Phase Space Research Articles

Basin entropy as an indicator of a bifurcation in a time-delayed system.

The basin entropy is a measure that quantifies, in a system that has two or more attractors, the predictability of a final state, as a function of the initial conditions. While the basin entropy has been demonstrated on a variety of multistable dynamical systems, to the best of our knowledge, it has not yet been tested in systems with a time delay, whose phase space is infinite dimensional because the initial conditions are functions defined in a time interval [-τ,0], where τ is the delay time. Here, we consider a simple time-delayed system consisting of a bistable system with a linear delayed feedback term. We show that the basin entropy captures relevant properties of the basins of attraction of the two coexisting attractors. Moreover, we show that the basin entropy can give an indication of the proximity of a Hopf bifurcation, but fails to capture the proximity of a pitchfork bifurcation. The Hopf bifurcation is detected because before the fixed points become unstable, a oscillatory, limit-cycle behavior appears that coexists with the fixed points. The new limit cycle modifies the structure of the basins of attraction, and this change is captured by basin entropy that reaches a maximum before the Hopf bifurcation. In contrast, the pitchfork bifurcation is not detected because the basins of attraction do not change as the bifurcation is approached. Our results suggest that the basin entropy can yield useful insights into the long-term predictability of time-delayed systems, which often have coexisting attractors.

Mixed (regular and chaotic) dynamics under the channeling of the high energy positrons in [100] direction of the silicon crystal and Podolskiy-Narimanov distribution

The character of motion (regular or chaotic) on the quantum level manifests itself in the statistics of the set of the system's energy levels. The completely regular case generates the sequence of the levels with exponential (Poisson) level spacing distribution; the completely chaotic one — with Wigner distribution. The most interesting case is the co-existence of the regular and chaotic motion domains in the phase space of the system under consideration. The assumption of independent generation of two level sequencies by one chaotic domain and by all regular domains leads to Berry-Robnik distribution. However, the presence of the chaotic motion domain in the phase space affects the levels produced by the regular domains via the so-called chaos-assisted tunneling (CAT) that leads to Podolskiy-Narimanov distribution of the level spacings. This distribution needs the mean amplitude of the tunnel transition and the relative contribution of the regular domains to the mean level density as the parameters. Using their estimations for the transverse motion of the high energy positrons channeling near [100] direction in the silicon crystal we found that Podolskiy-Narimanov distribution demonstrates the best agreement with the level spacing distribution.

Optimal semiclassical regularity of projection operators and strong Weyl law

Projection operators arise naturally as one-particle density operators associated to Slater determinants in fields such as quantum mechanics and the study of determinantal processes. In the context of the semiclassical approximation of quantum mechanics, projection operators can be seen as the analogue of characteristic functions of subsets of the phase space, which are discontinuous functions. We prove that projection operators indeed converge to characteristic functions of the phase space and that in terms of quantum Sobolev spaces, they exhibit the same maximal regularity as characteristic functions. This can be interpreted as a semiclassical asymptotic on the size of commutators in Schatten norms. Our study answers a question raised in Chong et al. [J. Eur. Math. Soc. (unpublished) (2024)] about the possibility of having projection operators as initial data. It also gives a strong convergence result in Sobolev spaces for the Weyl law in phase space.

Dynamical chaos and level splitting under the channeling of the high energy positrons in [100] direction of the silicon crystal

The motion of charged particles in a crystal in the axial channeling regime can be both regular and chaotic. The chaos in quantum case manifests itself in the statistical properties of the energy levels set. These properties have been studied previously for the electrons channeling along [110] direction of the silicon crystal, in the case when the classical motion was completely chaotic, as well as for the ones channeling along [100] direction, when the classical motion can be both regular and chaotic for the same energy depending on the initial conditions. Here we study the positrons channeling in [100] direction. This case is of special interest due to the substantial tunneling probability between dynamically isolated regular motion domains in the phase space. The interaction of the energy levels via tunneling distinctly changes the level spacing statistics. All transverse motion energy levels as well as corresponding stationary wave functions are computed numerically for the 30 GeV positrons channeling in [100] direction of the silicon crystal. The values of the matrix elements for the tunnel transitions areextractedfrom these data. These results confirm the chaos assistance for the tunneling and the level splitting. These values will be used in the further researches of the quantum chaos manifestations in the channeling phenomenon.

