615 research outputs found
Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations
A general type of nonlinear Fokker-Planck equation is derived directly from a
master equation, by introducing generalized transition rates. The H-theorem is
demonstrated for systems that follow those classes of nonlinear Fokker-Planck
equations, in the presence of an external potential. For that, a relation
involving terms of Fokker-Planck equations and general entropic forms is
proposed. It is shown that, at equilibrium, this relation is equivalent to the
maximum-entropy principle. Families of Fokker-Planck equations may be related
to a single type of entropy, and so, the correspondence between well-known
entropic forms and their associated Fokker-Planck equations is explored. It is
shown that the Boltzmann-Gibbs entropy, apart from its connection with the
standard -- linear Fokker-Planck equation -- may be also related to a family of
nonlinear Fokker-Planck equations.Comment: 19 pages, no figure
Option Pricing Formulas based on a non-Gaussian Stock Price Model
Options are financial instruments that depend on the underlying stock. We
explain their non-Gaussian fluctuations using the nonextensive thermodynamics
parameter . A generalized form of the Black-Scholes (B-S) partial
differential equation, and some closed-form solutions are obtained. The
standard B-S equation () which is used by economists to calculate option
prices requires multiple values of the stock volatility (known as the
volatility smile). Using which well models the empirical distribution
of returns, we get a good description of option prices using a single
volatility.Comment: final version (published
On a generalization of the binomial distribution and its Poisson-like limit
We examine a generalization of the binomial distribution associated with a
strictly increasing sequence of numbers and we prove its Poisson-like limit.
Such generalizations might be found in quantum optics with imperfect detection.
We discuss under which conditions this distribution can have a probabilistic
interpretation.Comment: 17 pages, 6 figure
Nonequilibrium Probabilistic Dynamics of the Logistic Map at the Edge of Chaos
We consider nonequilibrium probabilistic dynamics in logistic-like maps
, at their chaos threshold: We first introduce many
initial conditions within one among intervals partitioning the phase
space and focus on the unique value for which the entropic form
{\it linearly} increases with
time. We then verify that vanishes like
[]. We finally exhibit a new finite-size
scaling, . This
establishes quantitatively, for the first time, a long pursued relation between
sensitivity to the initial conditions and relaxation, concepts which play
central roles in nonextensive statistical mechanics.Comment: Final version with new Title and small modifications. REVTeX, 8 pages
and 4 eps figure
On a generalised model for time-dependent variance with long-term memory
The ARCH process (R. F. Engle, 1982) constitutes a paradigmatic generator of
stochastic time series with time-dependent variance like it appears on a wide
broad of systems besides economics in which ARCH was born. Although the ARCH
process captures the so-called "volatility clustering" and the asymptotic
power-law probability density distribution of the random variable, it is not
capable to reproduce further statistical properties of many of these time
series such as: the strong persistence of the instantaneous variance
characterised by large values of the Hurst exponent (H > 0.8), and asymptotic
power-law decay of the absolute values self-correlation function. By means of
considering an effective return obtained from a correlation of past returns
that has a q-exponential form we are able to fix the limitations of the
original model. Moreover, this improvement can be obtained through the correct
choice of a sole additional parameter, . The assessment of its validity
and usefulness is made by mimicking daily fluctuations of SP500 financial
index.Comment: 6 pages, 4 figure
Polarisation measurements with a CdTe pixel array detector for Laue hard X-ray focusing telescopes
Polarimetry is an area of high energy astrophysics which is still relatively
unexplored, even though it is recognized that this type of measurement could
drastically increase our knowledge of the physics and geometry of high energy
sources. For this reason, in the context of the design of a Gamma-Ray Imager
based on new hard-X and soft gamma ray focusing optics for the next ESA Cosmic
Vision call for proposals (Cosmic Vision 2015-2025), it is important that this
capability should be implemented in the principal on-board instrumentation. For
the particular case of wide band-pass Laue optics we propose a focal plane
based on a thick pixelated CdTe detector operating with high efficiency between
60-600 keV. The high segmentation of this type of detector (1-2 mm pixel size)
and the good energy resolution (a few keV FWHM at 500 keV) will allow high
sensitivity polarisation measurements (a few % for a 10 mCrab source in 106s)
to be performed. We have evaluated the modulation Q factors and minimum
detectable polarisation through the use of Monte Carlo simulations (based on
the GEANT 4 toolkit) for on and off-axis sources with power law emission
spectra using the point spread function of a Laue lens in a feasible
configuration.Comment: 10 pages, 6 pages. Accepted for publication in Experimental Astronom
Logarithmic diffusion and porous media equations: a unified description
In this work we present the logarithmic diffusion equation as a limit case
when the index that characterizes a nonlinear Fokker-Planck equation, in its
diffusive term, goes to zero. A linear drift and a source term are considered
in this equation. Its solution has a lorentzian form, consequently this
equation characterizes a super diffusion like a L\'evy kind. In addition is
obtained an equation that unifies the porous media and the logarithmic
diffusion equations, including a generalized diffusion equation in fractal
dimension. This unification is performed in the nonextensive thermostatistics
context and increases the possibilities about the description of anomalous
diffusive processes.Comment: 5 pages. To appear in Phys. Rev.
Generating functions for generalized binomial distributions
In a recent article a generalization of the binomial distribution associated
with a sequence of positive numbers was examined. The analysis of the
nonnegativeness of the formal expressions was a key-point to allow to give them
a statistical interpretation in terms of probabilities. In this article we
present an approach based on generating functions that solves the previous
difficulties: the constraints of nonnegativeness are automatically fulfilled, a
complete characterization in terms of generating functions is given and a large
number of analytical examples becomes available.Comment: PDFLaTex, 27 pages, 5 figure
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