Phase Space Structure Research Articles

Non-Gaussian statistics from the generalized uncertainty principle

In many quantum gravity theories, there is the emergence of a generalized uncertainty principle (GUP), implying a minimal length of the order of the Planck length. From the statistical mechanics point of view, this prescription enters into the phase space structure by modifying the elementary cell volume, which becomes momentum-dependent. In this letter, it is pointed out that if one assumes that the total phase space volume is not affected by the minimum length prescription, the statistics that maximize the entropy are non-Gaussian but exhibit a quadratic correction over Gaussian statistics. The departure from Gaussian statistics is significant for high energies. To substantiate our point, we apply these statistics to the Unruh effect and the Jeans gravitational instability and show that—in these cases—non-Gaussian statistics produce the same effect as the GUP and capture the underlying physics behind it.

Dynamical description of a quintom cosmological model nonminimally coupled with gravity

In this work we have studied a cosmological model based on a quintom dark energy model non-minimally coupled with gravity, endowed with a specific potential energy of the exponential squared type. For this specific type of potential energy and non-minimal coupling, the dynamical properties are analyzed and the corresponding cosmological effects are discussed. Considering the linear stability method, we have investigated the dynamical properties of the phase space structure, determining the physically acceptable solutions. The analysis showed that in this model we can have various cosmological epochs, corresponding to radiation, matter domination, and de Sitter eras. Each solution is investigated from a physical and cosmological point of view, obtaining possible constraints of the model’s parameters. In principle the present cosmological setup represent a possible viable scalar tensor theory which can explain various transitional effects related to the behavior of the dark energy equation of state and the evolution of the Universe at large scales.

Quantitative Comparison of Predictabilities of Warm and Cold Events Using the Backward Nonlinear Local Lyapunov Exponent Method

The backward nonlinear local Lyapunov exponent method (BNLLE) is applied to quantify the predictability of warm and cold events in the Lorenz model. Results show that the maximum prediction lead times of warm and cold events present obvious layered structures in phase space. The maximum prediction lead times of each warm (cold) event on individual circles concentric with the distribution of warm (cold) regime events are roughly the same, whereas the maximum prediction lead time of events on other circles are different. Statistical results show that warm events are more predictable than cold events.

Transport controlled by Poincaré orbit topology in a driven inhomogeneous lattice gas

In periodic quantum systems which are both homogeneously tilted and driven, the interplay between drive and Bloch oscillations controls transport dynamics. Using a quantum gas in a modulated optical lattice, we show experimentally that inhomogeneity of the applied force leads to a rich new variety of dynamical behaviors controlled by the drive phase, from self-parametrically-modulated Bloch epicycles to adaptive driving of transport against a force gradient to modulation-enhanced monopole modes. Matching experimental observations to fit-parameter-free numerical predictions of time-dependent band theory, we show that these phenomena can be quantitatively understood as manifestations of an underlying inhomogeneity-induced phase space structure, in which topological classification of stroboscopic Poincar\'e orbits controls the transport dynamics.

Self-acceleration and energy channeling in the saturation of the ion-sound instability in a bounded plasma

A novel regime of the saturation of the Pierce-type ion-sound instability in a bounded ion-beam-plasma system is revealed in 1D particle-in-cell simulations. It is found that the saturation of the instability is mediated by the oscillating virtual anode potential structure. The periodically oscillating potential barrier separates the incoming beam ions into two groups. One component forms a supersonic beam, which is accelerated to an energy exceeding the energy of the initial cold ion beam. The other component is organized as a self-consistent phase space structure of trapped ions with a wide energy spread—the ion hole. The effective temperature (energy spread) of the ions trapped in the hole is lower than the initial beam energy. In the final stage, the ion hole expands over the whole system length.

Phase-space elementary information content of confined Dirac spinors

Reporting on the Wigner formalism for describing Dirac spinor structures through a covariant phase-space formulation, the quantum information quantifiers for purity and mutual information involving spin–parity (discrete) and position–momentum (continuous) degrees of freedom are consistently obtained. For Dirac spinor Wigner operators decomposed into Poincaré classes of $$\mathrm{SU}(2)\otimes \mathrm{SU}(2)$$ spinor couplings, a definitive expression for quantum purity is identified in a twofold way: firstly in terms of phase-space positively defined quantities and secondly in terms of the spin–parity traced-out-associated density matrix in the position coordinate representation, both derived from the original Lorentz-covariant phase-space Wigner representation. Naturally, such a structure supports the computation of relative (linear) entropies, respectively, associated with discrete (spin–parity) and continuous (position–momentum) degrees of freedom. The obtained theoretical tools are used for quantifying (relative) purities, mutual information as well as, by means of the quantum concurrence quantifier, the spin–parity quantum entanglement, for a charged fermion trapped by a uniform magnetic field which, by the way, has the phase-space structure completely described in terms of Laguerre polynomials associated with the quantized Landau levels. Our results can be read as the first step in the systematic computation of the elementary information content of Dirac-like systems exhibiting some kind of confining behavior.

Phase space analysis of the dynamics on a potential energy surface with an entrance channel and two potential wells.

