86,445 research outputs found
Order in glassy systems
A directly measurable correlation length may be defined for systems having a
two-step relaxation, based on the geometric properties of density profile that
remains after averaging out the fast motion. We argue that the length diverges
if and when the slow timescale diverges, whatever the microscopic mechanism at
the origin of the slowing down. Measuring the length amounts to determining
explicitly the complexity from the observed particle configurations. One may
compute in the same way the Renyi complexities K_q, their relative behavior for
different q characterizes the mechanism underlying the transition. In
particular, the 'Random First Order' scenario predicts that in the glass phase
K_q=0 for q>x, and K_q>0 for q<x, with x the Parisi parameter. The hypothesis
of a nonequilibrium effective temperature may also be directly tested directly
from configurations.Comment: Typos corrected, clarifications adde
From Knowledge, Knowability and the Search for Objective Randomness to a New Vision of Complexity
Herein we consider various concepts of entropy as measures of the complexity
of phenomena and in so doing encounter a fundamental problem in physics that
affects how we understand the nature of reality. In essence the difficulty has
to do with our understanding of randomness, irreversibility and
unpredictability using physical theory, and these in turn undermine our
certainty regarding what we can and what we cannot know about complex phenomena
in general. The sources of complexity examined herein appear to be channels for
the amplification of naturally occurring randomness in the physical world. Our
analysis suggests that when the conditions for the renormalization group apply,
this spontaneous randomness, which is not a reflection of our limited
knowledge, but a genuine property of nature, does not realize the conventional
thermodynamic state, and a new condition, intermediate between the dynamic and
the thermodynamic state, emerges. We argue that with this vision of complexity,
life, which with ordinary statistical mechanics seems to be foreign to physics,
becomes a natural consequence of dynamical processes.Comment: Phylosophica
Range entropy: A bridge between signal complexity and self-similarity
Approximate entropy (ApEn) and sample entropy (SampEn) are widely used for
temporal complexity analysis of real-world phenomena. However, their
relationship with the Hurst exponent as a measure of self-similarity is not
widely studied. Additionally, ApEn and SampEn are susceptible to signal
amplitude changes. A common practice for addressing this issue is to correct
their input signal amplitude by its standard deviation. In this study, we first
show, using simulations, that ApEn and SampEn are related to the Hurst exponent
in their tolerance r and embedding dimension m parameters. We then propose a
modification to ApEn and SampEn called range entropy or RangeEn. We show that
RangeEn is more robust to nonstationary signal changes, and it has a more
linear relationship with the Hurst exponent, compared to ApEn and SampEn.
RangeEn is bounded in the tolerance r-plane between 0 (maximum entropy) and 1
(minimum entropy) and it has no need for signal amplitude correction. Finally,
we demonstrate the clinical usefulness of signal entropy measures for
characterisation of epileptic EEG data as a real-world example.Comment: This is the revised and published version in Entrop
Predictability: a way to characterize Complexity
Different aspects of the predictability problem in dynamical systems are
reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy,
Shannon entropy and algorithmic complexity is discussed. In particular, we
emphasize how a characterization of the unpredictability of a system gives a
measure of its complexity. Adopting this point of view, we review some
developments in the characterization of the predictability of systems showing
different kind of complexity: from low-dimensional systems to high-dimensional
ones with spatio-temporal chaos and to fully developed turbulence. A special
attention is devoted to finite-time and finite-resolution effects on
predictability, which can be accounted with suitable generalization of the
standard indicators. The problems involved in systems with intrinsic randomness
is discussed, with emphasis on the important problems of distinguishing chaos
from noise and of modeling the system. The characterization of irregular
behavior in systems with discrete phase space is also considered.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports.
Related information at this http://axtnt2.phys.uniroma1.i
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