Broad Histogram: An Overview

Abstract

The Broad Histogram is a method allowing the direct calculation of the energy degeneracy $g(E)$. This quantity is independent of thermodynamic concepts such as thermal equilibrium. It only depends on the distribution of allowed (micro) states along the energy axis, but not on the energy changes between the system and its environment. Once one has obtained $g(E)$, no further effort is needed in order to consider different environment conditions, for instance, different temperatures, for the same system. The method is based on the exact relation between $g(E)$ and the microcanonical averages of certain macroscopic quantities $N^{\rm up}$ and $N^{\rm dn}$. For an application to a particular problem, one needs to choose an adequate instrument in order to determine the averages $<N^{\rm up}(E)>$ and $<N^{\rm dn}(E)>$, as functions of energy. Replacing the usual fixed-temperature canonical by the fixed-energy microcanonical ensemble, new subtle concepts emerge. The temperature, for instance, is no longer an external parameter controlled by the user, all canonical averages being functions of this parameter. Instead, the microcanonical temperature $T_{m}(E)$ is a function of energy defined from $g(E)$ itself, being thus an {\bf internal} (environment independent) characteristic of the system. Accordingly, all microcanonical averages are functions of $E$. The present text is an overview of the method. Some features of the microcanonical ensemble are also discussed, as well as some clues towards the definition of efficient Monte Carlo microcanonical sampling rules.