with the volume of the system.

Histogram techniques
--------------------

Even though histogram reweighting techniques have been known since long [@oldpaper], they have gained increasing popularity as a practical tool only quite recently [@r20]. The best performance is achieved [*near*]{} criticality, and in this sense the histogram reweighting technique is complementary to the use of improved estimators. It is a quite general technique of data analysis based on the simple idea of recording whole distribution functions, and not only their first few moments (e.g., the average energy and specific heat), as is usually done. The energy distribution ${\cal P}_{\beta_0}(E)$ (normalized to unit area) at inverse temperature $\beta_0$ can be written as $${\cal P}_{\beta_0}(E) = \rho(E) e ^{-\beta_0 E} / Z(\beta_0),
\label{eq:13}$$ where $\rho(E)$ is the density of states with energy $E$. It is then easy to see that an expectation value $\langle f(E) \rangle$ can in principle be calculated for any $\beta$ from $$\langle f(E) \rangle(\beta) = \frac{\int_0^{\infty} dE f(E) {\cal P}_{\beta_0}
(E) e^{-(\beta-\beta_0)E}}{\int_0^{\infty} dE {\cal P}_{\beta_0}
(E) e^{-(\beta-\beta_0)E}}.
\label{eq:14}$$ To keep the notation short we have suppressed the lattice-size dependence of ${\cal P}_{\beta_0}(E)$.

In practice the continuous energy has to be discretized into bins of size $\Delta E$, and one 