11.2.2: Differential Forms

Now suppose \(E_i\) does not depend on an external parameter, and \(E\) changes by a small amount \(dE\). We will calculate from the equations above the changes in the other quantities, keeping only first-order variations (i.e., neglecting terms like \((dE)^2\) which, for small enough \(dE\), are insignificantly small)

\(\begin{align*} 0 &= \displaystyle \sum_{i} dp_i \tag{11.13} \\ dE &= \displaystyle \sum_{i} E_i dp_i \tag{11.14} \end{align*}\)

\(\begin{align*}
dS &=k_{B} \sum_{i} \ln \Big (\frac{1}{p_{i}} \Big) dp_{i}+k_{B} \sum_{i} p_{i} d\Big[\ln \Big(\frac{1}{p_{i}}\Big)\Big] \\
&=k_{B} \sum_{i} \ln \Big(\frac{1}{p_{i}}\Big) d p_{i}-k_{B} \sum_{i}\Big(\frac{p_{i}}{p_{i}}\Big) d p_{i} \\
&=k_{B} \sum_{i} (\alpha+\beta E_{i}) d p_{i} \\
&=k_{B} \beta d E \tag{11.15} \\
d \alpha &=\Big(\frac{1}{k_{B}}\Big) d S-\beta d E-E d \beta \\
&=-E d \beta \tag{11.16} \\
d p_{i} &=p_{i}(-d \alpha-E_{i} d \beta) \\
&=-p_{i}(E_{i}-E) d \beta \tag{11.17}
\end{align*} \)

from which it is not difficult to show

\(\begin{align*}
d E &= -\Big(\sum_{i} p_{i}(E_{i}-E)^{2}\Big) d \beta \tag{11.18}\\
d S &= -k_{B} \beta\Big(\sum_{i} p_{i}(E_{i}-E)^{2}\Big) d \beta \tag{11.19}
\end{align*}\)

These equations may be used in several ways. Note that all first-order variations are expressed as a function of \(d\beta\) so it is natural to think of \(\beta\) as the independent variable. But this is not necessary; these equations remain valid no matter which change causes the other changes.

As an example of the insight gained from these equations, note that the formula relating \(dE\) and \(d\beta\), Equation 11.18, implies that if \(E\) goes up then \(\beta\) goes down, and vice versa.