Hierarchical structures in the phase space and fractional kinetics: I. Classical systems

This article is concerned with defining and characterizing hierarchical structures in smooth dynamical systems. We define transitions between levels in a dynamical hierarchy by smooth projective maps from a phase space on a lower level, with high dimensionality, to a phase space on a higher level, with lower dimensionality. It is required that each level describe a self-contained deterministic dynamical system. We show that a necessary and sufficient condition for a projective map to be a transition between levels in the hierarchy is that the kernel of the differential of the map is tangent to an invariant manifold with respect to the flow. The implications of this condition are discussed in detail. We demonstrate two different causal dependences between degrees of freedom, and how these relations are revealed when the dynamical system is transformed into global Jordan form. Finally these results are used to define functional components on different levels, interaction networks, and dynamical hierarchies.

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Turbulent power distribution in the local interstellar medium

Astronomy and Astrophysics ◽

10.1051/0004-6361/201834533 ◽

2019 ◽

Vol 627 ◽

pp. A112 ◽

Cited By ~ 2

Author(s):

P. M. W. Kalberla ◽

U. Haud

Keyword(s):

Phase Transitions ◽

Phase Space ◽

Interstellar Medium ◽

Power Distribution ◽

Single Channel ◽

Power Spectra ◽

Hierarchical Structures ◽

Space Distribution ◽

Neutral Medium ◽

Highly Correlated

Context. The interstellar medium (ISM) on all scales is full of structures that can be used as tracers of processes that feed turbulence. Aims. We used H I survey data to derive global properties of the angular power distribution of the local ISM. Methods. HI4PI observations on an nside = 1024 HEALPix grid and Gaussian components representing three phases, the cold, warm, and unstable lukewarm neutral medium (CNM, WNM, and LNM), were used for velocities |vLSR|≤ 25 km s−1. For high latitudes |b| > 20° we generated apodized maps. After beam deconvolution we fitted angular power spectra. Results. Power spectra for observed column densities are exceptionally well defined and straight in log-log presentation with 3D power law indices γ ≥−3 for the local gas. For intermediate velocity clouds (IVCs) we derive γ = −2.6 and for high velocity clouds (HVCs) γ = −2.0. Single-phase power distributions for the CNM, LNM, and WNM are highly correlated and shallow with γ ~−2.5 for multipoles l ≤ 100. Excess power from cold filamentary structures is observed at larger multipoles. The steepest single-channel power spectra for the CNM are found at velocities with large CNM and low WNM phase fractions. Conclusions. The phase space distribution in the local ISM is configured by phase transitions and needs to be described with three distinct different phases, being highly correlated but having distributions with different properties. Phase transitions cause locally hierarchical structures in phase space. The CNM is structured on small scales and is restricted in position-velocity space. The LNM as an interface to the WNM envelops the CNM. It extends to larger scales than the CNM and covers a wider range of velocities. Correlations between the phases are self-similar in velocity.

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Phase space theory of quantum–classical systems with nonlinear and stochastic dynamics

Annals of Physics ◽

10.1016/j.aop.2014.01.007 ◽

2014 ◽

Vol 343 ◽

pp. 16-26 ◽

Cited By ~ 1

Author(s):

Nikola Burić ◽

Duška B. Popović ◽

Milan Radonjić ◽

Slobodan Prvanović

Keyword(s):

Phase Space ◽

Stochastic Dynamics ◽

Space Theory ◽

Classical Systems ◽

Phase Space Theory

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DEFORMATION QUANTIZATION FOR COUPLED HARMONIC OSCILLATORS ON A GENERAL NONCOMMUTATIVE SPACE

Modern Physics Letters A ◽

10.1142/s0217732308023992 ◽

2008 ◽

Vol 23 (06) ◽

pp. 445-456 ◽

Cited By ~ 3

Author(s):

BINGSHENG LIN ◽

SICONG JING ◽

TAIHUA HENG

Keyword(s):