Cryptanalysis and construction of keyed strong S-Box based on random affine transformation matrix and 2D hyper chaotic map

As the only nonlinear module, S-Box (substitution box) is widely used in stream cipher, hash function, and so on. However, there exist some weaknesses in existing S-Boxes, such as fixed point, reverse fixed point and short iteration period rings. Furthermore, the S-Boxes in AES and SM4.0 were fixed, which makes them vulnerable to attack. To address these weaknesses, first, a non-degenerate 2D chaotic map (2D-NDCM) with ergodicity in phase space was constructed. Second, based on 2D-NDCM, the seed S-Boxes with high nonlinearity were constructed in batches by affine transformation and Boolean function, and then three possible weaknesses were eliminated to obtain strong keyed S-Boxes. Statistical analysis demonstrated the effectiveness and practicability of the proposed S-Box batch generating algorithm

Development of an Image Analysis Method for Pepperpot Emittance Monitors

At the RIKEN Nishina Center for Accelerator-Based Science, we developed a pepperpot emittance monitor using a method that can change the distance between the pepperpot mask and the screen. The accuracy can be improved by correctly identifying the beam’s position at a close range followed by the increase of the distance; however, if the distance is increased excessively, the position matching will be difficult. To solve this problem, we developed a method for tracking continuous changes on a screen using optical flow. Using this method, the distance between the pepperpot mask and screen could be extended without plotting in the wrong area in the phase space, and the emittance measurement accuracy was successfully improved by more than 10%. This development will enable beam transport simulations to be performed more accurately.

Measurement of the Z boson invisible width at [formula omitted] TeV with the ATLAS detector

A measurement of the invisible width of the Z boson using events with jets and missing transverse momentum is presented using 37 fb−1 of 13 TeV proton–proton data collected by the ATLAS detector in 2015 and 2016. The ratio of Z→inv to Z→ℓℓ events, where inv refers to non-detected particles and ℓ is either an electron or a muon, is measured and corrected for detector effects. Events with at least one energetic central jet with pT≥110 GeV are selected for both the Z→inv and Z→ℓℓ final states to obtain a similar phase space in the ratio. The invisible width is measured to be 506±2(stat.)±12(syst.) MeV and is the single most precise recoil-based measurement. The result is in agreement with the most precise determination from LEP and the Standard Model prediction based on three neutrino generations.

Determination of impact parameter for CEE with digi-input neural networks

The impact parameter characterizes the centrality in nucleus-nucleus collision geometry. The determination of impact parameters in real experiments is usually based on the reconstructed particle attributes or the derived event-level observables. For the scheduled Cooler-storage-ring External-target Experiment (CEE), the low beam energy reduces correlation between the impact parameter and charged particle multiplicity, which decreases the validity of the explicit determination methods. This work investigates a few neural network-based models that directly take the digitized signals from the external Time-of-flight detectors as input. The model with the best performance shows a mean absolute error of 0.479 fm with simulated U-U collisions at 0.5 AGeV. The performances of the models implemented with digi inputs are compared with reference models with phase space inputs, showing the capability of neural networks to handle the original but potentially interrelated digitized signal information.

Simple 3D PIC Analysis for Beam Phase Space Oscillation in RF Driven Negative Hydrogen Ion Source

Temporal oscillation of the negative hydrogen ion (H−) beam in the Radio Frequency (RF) ion source is investigated by a simple 3D3V Particle-In-Cell (PIC) model. The model solves electron, proton, and H− transport processes in the extraction region. The calculation domain is from the vicinity of the beam aperture to the ground electrode. To understand the relation between the plasma density oscillation and the extracted H− beam characteristics, the input electron and proton fluxes from the driver region are varied parametrically with the fundamental J-PARC RF frequency (2 MHz). The numerical results give an idea of the main physical processes between the oscillations of the plasma parameters and the extracted H− ion trajectories depending on the plasma meniscus shape in the different RF phases.

