69 research outputs found
Broad Histogram: An Overview
The Broad Histogram is a method allowing the direct calculation of the energy
degeneracy . 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 , 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 and the
microcanonical averages of certain macroscopic quantities and
. For an application to a particular problem, one needs to choose
an adequate instrument in order to determine the averages , 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 is a
function of energy defined from itself, being thus an {\bf internal}
(environment independent) characteristic of the system. Accordingly, all
microcanonical averages are functions of .
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: 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 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
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