Chaos and chaotic phase mixing in cuspy triaxial potentials

Subjecting a galactic potential to (possibly damped) nearly periodic, time-dependent variations can lead to large numbers of chaotic orbits experiencing systematic changes in energy, and the resulting chaotic phase mixing could play an important role in explaining such phenomena as violent relaxation. This paper focuses on the simplest case of spherically symmetric potentials subjected to strictly periodic driving with the aim of understanding precisely why orbits become chaotic and under what circumstances they will exhibit systematic changes in energy. Four unperturbed potentials V 0(r) were considered, each subjected to a time dependence of the form V(r, t) = V 0(r)(1 + m0 sin ωt). In each case, the orbits divide clearly into regular and chaotic, distinctions which appear absolute. In particular, transitions from regularity to chaos are seemingly impossible. Over finite time intervals, chaotic orbits subdivide into what can be termed ‘sticky’ chaotic orbits, which exhibit no large-scale secular changes in energy and remain trapped in the phase-space region where they started; and ‘wildly’ chaotic orbits, which do exhibit systematic drifts in energy as the orbits diffuse to different phase-space regions. This latter distinction is not absolute, transitions corresponding apparently to orbits penetrating a ‘leaky’ phase-space barrier. The three different orbit types can be identified simply in terms of the frequencies for which their Fourier spectra have the most power. An examination of the statistical properties of orbit ensembles as a function of driving frequency ω allows us to identify the specific resonances that determine orbital structure. Attention focuses also on how, for fixed amplitude m0, such quantities as the mean energy shift, the relative measure of chaotic orbits and the mean value of the largest Lyapunov exponent vary with driving frequency ω; and how, for fixed ω, the same quantities depend on m0.

The stability of the asteroid orbits has been studied by the method of surface of section. Families of simple symmetric periodic orbits of the asteroid and their stability have been computed and this served as a guide for the selection of the energy levels for the surface of section. In this way all possible cases for the structure of phase space have been obtained. It was found that the region in phase space around the resonant orbits at the resonances 1/3, 3/5, 5/7,.... is unstable, but small stability regions of doubly symmetric periodic orbits near the above resonances are also present. At the resonances 1/2, 2/3, 3/4, .... it was found that there exist two separate regions in phase space at about the same resonance 1/2, 2/3, 3/4,...., respectively, one being stable and the other unstable. At certain energy levels only the stable region appears. The above results are consistent with the observed distribution of the asteroids.

Quantification of surface acoustic wave induced chaotic mixing-flows in microfluidic wells

Whether thermal noise is colored or white is of fundamental importance. In conventional theory, thermal noise is usually treated as white noise so that there are no directional transportations in the asymmetrical systems without external inputs, since only the colored fluctuations with appropriate autocorrelation time length can lead to directional transportations in the asymmetrical systems. Here, on the basis of molecular dynamics simulations, we show that the autocorrelation time length of thermal noise in water is ~10 ps at room temperature, which indicates that thermal noise is not white in the molecular scale while thermal noise can be reasonably assumed as white in macro- and meso-scale systems. The autocorrelation time length of thermal noise is intrinsic, since the value is almost unchanged for different temperature coupling methods. Interestingly, the autocorrelation time of thermal noise is correlated with the lifetime of hydrogen bonds, suggesting that the finite autocorrelation time length of thermal noise mainly comes from the finite lifetime of the interactions between neighboring water molecules.

Abstract. Neural networks are able to approximate chaotic dynamical systems when provided with training data that cover all relevant regions of the system's phase space. However, many practical applications diverge from this idealized scenario. Here, we investigate the ability of feed-forward neural networks to (1) learn the behavior of dynamical systems from incomplete training data and (2) learn the influence of an external forcing on the dynamics. Climate science is a real-world example where these questions may be relevant: it is concerned with a non-stationary chaotic system subject to external forcing and whose behavior is known only through comparatively short data series. Our analysis is performed on the Lorenz63 and Lorenz95 models. We show that for the Lorenz63 system, neural networks trained on data covering only part of the system's phase space struggle to make skillful short-term forecasts in the regions excluded from the training. Additionally, when making long series of consecutive forecasts, the networks struggle to reproduce trajectories exploring regions beyond those seen in the training data, except for cases where only small parts are left out during training. We find this is due to the neural network learning a localized mapping for each region of phase space in the training data rather than a global mapping. This manifests itself in that parts of the networks learn only particular parts of the phase space. In contrast, for the Lorenz95 system the networks succeed in generalizing to new parts of the phase space not seen in the training data. We also find that the networks are able to learn the influence of an external forcing, but only when given relatively large ranges of the forcing in the training. These results point to potential limitations of feed-forward neural networks in generalizing a system's behavior given limited initial information. Much attention must therefore be given to designing appropriate train-test splits for real-world applications.

