scholarly journals Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches

Mathematics ◽  
2018 ◽  
Vol 6 (2) ◽  
pp. 17 ◽  
Author(s):  
Guoxing Lin
Keyword(s):  
Phase Space ◽  
Anomalous Diffusion ◽  
Real Space

scholarly journals Analyze PFG Anomalous Diffusion via Real Space and Phase Space Approaches

2017 ◽  
Author(s):  
Guoxing Lin
Keyword(s):  
Phase Space ◽  
Field Gradient ◽  
Anomalous Diffusion ◽  
Signal Intensity ◽  
Integration Method ◽  
Real Space ◽  
Direct Integration ◽  
Signal Attenuation ◽  
Function Type ◽  
Direct Integration Method

Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may not be neglected such as in clinical diffusion magnetic resonance imaging (MRI). Here, two significantly different methods are proposed to analyze PFG anomalous diffusion: the effective phase shift diffusion equation (EPSDE) method and an observing the signal intensity at the origin method. The EPSDE method describes the phase evolution in virtual phase space, while the method to observe the signal intensity at the origin describes the magnetization evolution in real space. However, these two approaches give the same general PFG signal attenuation including FGPW effect, which can be numerically evaluated by a direct integration method. The direct integration method is fast and without overflow. It is a convenient numerical evaluation method for Mittag-Leffler function type PFG signal attenuation. The methods here provide a clear view of spin evolution under field gradient, and their results will help the analysis of PFG anomalous diffusion.

SCALING LAWS IN STOCHASTIC SYSTEMS WITH ANOMALOUS DIFFUSION

2002 ◽  
Vol 02 (04) ◽  
pp. L273-L278 ◽  
Author(s):  
DMITRII KHARCHENKO
Keyword(s):  
Phase Space ◽  
Anomalous Diffusion ◽  
Stochastic System ◽  
Hurst Exponent ◽  
Special Class ◽  
Scaling Laws ◽  
Stochastic Systems ◽  
Diffusion Processes ◽  
Anomalous Behaviour ◽  
Proposed Model

We consider the stochastic system with an anomalous diffusion. According to the obtained relations between characteristics of diffusion processes the special class of models which exhibit the anomalous behaviour is considered. It was shown that indexes of super- and subdiffusion are related to the Hürst exponent which defines the properties of the phase space inherent to the proposed model of stochastic system.

scholarly journals EFFECT OF CLASSICAL TRAPPING IN QUANTUM CHAOS: AN ANOMALOUS DIFFUSION APPROACH

2008 ◽  
Vol 22 (30) ◽  
pp. 5261-5277 ◽  
Author(s):  
JIAO WANG ◽  
ANTONIO M. GARCIA-GARCIA
Keyword(s):  
Phase Space ◽  
Anomalous Diffusion ◽  
Quantum Chaos ◽  
Quantum Dynamics ◽  
Classical Case ◽  
Dynamical Localization ◽  
Classical Motion ◽  
Classical Phase Space ◽  
Breaking Time ◽  
The Impact

We study generic effects on the quantum dynamics of classical trapping-leaking mechanism by investigating in detail the 2δ-kicked rotors whose classical phase space is partitioned into momentum cells separated by trapping regions which slow down the motion. We focus on a range of parameters where the dynamics is generic, namely, the phase space has no stable islands. As a consequence of the trapping-leaking mechanism, we show that the classical motion is described by a process of anomalous diffusion. We investigate in detail the impact of the underlying classical anomalous diffusion on the quantum dynamics with special emphasis on the phenomenon of dynamical localization. Based on the study of the quantum density of probability, its second moment and the return probability, we identify a region of weak dynamical localization where the quantum diffusion is still anomalous but the diffusion rate is slower than in the classical case. Moreover, we examine how other relevant time scales, such as the quantum-classical breaking time and the one related to the beginning of full dynamical localization, are modified by the classical anomalous diffusion. Finally, we discuss the relevance of our results for understanding the role of classical cantori in quantum mechanics.

