Phase Space View Research Articles

The phase-space view of non-local gravity cosmology

We consider non-local Integral Kernel Theories of Gravity in a homogeneous and isotropic universe background as a possible scenario to drive the cosmic history. In particular, we investigate the cosmological properties of a gravitational action containing the inverse d'Alembert operator of the Ricci scalar proposed to improve Einstein's gravity at both high and low-energy regimes. In particular, the dynamics of a physically motivated non-local exponential coupling is analyzed in detail by recasting the cosmological equations as an autonomous system of first-order differential equations with dimensionless variables. Consequently, we study the phase-space domain and its critical points, investigating their stability and main properties. In particular, saddle points and late-time cosmological attractors are discussed in terms of the free parameters of the model. Finally, we discuss the main physical consequences of our approach in view of dark energy behavior and the ΛCDM model.

A Phase-space View of Cold-gas Properties of Virgo Cluster Galaxies: Multiple Quenching Processes at Work?

Abstract We investigate the cold-gas properties of massive Virgo galaxies (>109 M ⊙) at <3R 200 (R 200 is the radius where the mean interior density is 200 times the critical density) on the projected phase-space diagram with the largest archival data set to date to understand the environmental effects on galaxy evolution in the Virgo cluster. We find lower H i and H2 mass fractions and higher star formation efficiencies (SFEs) from H i and H2 in the Virgo galaxies than in the field galaxies for matched stellar masses; the Virgo galaxies generally follow the field relationships between the offset from the main sequence of the star-forming galaxies [Δ(MS)] and the gas fractions and SFEs, to the slight offset to lower gas fractions or higher SFEs compared to field galaxies at Δ(MS) < 0; lower gas fractions in galaxies with smaller clustocentric distance and velocity; and lower gas fractions in the galaxies in the W cloud, a substructure of the Virgo cluster. Our results suggest the cold-gas properties of some Virgo galaxies are affected by their environment at least at 3R 200 maybe via strangulation and/or preprocesses, and H i and H2 in some galaxies are removed by ram pressure at <1.5R 200. Our data cannot rule out the possibility of other processes such as strangulation and galaxy harassment accounting for gas reduction in some galaxies at <1.5R 200. Future dedicated observations of a mass-limited complete sample are required for definitive conclusions.

Synthetic Gaia Surveys from the FIRE Cosmological Simulations of Milky Way-mass Galaxies

Abstract With Gaia Data Release 2, the astronomical community is entering a new era of multidimensional surveys of the Milky Way. This new phase-space view of our Galaxy demands new tools for comparing observations to simulations of Milky Way-mass galaxies in a cosmological context, to test the physics of both dark matter and galaxy formation. We present ananke, a framework for generating synthetic phase-space surveys from high-resolution baryonic simulations, and use it to generate a suite of synthetic surveys resembling Gaia DR2 in data structure, magnitude limits, and observational errors. We use three cosmological simulations of Milky Way-mass galaxies from the Latte suite of the Feedback In Realistic Environments project, which feature self-consistent clustering of star formation in dense molecular clouds and thin stellar/gaseous disks in live cosmological halos with satellite dwarf galaxies and stellar halos. We select three solar viewpoints from each simulation to generate nine synthetic Gaia-like surveys. We sample synthetic stars by assuming each star particle (of mass 7070 M ⊙) represents a single stellar population. At each viewpoint, we compute dust extinction from the simulated gas metallicity distribution and apply a simple error model to produce a synthetic Gaia-like survey that includes both observational properties and a pointer to the generating star particle. We provide the complete simulation snapshot at z = 0 for each simulated galaxy. We describe data access points, the data model, and plans for future upgrades. These synthetic surveys provide a tool for the scientific community to test analysis methods and interpret Gaia data.