The squared Fourier transform of the optical absorption spectrum provides a very useful characterization of the intramolecular dynamics. In practice, most of the information content is in that portion of the transform whose magnitude is of the order of (1/N) of its value for time=0, where N is the number of eigenstates in the zero order nonstationary bright state which is optically accessed. If the highly resolved spectrum manifests inherent structures (‘‘clumps’’) at lower levels of resolution then each clump can be regarded, for the purpose of the analysis, as a bright state with its own survival probability. This offers a significant advantage. We discuss theoretically and provide computational examples how this can be implemented within a maximum entropy formalism. We determine both the density of the region in phase space sampled up to time t and its entropy. Analytically and computationally it is shown that the evolution in phase space is sequential. Also discussed is the structure of the Hamiltonian matrix which can give rise to a nested inherent spectra. It is argued that each time scale is characterized by its set of good constants of motion which decrease in number upon the transition to the next time regime.

Understanding the concept of dynamic aperture provides essential insights into nonlinear beam dynamics, beam losses, and the beam lifetime in circular particle accelerators. This comprehension is crucial for the functioning of modern hadron synchrotrons like the CERN Large Hadron Collider and the planning of future ones such as the Future Circular Collider. The dynamic aperture defines the extent of the region in phase space where the trajectories of charged particles are bounded over numerous revolutions, the actual number being defined by the physical application. Traditional methods for calculating the dynamic aperture depend on computationally demanding numerical simulations, which require tracking over multiple turns of numerous initial conditions appropriately distributed in phase space. Prior research has shown the efficiency of a multilayer perceptron network in forecasting the dynamic aperture of the CERN Large Hadron Collider ring, achieving a remarkable speed-up of up to 200-fold compared to standard numerical tracking tools. Building on recent advancements, we conducted a comparative study of various deep learning networks based on BERT, DenseNet, ResNet and VGG architectures. The results demonstrate substantial enhancements in the prediction of the dynamic aperture, marking a significant advancement in the development of more precise and efficient surrogate models of beam dynamics.

In this work, we present the first full NLO predictions for the process pp → μ+νμe+νejj at the LHC in a typical tri-boson phase space. The NLO corrections reach 50% at the level of the fiducial cross section and have a very different hierarchy with respect to vector-boson-scattering phase spaces. By comparing the cross section of the full off-shell process with the sum of contributing on-shell electroweak-boson production subchannels, we find that the process is dominated by WWW and WH production, while vector-boson-scattering topologies still play a non-negligible role. In addition, NLO QCD predictions matched to parton shower which are supplemented by approximate electroweak corrections are provided. For the fiducial cross section, the electroweak corrections turn out to be small but the QCD corrections reach 47%. For the inclusive cross section, matching to parton shower affects the predictions by 7%. However, for differential distributions corrections due to the parton shower can be much more sizeable, depending on the region of phase space.

The lifetime distribution of energized molecules is examined within the framework of classical description of the molecular dynamics and in the connection with model theories of unimolecular reactions. If the reacting molecule represents an ergodic system, the lifetime distribution function of the molecules represented immediately after activation by the points in the phase space region corresponding to the internal energy ϵ to ϵ + δϵ (RRKM model) may be introduced; in a non‐ergodic case, the distribution function may be defined in that part of this phase space region, the point of which satisfy a supplementary condition of decomposition (Slater's model). It is shown that allowing a rapid intramolecular redistribution of internal energy of the molecule in the originally harmonic model, we obtain lifetime densities approximating the exponential (random) density. Some general features of the lifetime distribution in an ensemble of molecules with a uniform phase space density after activation are deduced from the qualitative analysis of the phase space trajectories.

