measures the associated histogram $P_{\beta_0}(E) = {\cal P}_{\beta_0}(E)\Delta E$. Since the wings of $P_{\beta_0}(E)$ have large statistical errors, one expects eq. (\[eq:14\]) (or its obvious discrete modification) to give reliable results only for $\beta$ near $\beta_0$. If $\beta_0$ is near criticality, the distribution is relatively broad and the method works best. In this case reliable estimates from (\[eq:14\]) can be expected for $\beta$ values in an interval around $\beta_0$ of width $\propto L^{-1/\nu}$, i.e., just in the FSS region. The Fig. 1(a) shows three typical energy histograms measured at slightly different temperatures near the critical point for the lattice size $L$=48.

The information stored in $P_{\beta_0}(E)$ is not yet sufficient to calculate also the magnetic moments $\langle m^k \rangle (\beta)$ with $m = |\vec{m}|$ as function of $\beta$ from a single simulation at $\beta_0$. Conceptually, the simplest way to do so is to record the two-dimensional histogram $P_{\beta_0}(E,M)$, where $M = mV$ is the total magnetization (in general we will denote by small letters quantities per site, and the associated total quantities by the corresponding capital letters). Since disk space limitations prevented us from doing that[^2] we have measured instead the “microcanonical averages” $$\langle\!\langle m^k
\rangle\!\rangle(E) = \sum_M P_{\beta_0}(E,M) m^k 