Teaching Entropy from Phase Space Perspective: Connecting the Statistical and Thermodynamic Views Using a Simple One-Dimensional Model

Connecting the thermodynamic definition of entropy, dS = dQ/T (Clausius’s equation), with the statistical definition, S = kB ln Ω (Boltzmann’s equation), has been a persistent challenge in chemical education at the undergraduate level. Not meeting this challenge results in students taking away the meaning of entropy in a vague and subjective way as a measure of “disorder” or increase in number of configurations without any meaningful way of connecting it to heat. To address this challenge, we present a simple model that connects these two definitions. This approach relies centrally on emphasizing that the number of configurations, Ω, includes configurations in both real space and momentum space, collectively known as the phase space. Without including momentum configurations (i.e., how fast the particles move), connecting heat to entropy change is not possible. We construct the phase space for an ensemble of simple one-dimensional systems at equilibrium and show that delivery of heat dQ to the system results in an increase in the number of momentum configurations and consequently an expansion of the phase space area by dΩ. Relating dQ to dΩ is the linchpin between the two views and, when integrated, leads to Boltzmann’s equation. We further show that understanding entropy in terms of volume of phase space removes common ambiguities in teaching the subject. Among other examples, we show that understanding adiabatic compression, if treated using the usual approach, results in contradictions that are resolved if a phase space view is adopted. We propose this approach at the undergraduate physical chemistry and physics level.

The time delay between star formation quenching and morphological transformation of galaxies in clusters: a phase–space view of EDisCS

AbstractWe explore the possible effect of cluster environments on the structure and star formation histories of galaxies by analysing the projected phase–space (PPS) of intermediate-redshift clusters (0.4 ≤ z ≤ 0.8). HST I−band imaging data from the ESO Distant Cluster Survey (EDisCS) allow us to measure deviations of the galaxies’ light distributions from symmetric and smooth profiles using two parameters, Ares (‘asymmetry’) and RFF (residual flux fraction or ‘roughness’). Combining these structural parameters with age-sensitive spectral indicators ($H_{\delta \rm {A}}$, $H_{\gamma \rm {A}}$, and Dn4000), we establish that in all environments younger star-forming galaxies of all morphologies are ‘rougher’ and more asymmetric than older, more quiescent ones. Combining a subset of the EDisCS clusters, we construct a stacked PPS diagram and find a significant correlation between the position of the galaxies on the PPS and their stellar ages, irrespective of their morphology. We also observe an increasing fraction of galaxies with older stellar populations towards the cluster core, while the galaxies’ structural parameters (Ares and RFF) do not seem to segregate strongly with PPS. These results may imply that, under the possible influence of their immediate cluster environment, galaxies have their star formation suppressed earlier, while their structural transformation happens on a longer time-scale as they accumulate and age in the cluster cores.

Constant Temperature from Phase Space View of Ensembles of Excited States

Constant Temperature from Phase Space View of Ensembles of Excited States

Phase Space View of Ensembles of Excited States

Phase Space View of Ensembles of Excited States

Phase space view of quantum mechanical systems and Fisher information

Pennini and Plastino showed that the form of the Fisher information generated by the canonical distribution function reflects the intrinsic structure of classical mechanics. Now, a quantum mechanical generalization of the Pennini–Plastino theory is presented based on the thermodynamical transcription of the density functional theory. Comparing to the classical case, the phase-space Fisher information contains an extra term due to the position dependence of the temperature. However, for the special case of constant temperature, the expression derived bears resemblance to the classical one. A complete analogy to the classical case is demonstrated for the linear harmonic oscillator.

BUDHIES II: a phase-space view of H i gas stripping and star formation quenching in cluster galaxies

