11.2.3: Differential Forms with External Parameters

Now we want to extend these differential forms to the case where the constraint quantities depend on external parameters. In our magnetic-dipole example, the energy of each state depends on the externally applied magnetic field \(H\). Each \(E_i\) could be written in the form \(E_i(H)\) to emphasize this dependence. Thus the constraint could be written to show this dependence explicitly:

\(E = \displaystyle \sum_{i} p_iE_i(H) \tag{11.20}\)

Then all the quantities (\(p_i, \alpha, \beta\), and \(S\)) can be thought of as depending on both \(E\) and \(H\). In the case of our magnetic-dipole model, the energy \(E_i(H)\) happens to be proportional to \(H\) with a constant of proportionality that depends on \(i\) but not on \(H\). In other models, for other physical systems, \(E\) might depend on \(H\) or other parameters in different ways.

Consider what happens if both \(E\) and \(H\) vary slightly, by amounts \(dE\) and \(dH\), from the values used to calculate \(p_i\), \(\alpha\), \(\beta\), and \(S\). There will be small changes \(dp_i\), \(d\alpha\), \(d\beta\), and \(dS\) in those quantities which can be expressed in terms of the small changes \(dE\) and \(dH\). The changes due to \(dE\) have been calculated above. The changes due to \(dH\) enter through the change in the energies associated with each state, \(dE_i(H)\) (formulas like the next few could be derived for changes caused by any external parameter, not just the magnetic field).

\(\begin{align*}
0 &=\sum_{i} d p_{i} \tag{11.21} \\
d E &=\sum_{i} E_{i}(H) d p_{i}+\sum_{i} p_{i} d E_{i}(H) \tag{11.22} \\
d S &=k_{B} \beta d E-k_{B} \beta \sum_{i} p_{i} d E_{i}(H) \tag{11.23} \\
d \alpha &=-E d \beta-\beta \sum_{i} p_{i} d E_{i}(H) \tag{11.24} \\
d p_{i} &=-p_{i}(E_{i}(H)-E) d \beta-p_{i} \beta d E_{i}(H)+p_{i} \beta \sum_{j} p_{j} d E_{j}(H) \tag{11.25} \\
d E &=-\left [\sum_{i} p_{i}\left(E_{i}(H)-E\right )^{2}\right] d \beta+\sum_{i} p_{i}\left(1-\beta\left(E_{i}(H)-E\right)\right) d E_{i}(H) \tag{11.26} \\
d S &=-k_{B} \beta\left[\sum_{i} p_{i}\left(E_{i}(H)-E\right)^{2}\right] d \beta-k_{B} \beta^{2} \sum_{i} p_{i}\left(E_{i}(H)-E\right) d E_{i}(H) \tag{11.27}
\end{align*}\)

For the particular magnetic dipole model considered here, the terms involving \(dE_i(H)\) can be simplified by noting that each state’s energy \(E_i(H)\) is proportional to the parameter \(H\) and therefore

\(\begin{align*} dE_i(H) &= \Big(\dfrac{E_i(H)}{H}\Big) dH \tag{11.28} \\ \displaystyle \sum_{i} p_idE_i(H) &= \Big(\dfrac{E}{H}\Big)dH \tag{11.29} \end{align*} \)

so these formulas simplify to

\[\begin{align*}
0 &=\sum_{i} d p_{i} \tag{11.30}\\
d E &=\sum_{i} E_{i}(H) d p_{i}+\left(\frac{E}{H}\right) d H \tag{11.31}\\
d S &=k_{B} \beta d E-\left(\frac{k_{B} \beta E}{H}\right) d H \tag{11.32}\\
d \alpha &=-E d \beta-\left(\frac{\beta E}{H}\right) d H \tag{11.33}\\
d p_{i} &=-p_{i}\left(E_{i}(H)-E\right)(d \beta+\left(\frac{\beta}{H}\right) d H) \tag{11.34}\\
d E &=-\left[\sum_{i} p_{i}\left(E_{i}(H)-E\right)^{2}\right](d \beta+\left(\frac{\beta}{H}\right) d H)+\left(\frac{E}{H}\right) d H \tag{11.35}\\
d S &=-k_{B} \beta\left[\sum_{i} p_{i}\left(E_{i}(H)-E\right)^{2}\right](d \beta+\left(\frac{\beta}{H}\right) d H) \tag{11.36}
\end{align*} \nonumber \]

These formulas can be used to relate the trends in the variables. For example, the last formula shows that a one percent change in \(\beta\) produces the same change in entropy as a one percent change in \(H\).