the value of $W_\alpha$ does not change—the rate of change also depends on the values of $W_\gamma$—but it does immediately imply that if $W_\gamma$ is the constant function on phase space (representing the completely mixed state), then it is also constant in time. That is, the completely mixed state is unchanged by any Hamiltonian evolution (which is of course correct).

As in the preceding section, our main concern here is to identify constraints on the transition rates $r_{\alpha\gamma}$ that characterize actual Hamiltonian flows in phase space. We begin by inverting Eq. (\[rdef\]) so as to express the Hamiltonian function $H_\alpha$ in terms of the $r$’s, if indeed the given set of $r$ values is consistent with a Hamiltonian.

Starting with Eq. (\[rdef\]), we use the properties expressed in Eq. (\[properties\]) to get $$\sum_{\alpha\gamma} r_{\alpha\gamma} \hat{A}_\alpha \hat{A}_\gamma = -\frac{i}{\hbar} N^2\left( \sum_\beta
H_\beta \hat{A}_\beta - \frac{1}{N} \sum_\beta H_\beta \right).$$ Now multiply by $\hat{A}_\delta$ and take the trace to get $$\label{firsth}
\frac{i\hbar}{N^2} \sum_{\alpha\gamma} r_{\alpha\gamma} \Gamma_{\alpha\gamma\delta} = H_\delta - \frac{1}{N^2}\sum_\beta H_\beta,$$ which gives us the Hamiltonian function $H_\delta$ up to an additive constant. (The additive constant does not affect the dynamics.) By renaming indices and making use of the symmetries of $\Gamma$ and $r_{\alpha\gamma}$, 