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 $q$. A generalized form of the Black-Scholes (B-S) partial differential equation, and some closed-form solutions are obtained. The standard B-S equation ($q=1$) which is used by economists to calculate option prices requires multiple values of the stock volatility (known as the volatility smile). Using $q=1.5$ 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 $x_{t+1}=1-a|x_t|^z$, $(z>1)$ at their chaos threshold: We first introduce many initial conditions within one among $W>>1$ intervals partitioning the phase space and focus on the unique value $q_{sen}<1$ for which the entropic form $S_q \equiv \frac{1-\sum_{i=1}^{W} p_i^q}{q-1}$ {\it linearly} increases with time. We then verify that $S_{q_{sen}}(t) - S_{q_{sen}}(\infty)$ vanishes like $t^{-1/[q_{rel}(W)-1]}$ [$q_{rel}(W)>1$]. We finally exhibit a new finite-size scaling, $q_{rel}(\infty) - q_{rel}(W) \propto W^{-|q_{sen}|}$. 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, $q_{m}$. 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