In this paper, we unveil the geometrical template of phase space structures that governs transport in a Hamiltonian system described by a potential energy surface with an entrance/exit channel and two wells separated by an index-1 saddle. For the analysis of the nonlinear dynamics mechanisms, we apply the method of Lagrangian descriptors, a trajectory-based scalar diagnostic tool that is capable of providing a detailed phase space tomography of the interplay between the invariant manifolds of the system. Our analysis reveals that the stable and unstable manifolds of the two families of unstable periodic orbits (UPOs) that exist in the regions of the wells are responsible for controlling access to the potential wells of the trajectories that enter the system through the entrance/exit channel. We demonstrate that the heteroclinic and homoclinic connections that arise in the system between the manifolds of the families of UPOs characterize the branching ratio, a relevant quantity used to measure product distributions in chemical reaction dynamics.

Formalizing slow-roll inflation in scalar–tensor theories of gravitation

The viability of slow-roll approximation is examined by considering the structure of phase spaces in scalar-tensor theories of gravitation and the analysis is exemplified with a nonminimally coupled scalar field to the spacetime curvature. The slow-roll field equations are obtained in the Jordan frame in two ways: first using the direct generalization of the slow-roll conditions in the minimal coupling case to nonminimal one, and second, conformal transforming the slow-roll field equations in the Einstein frame to the Jordan frame and then applying the generalized slow-roll conditions. Two inflationary models governed by the potentials $V(\phi) \propto \phi^2$ and $V(\phi) \propto \phi^4$ are considered to compare the outcomes of two methods based on the analysis of $n_s$ and $r$ values in the light of recent observational data.

Perturbative calculation of energy levels for the Dirac equation with generalized momenta

We analyze a modified Dirac equation based on a noncommutative structure in phase space originating from a generalized uncertainty principle with a minimum length. The noncommutative structure induces generalized momenta and contributions to the energy levels of the standard Dirac equation. Applying techniques of perturbation theory, we find the lowest-order corrections to the energy levels and eigenfunctions of the Dirac equation in three dimensions for a spherically symmetric linear potential and for a square-well times triangular potential along one spatial dimension. We find that the corrections due to the noncommutative contributions may be of the same order as the relativistic ones, leading to an upper bound on the parameter fixing the minimum length induced by the generalized uncertainty principle.

Liouvillian integrability of the three-dimensional generalized Hénon–Heiles Hamiltonian

In this paper, we report about three cases of integrability in sense of Liouville for three-dimensional generalized Henon–Heiles Hamiltonian. This also allow to get explicitly integrals of motions for each case. On the other hand, this paper investigates the phase space structure numerically with Poincare surfaces of section and 3D projections which allow to verify that the analytical results are in agreement with the computations.

Effects of natural selection by fertility on the evolution of the dynamic modes of population number: bistability and multistability

A discrete-time model of a structured population dynamics considering both density-dependent regulation and natural selection is studied. In the model, the dynamics of two population groups corresponding to different development stages is described, and birth rate limitation is considered. Fertility is assumed to change during microevolution. The stability loss of nontrivial fixed points was shown to realise according to the Neimark–Sacker scenario and the Feigenbaum one. Bifurcations, dynamic modes and possible shifting for the proposed model are explored. Bistability and multistability are also revealed. The phase space structure of bistability and multistability areas in which a variation in population sizes or population genotype compositions can lead to shift in the dynamic modes is examined using attraction basins. The multistability of genetic structure dynamics is also investigated. The presence of multistability fundamentally increases the quantity and variety of possible evolutionary scenarios and makes them dependent on both parameter values and initial conditions. In particular, shifting dynamic mode in a population can occur because of an increase in an individual’s reproductive potential during natural evolution.

Quasitopological electromagnetism and black holes

In this paper we extend the quasitopological electromagnetism, recently introduced by H.-S. Liu et al. [arXiv:1907.10876], to arbitrary dimensions by introducing a fundamental $p$-form field. This allows us to construct new dyonic black hole solutions in odd dimensions, as well as regular $D$-dimensional black holes and solitons. The three-dimensional system consists of a Maxwell field interacting with a scalar field, leading to a deformation of the Ba\~nados-Teitelboim-Zanelli black hole. We present the general formulas defining the black hole solutions in arbitrary dimensions in Lovelock theory and explore the thermal properties of the asymptotically anti-de Sitter black holes in the gravitational framework of general relativity. In five dimensions, the latter black holes possess a rich phase space structure in the canonical ensemble, giving rise to as many as five different black hole phases at a fixed temperature, for a given range of the parameters.

Understanding the origin of extreme events in El Niño southern oscillation.

