4,492 research outputs found
Measuring information growth in fractal phase space
We look at chaotic systems evolving in fractal phase space. The entropy
change in time due to the fractal geometry is assimilated to the information
growth through the scale refinement. Due to the incompleteness, at any scale,
of the information calculation in fractal support, the incomplete normalization
is applied throughout the paper. It is shown that the
information growth is nonadditive and is proportional to the trace-form
so that it can be connected to several nonadditive
entropies. This information growth can be extremized to give, for
non-equilibrium systems, power law distributions of evolving stationary state
which may be called ``maximum entropic evolution''.Comment: 10 pages, 1 eps figure, TeX. Chaos, Solitons & Fractals (2004), in
pres
Generalized (c,d)-entropy and aging random walks
Complex systems are often inherently non-ergodic and non-Markovian for which
Shannon entropy loses its applicability. In particular accelerating,
path-dependent, and aging random walks offer an intuitive picture for these
non-ergodic and non-Markovian systems. It was shown that the entropy of
non-ergodic systems can still be derived from three of the Shannon-Khinchin
axioms, and by violating the fourth -- the so-called composition axiom. The
corresponding entropy is of the form and depends on two system-specific scaling exponents, and . This
entropy contains many recently proposed entropy functionals as special cases,
including Shannon and Tsallis entropy. It was shown that this entropy is
relevant for a special class of non-Markovian random walks. In this work we
generalize these walks to a much wider class of stochastic systems that can be
characterized as `aging' systems. These are systems whose transition rates
between states are path- and time-dependent. We show that for particular aging
walks is again the correct extensive entropy. Before the central part
of the paper we review the concept of -entropy in a self-contained way.Comment: 8 pages, 5 eps figures. arXiv admin note: substantial text overlap
with arXiv:1104.207
Fractal geometry, information growth and nonextensive thermodynamics
This is a study of the information evolution of complex systems by
geometrical consideration. We look at chaotic systems evolving in fractal phase
space. The entropy change in time due to the fractal geometry is assimilated to
the information growth through the scale refinement. Due to the incompleteness
of the state number counting at any scale on fractal support, the incomplete
normalization is applied throughout the paper, where is the
fractal dimension divided by the dimension of the smooth Euclidean space in
which the fractal structure of the phase space is embedded. It is shown that
the information growth is nonadditive and is proportional to the trace-form
which can be connected to several nonadditive
entropies. This information growth can be extremized to give power law
distributions for these non-equilibrium systems. It can also be used for the
study of the thermodynamics derived from Tsallis entropy for nonadditive
systems which contain subsystems each having its own . It is argued that,
within this thermodynamics, the Stefan-Boltzmann law of blackbody radiation can
be preserved.Comment: Final version, 10 pages, no figures, Invited talk at the
international conference NEXT2003, 21-28 september 2003, Villasimius
(Cagliari), Ital
Incomplete information and fractal phase space
The incomplete statistics for complex systems is characterized by a so called
incompleteness parameter which equals unity when information is
completely accessible to our treatment. This paper is devoted to the discussion
of the incompleteness of accessible information and of the physical
signification of on the basis of fractal phase space. is
shown to be proportional to the fractal dimension of the phase space and can be
linked to the phase volume expansion and information growth during the scale
refining process.Comment: 12 pages, 2 ps figure, Te
Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot
We introduce a new set of generalized Fokker-Planck equations that conserve
energy and mass and increase a generalized entropy until a maximum entropy
state is reached. The concept of generalized entropies is rigorously justified
for continuous Hamiltonian systems undergoing violent relaxation. Tsallis
entropies are just a special case of this generalized thermodynamics.
Application of these results to stellar dynamics, vortex dynamics and Jupiter's
great red spot are proposed. Our prime result is a novel relaxation equation
that should offer an easily implementable parametrization of geophysical
turbulence. This relaxation equation depends on a single key parameter related
to the skewness of the fine-grained vorticity distribution. Usual
parametrizations (including a single turbulent viscosity) correspond to the
infinite temperature limit of our model. They forget a fundamental systematic
drift that acts against diffusion as in Brownian theory. Our generalized
Fokker-Planck equations may have applications in other fields of physics such
as chemotaxis for bacterial populations. We propose the idea of a
classification of generalized entropies in classes of equivalence and provide
an aesthetic connexion between topics (vortices, stars, bacteries,...) which
were previously disconnected.Comment: Submitted to Phys. Rev.
Incomplete descriptions and relevant entropies
Statistical mechanics relies on the complete though probabilistic description
of a system in terms of all the microscopic variables. Its object is to derive
therefrom static and dynamic properties involving some reduced set of
variables. The elimination of the irrelevant variables is guided by the maximum
entropy criterion, which produces the probability law carrying the least amount
of information compatible with the relevant variables. This defines relevant
entropies which measure the missing information (the disorder) associated with
the sole variables retained in an incomplete description. Relevant entropies
depend not only on the state of the system but also on the coarseness of its
reduced description. Their use sheds light on questions such as the Second Law,
both in equilibrium an in irreversible thermodynamics, the projection method of
statistical mechanics, Boltzmann's \textit{H}-theorem or spin-echo experiment.Comment: flatex relevant_entropies.tex, 1 file Submitted to: Am. J. Phy
Coarse-grained distributions and superstatistics
We show an interesting connexion between the coarse-grained distribution
function arising in the theory of violent relaxation for collisionless stellar
systems (Lynden-Bell 1967) and the notion of superstatistics introduced
recently by Beck & Cohen (2003). We also discuss the analogies and differences
between the statistical equilibrium state of a multi-components
self-gravitating system and the metaequilibrium state of a collisionless
stellar system. Finally, we stress the important distinction between mixing
entropies, generalized entropies, H-functions, generalized mixing entropies and
relative entropies
Generalized entropies and open random and scale-free networks
We propose the concept of open network as an arbitrary selection of nodes of
a large unknown network. Using the hypothesis that information of the whole
network structure can be extrapolated from an arbitrary set of its nodes, we
use Renyi mutual entropies in different q-orders to establish the minimum
critical size of a random set of nodes that represents reliably the information
of the main network structure. We also identify the clusters of nodes
responsible for the structure of their containing network.Comment: talk at CTNEXT07 (July 2007
- âŠ