We investigate the effect of ram-pressure from the intracluster medium on the stripping of HI gas in galaxies in a massive, relaxed, X-ray bright, galaxy cluster at z=0.2 from the Blind Ultra Deep HI Environmental Survey (BUDHIES). We use cosmological simulations, and velocity vs. position phase-space diagrams to infer the orbital histories of the cluster galaxies. In particular, we embed a simple analytical description of ram-pressure stripping in the simulations to identify the regions in phase-space where galaxies are more likely to have been sufficiently stripped of their HI gas to fall below the detection limit of our survey. We find a striking agreement between the model predictions and the observed location of HI-detected and non-detected blue (late-type) galaxies in phase-space, strongly implying that ram-pressure plays a key role in the gas removal from galaxies, and that this can happen during their first infall into the cluster. However, we also find a significant number of gas-poor, red (early-type) galaxies in the infall region of the cluster that cannot easily be explained with our model of ram-pressure stripping alone. We discuss different possible additional mechanisms that could be at play, including the pre-processing of galaxies in their previous environment. Our results are strengthened by the distribution of galaxy colours (optical and UV) in phase-space, that suggests that after a (gas-rich) field galaxy falls into the cluster, it will lose its gas via ram-pressure stripping, and as it settles into the cluster, its star formation will decay until it is completely quenched. Finally, this work demonstrates the utility of phase-space diagrams to analyze the physical processes driving the evolution of cluster galaxies, in particular HI gas stripping.

Exact cosmological solutions from Hojman conservation quantities

We present a new approach to find exact solutions for cosmological models. By requiring the existence of a symmetry transformation vector for the equations of motion of the given cosmological model (without using either Lagrangian or Hamiltonian), one can find corresponding Hojman conserved quantities. With the help of these conserved quantities, the analysis of the cosmological model can be simplified. In the case of quintessence scalar–tensor models, we show that the Hojman conserved quantities exist for a wide range of V(ϕ)-potentials and allow to find exact solutions for the cosmic scale factor and the scalar field. Finally, we investigate the general cosmological behavior of solutions by adopting a phase-space view.

Scalar-tensor cosmologies with a potential in the general relativity limit: Phase space view

We consider the potential dominated era of Friedmann-Lemaitre-Robertson-Walker flat cosmological models in the framework of general Jordan frame scalar-tensor theories of gravity with arbitrary coupling functions, and focus upon the phase space of the scalar field. To study the regime suggested by the local weak field tests (i.e. close to the so-called limit of general relativity) we propose a nonlinear approximation scheme, solve for the phase trajectories, and provide a complete classification of possible phase portraits. We argue that the topology of trajectories in the nonlinear approximation is representative of those of the full system, and thus can tell for which scalar-tensor models general relativity functions as an attractor.

Moshinsky atom and density functional theory — A phase space view

The problem of two particles in a common harmonic oscillator potential interacting through harmonic oscillator forces is discussed in the Weyl–Wigner phase-space representation. The Wigner function of the system is an ordinary function of the phase-space constants of the motion. The density functional description of the system is touched upon. The functional that determines the confining harmonic potential from the particle density is also an ordinary function.

Propagation-invariant classification of sonar signals

Underwater sound waves can undergo frequency-dependent effects, such as dispersion and damping, that cause the wave to change as it propagates. In active sonar, these changes can be detrimental to automatic classification because the observed backscatter depends on the propagation environment and how far the wave has traveled. We aim to develop signal processing techniques to enable propagation-invariant classification. Toward that end, we have developed a local phase space view of propagation with dispersion and damping that is physically illuminating and suggests new features for classification. In this talk, we will present moment-like features extracted from the backscattered wave that are invariant to dispersion and damping, per mode. Environmental knowledge is not needed, beyond knowing the general form of the damping. Two forms of damping are considered. Classification results on simulations of the backscatter from different steel shells propagating in a Pekeris waveguide with damping and random variations will be presented. A dispersion-invariant matched-filter receiver approach will also be presented. [Supported by ONR Grant No. N00014-06-1-0009.]

The phase-space view of f(R) gravity

We study the geometry of the phase space of spatially flat Friedmann–Lemaitre–Robertson–Walker models in f(R) gravity, for a general form of the function f(R). The equilibrium points (de Sitter spaces) and their stability are discussed, and a comparison is made with the phase space of the equivalent scalar-tensor theory. New effective Lagrangians and Hamiltonians are also presented.

