Phase-space Distribution Function Research Articles

On the conundrum of deriving exact solutions from approximate master equations

We derive the exact time-evolution for a general quantum system under the influence of a bosonic bath that causes pure phase noise and demonstrate that for a Gaussian initial state of the bath, the exact result can be obtained also within a perturbative time-local master equation approach already in second order of the system–bath coupling strength. We reveal that this equivalence holds if the initial state of the bath can be mapped to a Gaussian phase-space distribution function. Moreover, we discuss the relation to the standard Bloch–Redfield approach.

A Unifying Framework for Self‐consistent Gravitational Lensing and Stellar Dynamics Analyses of Early‐Type Galaxies

Gravitational lensing and stellar dynamics are two independent methods, based solely on gravity, to study the mass distributions of galaxies. Both methods suffer from degeneracies, however, that are difficult to break. In this paper, we present a new framework that self-consistently unifies gravitational lensing and stellar dynamics. This approach breaks some of classical degeneracies that have limited their individual usage, in particular in the study of high-redshift galaxies. The methodology is based on the premise that, for any given galaxy potential, the mapping of both the unknown lensed source brightness distribution and the stellar phase-space distribution function on to the photometric and kinematic observables, can be cast as a single set of coupled linear equations. This set of linear equations is solved, maximizing the likelihood penalty function. The evidence penalty function, as derived from Bayesian statistics, subsequently allows the best potential-model parameters to be found and potential-model families, or other model assumptions (e.g. PSF), to be quantitatively ranked. We have implemented a fast algorithm that solves for the maximum-likelihood pixelized lensed source brightness distribution and the two-integral stellar phase-space distribution function f(E, L_z), assuming axisymmetric potentials. To make the method practical, we have devised a new Monte-Carlo approach to Schwarzschild's orbital superposition method, based on the superposition of two-integral (E and L_z) toroidal components, to find the maximum-likelihood two-integral distribution function in a matter of seconds in any axisymmetric potential. The non-linear parameters of the potential are subsequently found through a hybrid MCMC and Simplex optimization of the evidence. (Abridged)

Deexcitation of high-Rydberg-state atoms with a chirped train of half-cycle pulses

Encouraged by the experiments on production of antihydrogen atoms in high Rydberg states we have calculated the effect of deexcitation towards lower states by a chirped train of identical unidirectional half-cycle pulses. The calculations exploit both the one-dimensional and impulse approximations providing convenient analytical formulas for the Rydberg-to-Rydberg transition amplitudes. The calculated deexcitation is shown in terms of the mean value of localization of the Rydberg wave packet in the coordinate space, the Rydberg-state population distribution, the Husimi phase-space distribution function, and the probability density distribution, each of these measures vs the length of the applied train of half-cycle pulses. The results for chirped trains are compared with those for periodic trains and examples of higher deexcitation efficiency of the chirped trains are given.

Theory of thermostatted inhomogeneous granular fluids: A self-consistent density functional description

The authors present a study of the nonequilibrium statistical properties of a one dimensional hard-rod fluid dissipating energy via inelastic collisions and subject to the action of a Gaussian heat bath, simulating an external driving mechanism. They show that the description of the fluid based on the one-particle phase-space reduced distribution function, in principle necessary because of the presence of velocity dependent collisional dissipation, can be contracted to a simpler description in configurational space. Indeed, by means of a multiple-time-scale method the authors derive a self-consistent governing equation for the particle density distribution function. This equation is similar to the dynamic density functional equation employed in the study of colloids, but contains additional terms taking into account the inelastic nature of the fluid. Such terms cannot be derived from a Liapunov generating functional and contribute not only to the relaxational properties, but also to the nonequilibrium steady state properties. A validation of the theory against molecular dynamics simulations is presented in a series of cases, and good agreement is found.

A deterministic particle method for the Vlasov–Fokker–Planck equation in one dimension

The Vlasov–Fokker–Planck equation is a model for a collisional, electrostatic plasma. The approximation of this equation in one spatial dimension is studied. The equation under consideration is linear in that the electric field is given as a known function that is not internally consistent with the phase space distribution function. The approximation method applied is the deterministic particle method described in Wollman and Ozizmir [Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in one dimension, J. Comput. Phys. 202 (2005) 602–644]. For the present linear problem an analysis of the stability and convergence of the numerical method is carried out. In addition, computations are done that verify the convergence of the numerical solution. It is also shown that the long term asymptotics of the computed solution is in agreement with the steady state solution derived in Bouchut and Dolbeault [On long time asymptotics of the Vlasov–Fokker–Planck equation and of the Vlasov–Poisson–Fokker–Planck system with coulombic and Newtonian potentials, Differential Integral Equations 8(3) (1995) 487–514].

