scholarly journals Does the orbit-averaged theory require a scale separation between periodic orbit size and perturbation correlation length?

2013 ◽  
Vol 20 (10) ◽  
pp. 102306
Author(s):  
Wenlu Zhang ◽  
Zhihong Lin
Keyword(s):  
Periodic Orbit ◽  
Correlation Length ◽  
Scale Separation ◽  
Orbit Size

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

scholarly journals Bottlebrush polymers in the melt and polyelectrolytes in solution share common structural features

2020 ◽  
Vol 117 (10) ◽  
pp. 5168-5175 ◽  
Author(s):  
Joel M. Sarapas ◽  
Tyler B. Martin ◽  
Alexandros Chremos ◽  
Jack F. Douglas ◽  
Kathryn L. Beers
Keyword(s):  
Neutron Scattering ◽  
Correlation Length ◽  
Salt Concentration ◽  
Polymer Concentration ◽  
Structural Features ◽  
Peak Position ◽  
Correlation Peak ◽  
Grafting Density ◽  
Scattering Measurements ◽  
Polyelectrolyte Solutions

Uncharged bottlebrush polymer melts and highly charged polyelectrolytes in solution exhibit correlation peaks in scattering measurements and simulations. Given the striking superficial similarities of these scattering features, there may be a deeper structural interrelationship in these chemically different classes of materials. Correspondingly, we constructed a library of isotopically labeled bottlebrush molecules and measured the bottlebrush correlation peak position q*=2π/ξ by neutron scattering and in simulations. We find that the correlation length scales with the backbone concentration, ξ∼cBB−0.47, in striking accord with the scaling of ξ with polymer concentration cP in semidilute polyelectrolyte solutions (ξ∼cP−1/2). The bottlebrush correlation peak broadens with decreasing grafting density, similar to increasing salt concentration in polyelectrolyte solutions. ξ also scales with sidechain length to a power in the range of 0.35–0.44, suggesting that the sidechains are relatively collapsed in comparison to the bristlelike configurations often imagined for bottlebrush polymers.

scholarly journals Unconventional singularities and energy balance in frictional rupture

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Efim A. Brener ◽  
Eran Bouchbinder
Keyword(s):  
Energy Balance ◽  
Square Root ◽  
Earthquake Physics ◽  
Macroscopic Theory ◽  
Scale Separation ◽  
Rate Dependent ◽  
Theoretical Predictions ◽  
Spatially Extended ◽  
Singularity Order ◽  
Square Root Singularity

AbstractA widespread framework for understanding frictional rupture, such as earthquakes along geological faults, invokes an analogy to ordinary cracks. A distinct feature of ordinary cracks is that their near edge fields are characterized by a square root singularity, which is intimately related to the existence of strict dissipation-related lengthscale separation and edge-localized energy balance. Yet, the interrelations between the singularity order, lengthscale separation and edge-localized energy balance in frictional rupture are not fully understood, even in physical situations in which the conventional square root singularity remains approximately valid. Here we develop a macroscopic theory that shows that the generic rate-dependent nature of friction leads to deviations from the conventional singularity, and that even if this deviation is small, significant non-edge-localized rupture-related dissipation emerges. The physical origin of the latter, which is predicted to vanish identically in the crack analogy, is the breakdown of scale separation that leads an accumulated spatially-extended dissipation, involving macroscopic scales. The non-edge-localized rupture-related dissipation is also predicted to be position dependent. The theoretical predictions are quantitatively supported by available numerical results, and their possible implications for earthquake physics are discussed.

Periodic orbit expansions for the Lorentz gas

1994 ◽  
Vol 75 (3-4) ◽  
pp. 553-584 ◽  
Author(s):  
Gary P. Morris ◽  
Lamberto Rondoni
Keyword(s):  
Periodic Orbit ◽  
Lorentz Gas
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