Simple analysis of complex system safety based on Finite State Machine Network and phase space theory

Safety analysis of complex systems, such as aero-engines, has always been a challenging problem. In particular, the risk propagation caused by system functional interactions has become increasingly prominent, leading to the difficulty of traditional safety analysis methods. Existing methods are mostly based on manual experience and deduction with difficulty to completely analyze the whole system safety regions. In this paper, a model-based framework is proposed by combining Finite State Machine Network (FSMN) and phase space theory to perform safety analysis of complex systems from function-logic perspective. We first construct the system model and system state-transition network based on FSMN. Then combining with phase space theory, we identify system safety regions of the entire operating space from macro perspective, and investigate the system risk propagation from micro perspective. An aero-engine system is taken as an example to illustrate the proposed method. Our results show that this method has the ability to effectively identify the potential risk factors and explore emergent unknown risks, which can provide guidance for formulating safety testing.

Micro-macro consistency in multiscale modeling: Score-based model assisted sampling of fast/slow dynamical systems.

A valuable step in the modeling of multiscale dynamical systems in fields such as computational chemistry, biology, and materials science is the representative sampling of the phase space over long time scales of interest; this task is not, however, without challenges. For example, the long term behavior of a system with many degrees of freedom often cannot be efficiently computationally explored by direct dynamical simulation; such systems can often become trapped in local free energy minima. In the study of physics-based multi-time-scale dynamical systems, techniques have been developed for enhancing sampling in order to accelerate exploration beyond free energy barriers. On the other hand, in the field of machine learning (ML), a generic goal of generative models is to sample from a target density, after training on empirical samples from this density. Score-based generative models (SGMs) have demonstrated state-of-the-art capabilities in generating plausible data from target training distributions. Conditional implementations of such generative models have been shown to exhibit significant parallels with long-established-and physics-based-solutions to enhanced sampling. These physics-based methods can then be enhanced through coupling with the ML generative models, complementing the strengths and mitigating the weaknesses of each technique. In this work, we show that SGMs can be used in such a coupling framework to improve sampling in multiscale dynamical systems.

On the structural sensitivity of some diffusion-reaction models of population dynamics

In mathematical ecology, it is often assumed that properties of a mathematical model are robust to specific parameterization of functional responses, in particular preserving the bifurcation structure of the system, as long as different functions are qualitatively similar. This intuitive assumption has been challenged recently (Fussmann & Blasius, Community response to enrichment is highly sensitive to model structure, Biology Letters 1:9-12, 2005). Having considered the prey-predator system as a paradigm of nonlinear population dynamics, it has been shown that in fact both the bifurcation structure and the structure of the phase space can be rather different even when the component functions are apparently close to each other. However, these observations have so far been largely limited to nonspatial systems described by ODEs. In this paper, our main motivating interest is to investigate whether such structural sensitivity occurs in spatially explicit models of population dynamics, in particular those that are described by PDEs. We consider a prey-predator model described by a system of two nonlinear reaction–diffusion-advection equations where the predation term is parameterized by three different yet numerically close functions. Using some analytical tools along with numerical simulations, we show that the properties of spatiotemporal dynamics are rather different between the three cases, so that patterns observed for one parameterization may not occur for the other two ones.

