181 research outputs found
Correction algorithm for finite sample statistics
Assume in a sample of size M one finds M_i representatives of species i with
i=1...N^*. The normalized frequency p^*_i=M_i/M, based on the finite sample,
may deviate considerably from the true probabilities p_i. We propose a method
to infer rank-ordered true probabilities r_i from measured frequencies M_i. We
show that the rank-ordered probabilities provide important informations on the
system, e.g., the true number of species, the Shannon- and the Renyi-entropies.Comment: 11 pages, 9 figure
Brownian Particles far from Equilibrium
We study a model of Brownian particles which are pumped with energy by means
of a non-linear friction function, for which different types are discussed. A
suitable expression for a non-linear, velocity-dependent friction function is
derived by considering an internal energy depot of the Brownian particles. In
this case, the friction function describes the pumping of energy in the range
of small velocities, while in the range of large velocities the known limit of
dissipative friction is reached. In order to investigate the influence of
additional energy supply, we discuss the velocity distribution function for
different cases. Analytical solutions of the corresponding Fokker-Planck
equation in 2d are presented and compared with computer simulations. Different
to the case of passive Brownian motion, we find several new features of the
dynamics, such as the formation of limit cycles in the four-dimensional
phase-space, a large mean squared displacement which increases quadratically
with the energy supply, or non-equilibrium velocity distributions with
crater-like form. Further, we point to some generalizations and possible
applications of the model.Comment: 10 pages, 12 figure
Self-Organization, Active Brownian Dynamics, and Biological Applications
After summarizing basic features of self-organization such as entropy export,
feedbacks and nonlinear dynamics, we discuss several examples in biology. The
main part of the paper is devoted to a model of active Brownian motion that
allows a stochastic description of the active motion of biological entities
based on energy consumption and conversion. This model is applied to the
dynamics of swarms with external and interaction potentials. By means of
analytical results, we can distiguish between translational, rotational and
amoebic modes of swarm motion. We further investigate swarms of active Brownian
particles interacting via chemical fields and demonstrate the application of
this model to phenomena such as biological aggregation and trail formation in
insects.Comment: 22 pages, 9 multipart figures (minor changes after vers.1), For
related papers see http://www.ais.fraunhofer.de/~frank/papers.htm
Guessing probability distributions from small samples
We propose a new method for the calculation of the statistical properties, as
e.g. the entropy, of unknown generators of symbolic sequences. The probability
distribution of the elements of a population can be approximated by
the frequencies of a sample provided the sample is long enough so that
each element occurs many times. Our method yields an approximation if this
precondition does not hold. For a given we recalculate the Zipf--ordered
probability distribution by optimization of the parameters of a guessed
distribution. We demonstrate that our method yields reliable results.Comment: 10 pages, uuencoded compressed PostScrip
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