Broad Histogram: Tests for a Simple and Efficient Microcanonical Simulator

The Broad Histogram Method (BHM) allows one to determine the energy degeneracy g(E), i.e. the energy spectrum of a given system, from the knowledge of the microcanonical averages and of two macroscopic quantities Nup and Ndn defined within the method. The fundamental BHM equation relating g(E) to the quoted averages is exact and completely general for any conceivable system. Thus, the only possible source of numerical inaccuracies resides on the measurement of the averages themselves. In this text, we introduce a Monte Carlo recipe to measure microcanonical averages. In order to test its performance, we applied it to the Ising ferromagnet on a 32x32 square lattice. The exact values of g(E) are known up to this lattice size, thus it is a good standard to compare our numerical results with. Measuring the deviations relative to the exactly known values, we verified a decay proportional to 1/sqrt(counts), by increasing the counter (counts) of averaged samples over at least 6 decades. That is why we believe this microcanonical simulator presents no bias besides the normal statistical fluctuations. For counts~10**10, we measured relative deviations near 10**(-5) for both g(E) and the specific heat peak, obtained through BHM relation.Comment: 9 pages, plain tex, 3 PS figure

Broad Histogram: An Overview

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 $$and$$, 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.Comment: 32 pages, tex, 3 PS figure

Broad Histogram Relation Is Exact

The Broad Histogram is a method designed to calculate the energy degeneracy g(E) from microcanonical averages of certain macroscopic quantities Nup and Ndn. These particular quantities are defined within the method, and their averages must be measured at constant energy values, i.e. within the microcanonical ensemble. Monte Carlo simulational methods are used in order to perform these measurements. Here, the mathematical relation allowing one to determine g(E) from these averages is shown to be exact for any statistical model, i.e. any energy spectrum, under completely general conditions. We also comment about some troubles concerning the measurement of the quoted microcanonical averages, when one uses a particular approach, namely the energy random walk dynamics. These troubles appear when movements corresponding to different energy jumps are performed using the same probability, and also when the correlations between successive averaging states are not adequately treated: they have nothing to do with the method itself.Comment: 10 pages, tex, 1 figure to appear in Eur. Phys. J.

Rich or poor: Who should pay higher tax rates?

A dynamic agent model is introduced with an annual random wealth multiplicative process followed by taxes paid according to a linear wealth-dependent tax rate. If poor agents pay higher tax rates than rich agents, eventually all wealth becomes concentrated in the hands of a single agent. By contrast, if poor agents are subject to lower tax rates, the economic collective process continues forever.Comment: 5 pages, 3 figure

The first shall be last: selection-driven minority becomes majority

Street demonstrations occur across the world. In Rio de Janeiro, June/July 2013, they reach beyond one million people. A wrathful reader of \textit{O Globo}, leading newspaper in the same city, published a letter \cite{OGlobo} where many social questions are stated and answered Yes or No. These million people of street demonstrations share opinion consensus about a similar set of social issues. But they did not reach this consensus within such a huge numbered meetings. Earlier, they have met in diverse small groups where some of them could be convinced to change mind by other few fellows. Suddenly, a macroscopic consensus emerges. Many other big manifestations are widespread all over the world in recent times, and are supposed to remain in the future. The interesting questions are: 1) How a binary-option opinion distributed among some population evolves in time, through local changes occurred within small-group meetings? and 2) Is there some natural selection rule acting upon? Here, we address these questions through an agent-based model.Comment: 8 pages, 4 figures, accepted for publication in Physica