Research and development of a muon entrance trigger for the muEDM experiment at PSI

The muEDM experiment at the Paul Scherrer Institute (PSI) in Switzerland aims to probe the muon electric dipole moment (EDM) using the frozen-spin technique in a compact storage ring, with a sensitivity of 6×10−23 e⋅cm. A fast entrance detector is expected to work in concert with a magnetic pulse generator to direct muons into the desired orbit. Simultaneously, the entrance detector is designed to veto muons that exceed the apparatus's admittance without introducing significant multiple scatterings. We developed a prototype entrance trigger detector consisting of a thin scintillator for detecting incoming muons and four wall scintillators as veto detectors. The prototype was tested at 27.5 MeV/c at the πE1 beamline at PSI. A total of 7×105 events were collected, which were read out by SiPMs coupled to the plastic scintillators under two different beam tunes. These events were analyzed to characterize the detector's performance, which was also cross-checked with Monte Carlo simulations that took into account the beam phase space and scintillation processes.

Performance and painting scheme considerations for the CSNS injection

The accelerator complex of China Spallation Neutron Source (CSNS) consists of an 80 MeV H- Linac and a 1.6 GeV proton rapid cycling synchrotron (RCS). The beam injection is one of the most important issues for the CSNS accelerator complex. In this paper, the injection methods have been comprehensively studied, including phase space painting and H- stripping. By using the design scheme of the anti-correlated painting, the beam power has successfully reached 50 kW. However, some difficulties have been found in the higher power beam commissioning. In order to solve these key problems, flexibility in the CSNS design has been exploited to implement the correlated painting by using the rising current curve of the pulse power supply. The effectiveness of the new method has been verified in the simulation and beam commissioning. By using the new method, the beam power on the target has successfully risen to the design value. Secondly, according to the CSNS beam commissioning experience, based on the present design of the injection system, a new idea is proposed to perform both correlated and anti-correlated painting in the CSNS. By adopting an additional vertical shift bump generated by four additional alternating current (AC) magnets in the injection region, the local orbit can be manipulated to meet the requirement of correlated painting. The new method can not only perform the correlated painting, but also optimize the anti-correlated painting. The simulation study shows that the new method works well, and both correlated and anti-correlated painting methods are well performed.

The Child Langmuir Illusion

A Particle in Cell (PIC) model of ion beam extraction from a plasma provides strong evidence that the origin of the observed V(3/2) relationship between the applied extraction voltage and beam current is caused by meniscus shape and collimation on the extraction electrode, not by space charge limited extraction. Excellent correlation between the PIC model, a gun code and experiment is shown for 100 µA Ar+ beams. The s-shape in the emittance phase space is shown to be caused by the radial electric field created by the external edge of the plasma electrode.

Phase Space Graph Convolutional Network for Chaotic Time Series Learning

Phase Space Graph Convolutional Network for Chaotic Time Series Learning

Ghost dark energy in Tsallis and Barrow cosmology

According to the thermodynamics-gravity conjecture, any modification to the entropy expression leads to the modified cosmological field equations. Based on this, we investigate the cosmological consequences of the modified Friedmann equations when the entropy associated with the horizon is in the form of Tsallis/Barrow entropy and the dark energy (DE) component is in the form of ghost dark energy (GDE). We perform a dynamical system analysis and see that the Tisallis GDE(TGDE) and Barrow GDE(BGDE) can exhibit a correct phase space evolution for suitable range of the free parameters(0<Δ<1 for BGDE and β<3/2 for TGDE). We find that in BGDE and TGDE (with Q=3b2H(ρD+ρm)), there exist an early radiation dominated phase of expansion which is absent for GDE model in standard cosmology. It is worth mentioning that seeking an unstable phase of matter dominated in TGDE leads to a new constraint on β (<3/2). Using the resulting range of free parameters from latter step we find that both models are capable to explain the cosmic evolution from deceleration to an accelerated phase. We observe that increasing Δ (β) parameters leads to a delay in the cosmic phase transition. We conclude that BGDE and TGDE are viable cosmological models which predict a consistent phase transition of the cosmic expansion for suitable ranges of the parameters. We also calculate the squared sound speed for both models and find out that they are unstable against perturbations. Next, we proceed the statefinder analysis and see that both models are significantly distinct from ΛCDM as well as with respect to each other at past epochs, while both of them catch the standard cosmology at far future. We also explore the impacts of the non-extensive parameters on the density perturbations and find out that the pattern of density contrast (δ) evolution in TGDE/BGDE is distinct from GDE in standard cosmology.

