22,647 research outputs found
Spatial Measures of Urban Systems: from Entropy to Fractal Dimension
A type of fractal dimension definition is based on the generalized entropy
function. Both entropy and fractal dimension can be employed to characterize
complex spatial systems such as cities and regions. Despite the inherent
connect between entropy and fractal dimension, they have different application
scopes and directions in urban studies. This paper focuses on exploring how to
convert entropy measurement into fractal dimension for the spatial analysis of
scale-free urban phenomena using ideas from scaling. Urban systems proved to be
random prefractal and multifractals systems. The entropy of fractal cities
bears two typical properties. One is the scale dependence. Entropy values of
urban systems always depend on the scales of spatial measurement. The other is
entropy conservation. Different fractal parts bear the same entropy value. Thus
entropy cannot reflect the spatial heterogeneity of fractal cities in theory.
If we convert the generalized entropy into multifractal spectrums, the problems
of scale dependence and entropy homogeneity can be solved to a degree for urban
spatial analysis. The essence of scale dependence is the scaling in cities, and
the spatial heterogeneity of cities can be characterized by multifractal
scaling. This study may be helpful for the students to describe and understand
spatial complexity of cities.Comment: 27 page, 9 figure, 5 table
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
Gibbs' theorem for open systems with incomplete statistics
Gibbs' theorem, which is originally intended for canonical ensembles with
complete statistics has been generalized to open systems with incomplete
statistics. As a result of this generalization, it is shown that the stationary
equilibrium distribution of inverse power law form associated with the
incomplete statistics has maximum entropy even for open systems with energy or
matter influx. The renormalized entropy definition given in this paper can also
serve as a measure of self-organization in open systems described by incomplete
statistics.Comment: 6 pages, accepted to Chaos, Solitons and Fractal
Hamiltonian dynamics, nanosystems, and nonequilibrium statistical mechanics
An overview is given of recent advances in nonequilibrium statistical
mechanics on the basis of the theory of Hamiltonian dynamical systems and in
the perspective provided by the nanosciences. It is shown how the properties of
relaxation toward a state of equilibrium can be derived from Liouville's
equation for Hamiltonian dynamical systems. The relaxation rates can be
conceived in terms of the so-called Pollicott-Ruelle resonances. In spatially
extended systems, the transport coefficients can also be obtained from the
Pollicott-Ruelle resonances. The Liouvillian eigenstates associated with these
resonances are in general singular and present fractal properties. The singular
character of the nonequilibrium states is shown to be at the origin of the
positive entropy production of nonequilibrium thermodynamics. Furthermore,
large-deviation dynamical relationships are obtained which relate the transport
properties to the characteristic quantities of the microscopic dynamics such as
the Lyapunov exponents, the Kolmogorov-Sinai entropy per unit time, and the
fractal dimensions. We show that these large-deviation dynamical relationships
belong to the same family of formulas as the fluctuation theorem, as well as a
new formula relating the entropy production to the difference between an
entropy per unit time of Kolmogorov-Sinai type and a time-reversed entropy per
unit time. The connections to the nonequilibrium work theorem and the transient
fluctuation theorem are also discussed. Applications to nanosystems are
described.Comment: Lecture notes for the International Summer School Fundamental
Problems in Statistical Physics XI (Leuven, Belgium, September 4-17, 2005
On observability of Renyi's entropy
Despite recent claims we argue that Renyi's entropy is an observable
quantity. It is shown that, contrary to popular belief, the reported domain of
instability for Renyi entropies has zero measure (Bhattacharyya measure). In
addition, we show the instabilities can be easily emended by introducing a
coarse graining into an actual measurement. We also clear up doubts regarding
the observability of Renyi's entropy in (multi--)fractal systems and in systems
with absolutely continuous PDF's.Comment: 18 pages, 1 EPS figure, REVTeX, minor changes, accepted to Phys. Rev.
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
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