/P_{\beta_0}(E),
\label{eq:14a}$$ where we have used the trivial relation $\sum_M P_{\beta_0}(E,M) = P_{\beta_0}(E)$. In practice this can be done simply by accumulating the measurements of $m^k$ in different slots or bins according to the energy of the configuration and normalizing at the end by the total number of hits of each energy bin. Clearly, once $\langle\!\langle m^k \rangle\!\rangle(E)$ is determined, this is a special case of $f(E)$ in eq.(\[eq:14\]), so that $$\langle m^k \rangle(\beta) = \frac{
\sum_E \langle\!\langle m^k \rangle\!\rangle(E) P_{\beta_0}(E)
e^{-(\beta - \beta_0)E}}
{\sum_E P_{\beta_0}(E) e^{-(\beta - \beta_0)E}} .
\label{eq:14b}$$ Similar to $\rho(E)$ in (\[eq:13\]), theoretically the microcanonical averages $\langle\!\langle m^k \rangle\!\rangle(E) $ do not depend on the temperature at which the simulation is performed. Due to the limited statistics in the wings of $P_{\beta_0}(E)$, however, there is only a finite range around $E_0 \equiv \langle E \rangle (\beta_0)$ where one can expect reasonable results for $\langle\!\langle m^k \rangle\!\rangle(E) $. Outside of this range it simply can happen (and does happen) that there are no events to be averaged. This is illustrated in Fig. 1(b), where $\langle\!\langle m \rangle\!\rangle(E) $ is plotted versus energy as obtained from the three runs which gave the energy histograms in Fig. 1(a). We 