We investigate a low-dimensional slow-fast model to understand the dynamical origin of El Niño southern oscillation. A close inspection of the system dynamics using several bifurcation plots reveals that a sudden large expansion of the attractor occurs at a critical system parameter via a type of interior crisis. This interior crisis evolves through merging of a cascade of period-doubling and period-adding bifurcations that leads to the origin of occasional amplitude-modulated extremely large events. More categorically, a situation similar to homoclinic chaos arises near the critical point; however, atypical global instability evolves as a channellike structure in phase space of the system that modulates variability of amplitude and return time of the occasional large events and makes a difference from the homoclinic chaos. The slow-fast timescale of the low-dimensional model plays an important role on the onset of occasional extremely large events. Such extreme events are characterized by their heights when they exceed a threshold level measured by a mean-excess function. The probability density of events' height displays multimodal distribution with an upper-bounded tail. We identify the dependence structure of interevent intervals to understand the predictability of return time of such extreme events using autoregressive integrated moving average model and box-plot analysis.

Bifurcation analysis of nonlinear Hamiltonian dynamics in the Fermilab Integrable Optics Test Accelerator

The Integrable Optics Test Accelerator (IOTA) is a novel storage ring at Fermi National Accelerator Laboratory designed (in part) to investigate the dynamics of beams in the presence of highly nonlinear transverse focusing fields that generate integrable single-particle motion with a large spread in the intrinsic betatron tunes. We describe how contemporary geometrical methods from the theory of integrable Hamiltonian systems may be used to locate all critical separatrixlike structures in the 4D transverse phase space, and to construct a complete analysis of the dynamical bifurcations of the system. Application of these techniques results in a global picture of the nominal on-energy transverse dynamics, revealing a rich diversity of accessible dynamical behavior. Similar techniques may be applied to future facilities that exploit the concept of nonlinear integrable optics.

Clustering stellar pairs to detect extended stellar structures

AbstractGaia data allows for search for extended stellar structures in phase (coordinates plus velocities) space. We describe a method of using DBSCAN clustering algorithm, which is used to group closely-packed-together data points, to a list of preliminary selected pairs of stars, with parameters expected to be found within stellar streams and comoving groups: loose structures in which stars are not gravitationally bound, but do share motion and evolutionary properties. To test our approach, we construct a model population of background stars, and use pair-constructing and clustering algorithms on it. Results show that transitioning to a list of pairs sharply reveals structures not presented in background model, which then become more apparent targets in coordinate-velocity phase space for DBSCAN algorithm thanks to now increased relative density of the extended stellar structure.

Constrained dynamics: generalized Lie symmetries, singular Lagrangians, and the passage to Hamiltonian mechanics

Guided by the symmetries of the Euler–Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation—called second-order, Euler–Lagrange vector fields (SOELVFs)—with integral flows that have this symmetry are determined. Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always possible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to the Hamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space needed for their evolution. In particular, the primary Hamiltonian constraints can be constructed from vectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show that the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this way equivalent.

The phase space mechanism for selectivity in a symmetric potential energy surface with a post-transition-state bifurcation

Chemical selectivity is a phenomenon displayed by potential energy surfaces (PES) that is relevant for many organic chemical reactions whose PES feature a valley-ridge inflection point in the region between two sequential index-1 saddles. In this letter we describe the underlying dynamical phase space mechanism that qualitatively determines the product distributions resulting from bifurcating reaction pathways. We show that selectivity is a consequence of the heteroclinic and homoclinic connections established between the invariant manifolds of the families of unstable periodic orbits (UPOs) present in the system. The geometry of the homoclinic and heteroclininc connections is determined using the technique of Lagrangian descriptors, a trajectory-based scalar technique with the capability of unveiling the geometrical template of phase space structures that characterizes transport.

Revealing roaming on the double Morse potential energy surface with Lagrangian descriptors

In this paper, we analyse the phase space structure of the roaming dynamics in a 2 degree of freedom potential energy surface consisting of two identical planar Morse potentials separated by a distance. This potential energy surface was previously studied in Carpenter B K et al (2018 Regul. Chaotic Dyn. 23 60–79), and it has two potential wells surrounded by an unbounded flat region containing no critical points. We study the phase space mechanism for the transference between the wells using the method of Lagrangian descriptors.

Exploring isomerization dynamics on a potential energy surface with an index-2 saddle using lagrangian descriptors

In this paper we explore the phase space structures governing isomerization dynamics on a potential energy surface with four wells and an index-2 saddle. For this model, we analyze the influence that coupling both degrees of freedom of the system and breaking the symmetry of the problem have on the geometrical template of phase space structures that characterizes reaction. To achieve this goal we apply the method of Lagrangian descriptors, a technique with the capability of unveiling the key invariant manifolds that determine transport processes in nonlinear dynamical systems. This approach reveals with extraordinary detail the intricate geometry of the isomerization routes interconnecting the different potential wells, and provides us with valuable information to distinguish between initial conditions that undergo sequential and concerted isomerization.

Open-string non-associativity in an R-flux background

We derive the commutation relations for open-string coordinates on D-branes in non-geometric background spaces. Starting from D0-branes on a three-dimensional torus with H -flux, we show that open strings with end points on D3-branes in a three-dimensional R-flux background exhibit a non-associative phase-space algebra, which is similar to the non-associative R-flux algebra of closed strings. Therefore, the effective open-string gauge theory on the D3-branes is expected to be a non-associative gauge theory. We also point out differences between the non-associative phase space structure of open and closed strings in non-geometric backgrounds, which are related to the different structure of the world-sheet commutators of open and closed strings.