The phase-space view of conservation laws

The phase-space view of conservation laws

Quantum transport through cantori

We study the effect of classical cantori in quantum mechanics, extending previous results by several groups. We find that cantori form exponential barriers to quantum transport not only when Planck's constant exceeds the flux through the cantorus but also when it is smaller than the flux. The mechanism of localization in the two cases is different, and we describe the switch from dynamical localization to a mechanism we call "retunneling" as Planck's constant increases. We investigate the Planck's over 2pi dependence of the exponential decay for retunneling and find that the Planck's over 2pi(-0.66) coefficient found previously at criticality appears to hold also away from criticality provided piPlanck's over 2pi is large enough compared to the flux. Numerical evidence as well as an analytic argument are given. Our final contribution to this subject is a phase space view of cantori in quantum mechanics. We illustrate our results using the whisker map.

Orbiting Complex Formation in Na+/N2 Collisions: A Phase Space View

In this paper we investigate the dynamics of orbiting complex formation in Na+/N2 collisions using classical trajectories. We first demonstrate that the rigid-rotor approximation is appropriate for computing complex formation probabilities and lifetimes, as the NZ vibration is weakly coupled to the other degrees of freedom. For a collision ensemble with zero impact parameter and total angular momentum J = 0, complex formation probabilities exhibit oscillatory behavior with E. Several different representations of the phase space structure are compared, and it is found that the (yp,,) surface of section provides most insight into the complex formation dynamics. Oscillatory complex formation probabilities and the influence of potential anisotropy on complex formation rates are analyzed in terms of the (y,pv) section.

Decoupling surface analysis of classical irregular scattering and clarification of its icicle structure

Irregular scattering in molecular inelastic collision is analyzed classical mechanically by a novel method called ‘‘decoupling surface analysis.’’ Effective Hamiltonian of this analysis provides a phase space view of collision processes analogous to the Poincaré section of coupled-oscillator systems. In this phase space view irregular scattering occurs in a stochastic layer formed around separatrix connected to resonance structure of the effective Hamiltonian. This circumstance is parallel to that in the coupled-oscillator systems, in which stochastic motion is known to be connected to nonlinear resonance. The resonance structure in collision indicates trapping of classical trajectories in a certain dynamical well. The decoupling surface analysis suggests that the dynamical well is formed by a dip of stability exponents of trajectories as a function of time. By using a prototypical model exhibiting irregular scattering, a formal theoretical treatment is developed to analyze the structure of the fractal, termed icicle structure, observed in the plot of final vibrational action against the initial vibrational phase angle.

Vibration–rotation interaction in the rigid bender: A quantum mechanical phase space view

In this paper, we examine classical–quantum correspondence in a system with strong vibration–rotation interaction. We study the quantum mechanics of a two degree of freedom rigid bender Hamiltonian, previously considered in the context of classical mechanics by Ezra [Chem. Phys. Lett. 127, 492 (1986)] and by Frederick and McClelland [J. Chem. Phys. 84, 4347 (1986)]. Eigenfunctions and eigenvalues of this system are calculated by matrix diagonalization in a harmonic oscillator–rigid rotor basis. The quantum mechanics are compared with the classical mechanics by visualizing the quantum eigenstates on a surface of section, defined in terms of the coherent states of the rotor and oscillator. We find clear connections between eigenstate structure on the quantum surface of section and features of the classical phase space, such as nonlinear resonance, period doubling, and chaos.

Some properties of the Husimi function

The Husimi function provides a phase-space view of quantum systems. This paper considers a number of its properties, including ways in which it can be expanded in terms of basis functions. A spectral or ‘‘natural’’ expansion and an expansion analogous to the Carlson–Keller expansion in terms of coordinate-density momentum-density products are considered, as is a method for separating the angular dependence of the momentum. There is a set of functions in phase space having the same overlap properties as the initial orbital basis in terms of which the charge density matrix is expressed. A Husimi matrix is defined and a scalar product in the space containing such matrices as elements is introduced. The connection with the vector space of density matrices is examined. The Husimi function is related to the Husimi matrix in a way analogous to the relationship between a density matrix and a density, but there is a major difference in that this map can always be inverted. For a harmonic oscillator basis the phase space basis functions, in terms of which the Husimi function is expressed, span a linear space but do not provide a complete set; their products provide a linearly independent set that is complete. It is suggested that similar behavior can be expected for other basis sets.