scholarly journals Dynamical Structures in the Galactic Disk

2013 ◽  
Vol 9 (S298) ◽  
pp. 105-116
Author(s):  
Alice C. Quillen
Keyword(s):  
Phase Space ◽  
Galactic Disk ◽  
Past History ◽  
Real Space ◽  
Local Velocity ◽  
Spiral Density Waves ◽  
History Of ◽  
Pattern Speed ◽  
Orbital Periods ◽  
Space Phase

AbstractResonances with spiral density waves or the Galactic bar can cause structure in local velocity distributions (also known as phase space). Because resonances are narrow, it is possible to place tight constraints on a pattern speed or the shape of the underlying gravitational potential.Interference between multiple spiral patterns can cause localized bursts of star formation and discontinuities or kinks in the spiral arm morphology. Numerical simulations suggest that boundaries between different dominant patterns in the disk can manifest in local velocity distributions as gaps that are associated with specific orbital periods or angular momentum values. Recent studies have detected age gradients that may be associated with the appearance of spiral features such as armlets and spurs.When patterns grow or vary in speed, resonances can be swept through the disk causing changes in the velocity distributions. Evidence of resonance capture or resonance crossing can be used to place constraints on the past history of patterns in the disk. The X-shaped Galactic bulge may have been caused by stars captured into vertical resonance with the Bar.Disturbances in the Galactic disk, such as from a past merger, can cause an uneven distribution of disk stars in action angles. Subsequently the stellar distribution becomes more tightly wound in phase space. Phase wrapping can cause a series of shell like features in either real space or in a local velocity distribution. The spacing between features is dependent on the time since the disturbance.

scholarly journals Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping

2020 ◽  
Vol 102 (4) ◽  
pp. 2385-2398
Author(s):  
D. Eeltink ◽  
A. Armaroli ◽  
C. Luneau ◽  
H. Branger ◽  
M. Brunetti ◽  
...  
Keyword(s):  
Phase Space ◽  
Phase Shift ◽  
Symmetry Breaking ◽  
Fundamental Frequency ◽  
Deep Water ◽  
Water Waves ◽  
Plane Waves ◽  
Real Space ◽  
Separatrix Crossing ◽  
Dysthe Equation

AbstractWe theoretically and experimentally examine the effect of forcing and damping on systems that can be described by the nonlinear Schrödinger equation (NLSE), by making use of the phase-space predictions of the three-wave truncation. In the latter, the spectrum is truncated to only the fundamental frequency and the upper and lower sidebands. Our experiments are performed on deep water waves, which are better described by the higher-order NLSE, the Dysthe equation. We therefore extend our analysis to this system. However, our conclusions are general for NLSE systems. By means of experimentally obtained phase-space trajectories, we demonstrate that forcing and damping cause a separatrix crossing during the evolution. When the system is damped, it is pulled outside the separatrix, which in the real space corresponds to a phase-shift of the envelope and therefore doubles the period of the Fermi–Pasta–Ulam–Tsingou recurrence cycle. When the system is forced by the wind, it is pulled inside the separatrix, lifting the phase-shift. Furthermore, we observe a growth and decay cycle for modulated plane waves that are conventionally considered stable. Finally, we give a theoretical demonstration that forcing the NLSE system can induce symmetry breaking during the evolution.

Corrigenda

1961 ◽  
Vol 262 (1311) ◽  
pp. 435-435
Keyword(s):  
Phase Space ◽  
Energy Density ◽  
Point Source ◽  
Flux Density ◽  
Real Space ◽  
Relative Energy

Proc. Roy. Soc. A, 261, 423-434 (Haselgrove, Haselgrove & Jennison) In §2 the quantity I (x, u) is described as a flux density. It is in fact the energy density in phase space. The quantity I/I 0 appearing in §4 onwards is the relative energy density in real space due to a point source.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
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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