488 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
Linear instability and statistical laws of physics
We show that a meaningful statistical description is possible in conservative
and mixing systems with zero Lyapunov exponent in which the dynamical
instability is only linear in time. More specifically, (i) the sensitivity to
initial conditions is given by with
; (ii) the statistical entropy in the infinitely fine graining limit (i.e., {\it
number of cells into which the phase space has been partitioned} ),
increases linearly with time only for ; (iii) a nontrivial,
-generalized, Pesin-like identity is satisfied, namely the . These facts (which are
in analogy to the usual behaviour of strongly chaotic systems with ), seem
to open the door for a statistical description of conservative many-body
nonlinear systems whose Lyapunov spectrum vanishes.Comment: 7 pages including 2 figures. The present version is accepted for
publication in Europhysics Letter
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
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
Equivalence among different formalisms in the Tsallis entropy framework
In a recent paper [Phys. Lett. A {\bf335}, 351 (2005)] the authors discussed
the equivalence among the various probability distribution functions of a
system in equilibrium in the Tsallis entropy framework. In the present letter
we extend these results to a system which is out of equilibrium and evolves to
a stationary state according to a nonlinear Fokker-Planck equation. By means of
time-scale conversion, it is shown that there exists a ``correspondence'' among
the self-similar solutions of the nonlinear Fokker-Planck equations associated
with the different Tsallis formalisms. The time-scale conversion is related to
the corresponding Lyapunov functions of the respective nonlinear Fokker-Planck
equations.Comment: 20 pages, 1 figure, Elsart macro style, version accepted on Physica
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.
Nonextensive Entropies derived from Form Invariance of Pseudoadditivity
The form invariance of pseudoadditivity is shown to determine the structure
of nonextensive entropies. Nonextensive entropy is defined as the appropriate
expectation value of nonextensive information content, similar to the
definition of Shannon entropy. Information content in a nonextensive system is
obtained uniquely from generalized axioms by replacing the usual additivity
with pseudoadditivity. The satisfaction of the form invariance of the
pseudoadditivity of nonextensive entropy and its information content is found
to require the normalization of nonextensive entropies. The proposed principle
requires the same normalization as that derived in [A.K. Rajagopal and S. Abe,
Phys. Rev. Lett. {\bf 83}, 1711 (1999)], but is simpler and establishes a basis
for the systematic definition of various entropies in nonextensive systems.Comment: 16 pages, accepted for publication in Physical Review
Nonlinear equation for anomalous diffusion: unified power-law and stretched exponential exact solution
The nonlinear diffusion equation is analyzed here, where , and , and are real parameters.
This equation unifies the anomalous diffusion equation on fractals ()
and the spherical anomalous diffusion for porous media (). Exact
point-source solution is obtained, enabling us to describe a large class of
subdiffusion (), normal diffusion () and
superdiffusion (). Furthermore, a thermostatistical basis
for this solution is given from the maximum entropic principle applied to the
Tsallis entropy.Comment: 3 pages, 2 eps figure
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