A discrete version of Liouville’s theorem on conformal maps

Liouville’s theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally equivalent if they are combinatorially equivalent and the lengths of corresponding edges are related by scale factors associated with the vertices.

Entropic Uncertainty Principle, Partition Function and Holographic Principle derived from Liouville's Theorem

An entropic version of Liouville’s theorem is defined in terms of the conjugate variables (“hyperbolic position” and “entropic momentum”) of an entropic Hamiltonian. It is used to derive the Holographic Principle as applied to holomorphic structures that represent maximum entropy configurations. The Bekenstein-Hawking expression for black hole entropy is a consequence. Based on the entropic commutator derived from Liouville’s theorem and the same entropic conjugate variables, an entropic Uncertainty Principle (in units of Boltzmann’s constant) isomorphic to the kinematic Uncertainty Principle (in units of Planck’s constant) is also derived. These formal developments underpin the previous treatment of Quantitative Geometrical Thermodynamics (QGT) which has established (entirely on geometric entropy grounds) the stability of the double-helix, the double logarithmic spiral, and the sphere. Since in the QGT formalism the Boltzmann and Planck constants are quanta of quantities orthogonal to each other in Minkowski spacetime, a solution of the Schrödinger Equation is demonstrated isomorphic to a probability term of an entropic Partition Function, where both are defined by path integrals obeying the stationary principle: this isomorphism represents an important symmetry of the formalism. The geometry of a holomorphic structure must also exhibit at least C2 symmetry.

Phase-space dynamics and ergodicity

This chapter provides the mathematical justification for the theory of equilibrium statistical mechanics. A Hamiltonian system which is ergodic is shown to have time-average behavior equal to the average behavior in the energy shell. Liouville’s theorem is used to justify the use of phase-space volume in taking this average. Exercises explore the breakdown of ergodicity in planetary motion and in dissipative systems, the application of Liouville’s theorem by Crooks and Jarzynski to non-equilibrium statistical mechanics, and generalizations of statistical mechanics to chaotic systems and to two-dimensional turbulence and Jupiter’s great red spot.

A Survey on High-energy Protons Response to Geomagnetic Storm in the Inner Radiation Belt

RBSPA observations suggest that the inner radiation belt high energy proton fluxes drop significantly during the storm main phase and recover in parallel to as the SYM-H index [Xu et al., 2019]. A natural problem arises: are these storm‐time proton flux variations in response to the magnetic field modifications adiabatic? Based on Liouville's theorem and conservation of the first and third adiabatic invariants, the fully adiabatic effects of high energy protons in the inner radiation belt have been quantitatively evaluated. Two case studies show that theoretically calculated, adiabatic flux decreases are in good agreement with RBSPA observations. Statistical survey of 67 geomagnetic storms which occurred in 2013–2016 has been conducted. The results confirm that the fully adiabatic response constitutes the main contribution 90 % to the changes in high energy protons in inner radiation belt during the storm main and recovery phases. It indicates that adiabatic invariants of the inner belt high energy protons are well preserved for majority of storms. Phase space density results also support adiabatic effect controls the varication of high energy protons especially for small and medium geomagnetic storms. Non-adiabatic effects could play important role for the most intense storms with fast changes in magnetic configuration.

Оптимизация масс-сепараторов

The optimization of separator masses in this paper is based on the realization of the linearity of the system under study and the consequences of Liouville's theorem. The properties of several mass separators with an ion energy of 30 keV and a beam emittance of 4 mm*mrad are considered. Focusing is provided by an aberration-free lens and a magnetic corrector. Phase diagrams along the beam path are in the form of parallelograms, which indicates the absence of geometric aberrations. For each of the separators, the resolution calculated in the linear approximation coincides with the simulation results. It is shown that a mass separator based on a magnet with a rotation angle of 54.70 has a resolution of about 5000, and a separator based on two magnets with a rotation angle of 450 and 900, respectively, has a resolution of 14000-15000.