Phase Space ◽

Deformation Quantization ◽

Harmonic Oscillators ◽

Energy Spectra ◽

Noncommutative Space ◽

Wigner Functions ◽

Noncommutative Phase Space ◽

Classical Systems ◽

Coupled Harmonic Oscillators ◽

Special Cases

Deformation quantization is a powerful tool to quantize some classical systems especially in noncommutative space. In this work we first show that for a class of special Hamiltonian one can easily find relevant time evolution functions and Wigner functions, which are intrinsic important quantities in the deformation quantization theory. Then based on this observation we investigate a two-coupled harmonic oscillators system on the general noncommutative phase space by requiring both spatial and momentum coordinates do not commute each other. We derive all the Wigner functions and the corresponding energy spectra for this system, and consider several interesting special cases, which lead to some significant results.

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A one-dimensional self-gravitating stellar gas

Symposium - International Astronomical Union ◽

10.1017/s0074180900105261 ◽

1966 ◽

Vol 25 ◽

pp. 46-48 ◽

Cited By ~ 2

Author(s):

M. Lecar

Keyword(s):

Gravitational Field ◽

Phase Space ◽

Motion Picture ◽

Space Distribution ◽

Two Dimensional ◽

One Dimensional ◽

Phase Space Distribution ◽

Dimensional Phase Space ◽

The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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Relationships between hierarchical structure and properties in macromolecular systems

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100126950 ◽

1987 ◽

Vol 45 ◽

pp. 440-443

Author(s):

E. Baer

Keyword(s):

Uniaxial Tension ◽

Hierarchical Structure ◽

Inorganic Material ◽

Hierarchical Structures ◽

Bone Collagen ◽

Structure And Properties ◽

Connective Tissues ◽

Scale Level ◽

Fibrous Protein ◽

Macromolecular Systems

The most advanced macromolecular materials are found in plants and animals, and certainly the connective tissues in mammals are amongst the most advanced macromolecular composites known to mankind. The efficient use of collagen, a fibrous protein, in the design of both soft and hard connective tissues is worthy of comment. Very crudely, in bone collagen serves as a highly efficient binder for the inorganic hydroxyappatite which stiffens the structure. The interactions between the organic fiber of collagen and the inorganic material seem to occur at the nano (scale) level of organization. Epitatic crystallization of the inorganic phase on the fibers has been reported to give a highly anisotropic, stress responsive, structure. Soft connective tissues also have sophisticated oriented hierarchical structures. The collagen fibers are “glued” together by a highly hydrated gel-like proteoglycan matrix. One of the simplest structures of this type is tendon which functions primarily in uniaxial tension as a reinforced elastomeric cable between muscle and bone.

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Biomimetic materials: An introduction

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100129735 ◽

1992 ◽

Vol 50 (2) ◽

pp. 1020-1021 ◽

Cited By ~ 1

Author(s):

M. Sarikaya ◽

J. T. Staley ◽

I. A. Aksay

Keyword(s):

Self Assembly ◽

Structural Control ◽

Hierarchical Structures ◽

Ceramic Composites ◽

Advanced Materials ◽

Length Scale ◽

Spider Webs ◽

Biomimetic Materials ◽

Synthetic Materials ◽

All Ceramic

Biomimetics is an area of research in which the analysis of structures and functions of natural materials provide a source of inspiration for design and processing concepts for novel synthetic materials. Through biomimetics, it may be possible to establish structural control on a continuous length scale, resulting in superior structures able to withstand the requirements placed upon advanced materials. It is well recognized that biological systems efficiently produce complex and hierarchical structures on the molecular, micrometer, and macro scales with unique properties, and with greater structural control than is possible with synthetic materials. The dynamism of these systems allows the collection and transport of constituents; the nucleation, configuration, and growth of new structures by self-assembly; and the repair and replacement of old and damaged components. These materials include all-organic components such as spider webs and insect cuticles (Fig. 1); inorganic-organic composites, such as seashells (Fig. 2) and bones; all-ceramic composites, such as sea urchin teeth, spines, and other skeletal units (Fig. 3); and inorganic ultrafine magnetic and semiconducting particles produced by bacteria and algae, respectively (Fig. 4).

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