\Big ) \tag{11.10} \\
p_{i} &=e^{-\alpha} e^{-\beta E_{i}} \tag{11.11} \\
\alpha &=\ln \Big (\sum_{i} e^{-\beta E_{i}}\Big ) \\
&=\frac{S}{k_{B}}-\beta E \tag{11.12}
\end{align*}\)