scholarly journals Transport in the nonergodic extended phase of interacting quasiperiodic systems

2020 ◽  
Vol 102 (22) ◽  
Author(s):  
Soumi Ghosh ◽  
Jyotsna Gidugu ◽  
Subroto Mukerjee
Keyword(s):  
Extended Phase ◽  
Quasiperiodic Systems

Shape Dynamics and the Linking Theory

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Line Element ◽  
Hamiltonian Formulation ◽  
Operational Definition ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Problem Of Time ◽  
Definition Of ◽  
Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

Autonomous Geometrical Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Contact Structure ◽  
Invariant Tori ◽  
Time Dependent ◽  
Contact Structures ◽  
Natural Setting ◽  
Adiabatic Invariance ◽  
Jet Bundle ◽  
Integrability Theorem ◽  
Extended Phase

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

Other topics

2018 ◽  
Author(s):  
Ted Janssen ◽  
Gervais Chapuis ◽  
Marc de Boissieu
Keyword(s):  
Photonic Crystals ◽  
Crystal Morphology ◽  
Three Dimensional ◽  
Icosahedral Quasicrystal ◽  
Crystal Surfaces ◽  
Decagonal Quasicrystal ◽  
System A ◽  
Fundamental Law ◽  
Quasiperiodic Systems ◽  
Aperiodic Crystals

The law of rational indices to describe crystal faces was one of the most fundamental law of crystallography and is strongly linked to the three-dimensional periodicity of solids. This chapter describes how this fundamental law has to be revised and generalized in order to include the structures of aperiodic crystals. The generalization consists in using for each face a number of integers, with the number corresponding to the rank of the structure, that is, the number of integer indices necessary to characterize each of the diffracted intensities generated by the aperiodic system. A series of examples including incommensurate multiferroics, icosahedral crystals, and decagonal quaiscrystals illustrates this topic. Aperiodicity is also encountered in surfaces where the same generalization can be applied. The chapter discusses aperiodic crystal morphology, including icosahedral quasicrystal morphology, decagonal quasicrystal morphology, and aperiodic crystal surfaces; magnetic quasiperiodic systems; aperiodic photonic crystals; mesoscopic quasicrystals, and the mineral calaverite.

Explicit symplectic-like integration with corrected map for inseparable Hamiltonian

2020 ◽  
Vol 501 (1) ◽  
pp. 1511-1519
Author(s):  
Junjie Luo ◽  
Weipeng Lin ◽  
Lili Yang
Keyword(s):  
Phase Space ◽  
Space Structure ◽  
Eighth Order ◽  
Time Step ◽  
Extended Phase ◽  
Energy Error ◽  
Phase Space Structure ◽  
Compact Binary ◽  
Three Body

ABSTRACT Symplectic algorithms are widely used for long-term integration of astrophysical problems. However, this technique can only be easily constructed for separable Hamiltonian, as preserving the phase-space structure. Recently, for inseparable Hamiltonian, the fourth-order extended phase-space explicit symplectic-like methods have been developed by using the Yoshida’s triple product with a mid-point map, where the algorithm is more effective, stable and also more accurate, compared with the sequent permutations of momenta and position coordinates, especially for some chaotic case. However, it has been found that, for the cases such as with chaotic orbits of spinning compact binary or circular restricted three-body system, it may cause secular drift in energy error and even more the computation break down. To solve this problem, we have made further improvement on the mid-point map with a momentum-scaling correction, which turns out to behave more stably in long-term evolution and have smaller energy error than previous methods. In particular, it could obtain a comparable phase-space distance as computing from the eighth-order Runge–Kutta method with the same time-step.

scholarly journals Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

2021 ◽  
Vol 87 (3) ◽  
Author(s):  
Nicolas Crouseilles ◽  
Paul-Antoine Hervieux ◽  
Yingzhe Li ◽  
Giovanni Manfredi ◽  
Yajuan Sun
Keyword(s):  
Phase Space ◽  
Maxwell Equations ◽  
Stimulated Raman Scattering ◽  
Extended Phase ◽  
Five Dimensions ◽  
Particle In Cell ◽  
Spin Effects ◽  
Dimensional Phase Space ◽  
Spin Polarized ◽  
Underdense Plasma

We propose a numerical scheme to solve the semiclassical Vlasov–Maxwell equations for electrons with spin. The electron gas is described by a distribution function $f(t,{\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ that evolves in an extended 9-dimensional phase space $({\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ , where $\boldsymbol s$ represents the spin vector. Using suitable approximations and symmetries, the extended phase space can be reduced to five dimensions: $(x,{{p_x}}, {\boldsymbol s})$ . It can be shown that the spin Vlasov–Maxwell equations enjoy a Hamiltonian structure that motivates the use of the recently developed geometric particle-in-cell (PIC) methods. Here, the geometric PIC approach is generalized to the case of electrons with spin. Total energy conservation is very well satisfied, with a relative error below $0.05\,\%$ . As a relevant example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, where the electrons are partially or fully spin polarized. It is shown that the Raman instability is very effective in destroying the electron polarization.

TRANSITION STATE THEORY IN EXTENDED PHASE SPACE

1993 ◽  
Vol 07 (19) ◽  
pp. 1263-1268
Author(s):  
H. DEKKER ◽  
A. MAASSEN VAN DEN BRINK
Keyword(s):  
Phase Space ◽  
Transition State Theory ◽  
Unstable Mode ◽  
Planck Equation ◽  
State Theory ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Theoretical Concepts ◽  
Rate Processes ◽  
Mode Energy

Turnover theory (of the escape Γ) à la Grabert will be based solely on Kramers' Fokker–Planck equation for activated rate processes. No recourse to a microscope model or Langevin dynamics will be made. Apart from the unstable mode energy E, the analysis requires new theoretical concepts such as a constrained Gaussian transformation (CGT) and dynamically extended phase space (EPS).

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