The charged particle multiplicity distribution has been studied for non-single-diffractive π+ p andpp collisions at\(\sqrt s = 22\) GeV, for full phase space as well as for intervals in rapidity, azimuthal angle and transverse momentum. In general, the multiplicity distribution is well described by a negative binomial. From comparison of the distribution for negative or positive particles to that of all charged particles, cascading is favoured as an interpretation over stimulated emission. Interesting consequences follow from a comparison of our results to those at collider energies and toe + e − data at comparable energy. Furthermore, evidence is given that the multiplicity distribution is not exactly of negative binomial type in every (connected or disconnected) phase space region.

The first computation of Higgs production in association with three jets at NLO in QCD has recently been performed using the effective theory, where the top quark is treated as an infinitely heavy particle and integrated out. This approach is restricted to the regions in phase space where the typical scales are not larger than the top quark mass. Here we investigate this statement at a quantitative level by calculating the leading-order contributions to the production of a Standard Model Higgs boson in association with up to three jets taking full top-quark and bottom-quark mass dependence into account. We find that the transverse momentum of the hardest particle or jet plays a key role in the breakdown of the effective theory predictions, and that discrepancies can easily reach an order of magnitude for transverse momenta of about 1 TeV. The impact of bottom-quark loops is found to be visible in the small transverse momentum region, leading to corrections of up to 5 percent. We further study the impact of mass corrections when VBF selection cuts are applied and when the center-of-mass energy is increased to 100 TeV.

The long-term stability of the evolution of two-planet systems is considered by using the general three body problem (GTBP). Our study is focused on the stability of systems with adjacent orbits when at least one of them is highly eccentric. In these cases, in order for close encounters, which destabilize the planetary systems, to be avoided, phase protection mechanisms should be considered. Additionally, since the GTBP is a non-integrable system, chaos may also cause the destabilization of the system after a long time interval. By computing dynamical maps, based on Fast Lyapunov Indicator, we reveal regions in phase space with stable orbits even for very high eccentricities (e>0.5). Such regions are present in mean motion resonances (MMR). We can determine the position of the exact MMR through the computation of families of periodic orbits in a rotating frame. Elliptic periodic orbits are associated with the presence of apsidal corotation resonances (ACR). When such solutions are stable, they are associated with neighbouring domains of initial conditions that provide long-term stability. We apply our methodology so that the evolution of planetary systems of highly eccentric orbits is assigned to the existence of such stable domains. Particularly, we study the orbital evolution of the extrasolar systems HD 82943, HD 3651, HD 7449, HD 89744 and HD 102272 and discuss the consistency between the orbital elements provided by the observations and the dynamical stability.

Synchronization of coupled equations of Morris–Lecar model

Enhanced backscatter effects have long been predicted theoretically and experimentally demonstrated. The reciprocity of a turbulent channel generates a group of paired rays with identical trajectory and phase information that lea ds to a region in phase space with double intensity and scintillation index. Though simulation work based on phase screen models ha s demonstrated the existence of the phenomenon, few experimental results have been published describing its characteristics , and possible applications of the enhanced backscatter phenomenon are still unclear. W ith the development of commerciall y available high powered lasers and advanced cameras with high frame rates, we have successfully captured the enhanced backscatter effect s from different reflection surfaces. In addition to static observations , we have also tilted and pre -distorted the transmitted beam at various frequencies to track the dynamic properties of the enhanced backscatter phenomenon to verify its possible applic ation in guidance and beam and image correction through atmospheric turbulence. In this paper, experiment al results will be described, and discussions on the principle and applications of the phenomenon will be included. Enhanced backscatter effects are be st observed in certain levels of turbulence (C

We derive necessary conditions for finite systems to exhibit two locally stable states with respect to fluctuations in the microcanonical and canonical ensembles. The requirements are, respectively, that there be two inflections in the logarithm of the probability distribution of the energy or the number of particles in the canonical or grand canonical ensembles. The thermodynamics associated with the whole phase space may be related to quantities averaged over regions of phase space and are consistent with van Hove's theorem.