From Snyder space-times to doubly κ-dependent Yang quantum phase spaces and their generalizations

We propose the doubly κ-dependent Yang quantum phase space which describes the generalization of D=4 Yang model. We postulate that such model is covariant under the generalized Born map, what permits to derive this new model from the earlier proposed κ-Snyder model. Our model of D=4 relativistic Yang quantum phase space depends on five deformation parameters which form two Born map-related dimensionful pairs: (M,R) specifying the standard Yang model and (κ,κ˜) characterizing the Born-dual κ-dependence of quantum space-time and quantum fourmomenta sectors; fifth parameter ρ is dimensionless and Born-selfdual. In the last section, we propose the Kaluza-Klein generalization of D=4 Yang model and the new quantum Yang models described algebraically by quantum-deformed oˆ(1,5) algebras.

Impact of Planetary Parameters on Water Clouds Microphysics

Potentially habitable exoplanets are targets of great interest for the James Webb Space Telescope and upcoming mission concepts such as the Habitable Worlds Observatory. Clouds strongly affect climate and habitability, but predicting their properties is difficult. In Global Climate Models (GCMs), especially those aiming at simulating Earth, cloud microphysics is often crudely approximated by assuming that all cloud particles have a single, constant size or a prescribed size distribution and that all clouds in a grid cell are identical. For exoplanets that range over a large phase space of planetary properties, this method could result in large errors. In this work, our goal is to determine how cloud microphysics on terrestrial exoplanets, whose condensable is mainly water vapor, depend on aerosol properties and planetary parameters such as surface pressure, surface gravity, and incident stellar radiation. We use the Community Aerosol and Radiation Model for Atmospheres as a 1D microphysical model to simulate the formation and evolution of clouds including the processes of nucleation, condensation, evaporation, coagulation, and vertical transfer. In these 1D idealized experiments, we find that the parameters that determine the macrophysical thermal structure of the atmospheres, including surface pressure and stellar flux, impact cloud radiative effect (CRE) most significantly. Parameters such as gravity and number density of aerosols working as cloud condensation nuclei affect the microphysical processes of cloud formation, including activation and vertical transfer. They also have a significant, though weaker effect on CRE. This work motivates the development of more accurate GCM cloud schemes and should aid in the interpretation of future observations.

Prompt extinction of bump-on-tail energy spectra for radiation belt electron phase space density

The bump-on-tail electron flux energy spectral distributions in the Earth's radiation belt are widely observed and have been proven to be generated by plasmaspheric hiss-led wave–particle interactions. Here, we report and discuss the prompt disappearance of them induced by the high solar wind dynamic pressure (Pdyn) combined with substorms on 12–13 March 2014. To remove the potential adiabatic influence, the electron fluxes are all converted to phase space density in the TS04 geomagnetic coordinate. Immediately after the Pdyn reached its maximum accompanied by the intensifying of substorms, the initial bump-on-tail spectra of the phase space density at L* (the third adiabatic invariant) from 3.8 to 4.4 were destroyed and rapidly smoothed to the exponential distributions, with a distinct ascent for lower energy electrons and a sharp decline for higher energy electrons. The balance line between the source and loss processes was located at μ = 50–200 MeV/G (μ is the first adiabatic invariant), which is sensitive to the spatial location. Further investigation of the temporal evolution of the phase space density along with L* and the plasma wave distributions provided evidence that the loss processes dominated at higher L* for electrons at μ ≳ 100 MeV/G, while the source processes became stronger at lower L* for electrons at μ ≲ 10 MeV/G. Our study provides new insight into the rapid change of the radiation belt electron spectral types in response to solar wind disturbances, which could be fully considered in further radiation belt particle modeling.