The Quantum-Classical Comparison of the Arrival-Time Distribution Through the Probability Current

We consider the arrival time distribution defined through the quantum probability current for a Gaussian wave packet representing free particles in quantum mechanics in order to explore the issue of the classical limit of arrival time. We formulate the classical analogue of the arrival time distribution for an ensemble of free particles represented by a phase space distribution function evolving under the classical Liouville's equation. The classical probability current so constructed matches with the quantum probability current in the limit of minimum uncertainty. Further, it is possible to show in general that smooth transitions from the quantum mechanical probability current and the mean arrival time to their respective classical values are obtained in the limit of large mass of the particles.

Mid-plane sedimentation of large solid bodies in turbulent protoplanetary discs

We study the vertical settling of solid bodies in a turbulent protoplanetary disc. We consider the situation when the coupling to the gas is weak or equivalently when the particle stopping time τ st due to friction with the gas is long compared to the orbital time-scale Ω -1 . An analytical model, which takes into account the stochastic nature of the sedimentation process using a Fokker-Planck equation for the particle distribution function in phase space, is used to obtain the vertical scaleheight of the solid layer as a function of the vertical component of the turbulent gas velocity correlation function and the particle stopping time. This is found to be of the same form as the relation obtained for strongly coupled particles in previous work. We compare the predictions of this model with results obtained from local shearing box magnetohydrodynamic simulations of solid particles embedded in a vertically stratified disc in which there is turbulence driven by the magnetorotational instability. We find that the ratio of the dust disc thickness to the gas disc thickness satisfies H d /H = 0.08 (Ωτ st ) -1/2 , which is in very good agreement with the analytical model. By discussing the conditions for gravitational instability in the outer regions of protoplanetary discs in which there is a similar level of turbulence, we find that bodies in the size range 50-600 m can aggregate to form Kuiper belt-like objects with characteristic radii ranging from tens to hundreds of kilometres.

Coupling, degeneracy breaking and isolation of Weibel modes in relativistic plasmas: II. Specific examples

Recently, a general proof was given (Tautz et al 2006 J. Phys. A: Math. Gen. 39 13831) that for an asymmetric relativistic particle phase-space distribution function, and in the absence of a homogeneous background magnetic field, any unstable linear Weibel modes are isolated, i.e. restricted to discrete wavenumbers. In this paper, for a specific distribution function consisting of mono-energetic counterstreaming electron and positron beams, growth rates and associated wavenumbers for the isolated modes are calculated, proving the existence of discrete values for unstable wavenumbers. Furthermore, electrostatic and electromagnetic Weibel modes are investigated for monoenergetic counterstreaming plasmas, yielding constraints to the momentum components that have to be fulfilled in order to have unstable wave modes.

Coupling, degeneracy breaking and isolation of Weibel modes in relativistic plasmas: I. General theory

A general proof is given that for an asymmetric particle phase-space distribution function, and in the absence of a homogeneous background magnetic field, any unstable linear Weibel modes are isolated, i.e., restricted to discrete wavenumbers. Starting from the linearized relativistic Vlasov equation it is shown that, unless the asymmetry in the distribution function is precisely zero, the broad ranges of unstable wavenumbers occurring for symmetric distribution functions are reduced to discrete, isolated wavenumbers for which unstable modes can exist. For asymmetric plasmas, electrostatic and electromagnetic wave modes are coupled to each other and the degeneracy of the two electromagnetic wave modes (that holds for symmetric distributions) is therefore broken.

2D Fokker-Planck models of rotating clusters

Globular clusters rotate significantly, and with the increasing amount of detailed morphological and kinematical data obtained in recent years on galactic globular clusters many interesting features show up. We show how our theoretical evolutionary models of rotating clusters can be used to obtain fits, which at least properly model the overall rotation and its implied kinematics in full 2D detail (dispersions, rotation velocities). Our simplified equal mass axisymmetric rotating model provides detailed two-dimensional kinematical and morphological data for star clusters. The degree of rotation is not dominant in energy, but also non-negligible for the phase-space distribution function, shape and kinematics of clusters. Therefore, the models are well applicable for galactic globular clusters. Since previously published papers on that matter by us made it difficult to do detailed comparisons with observations, we provide a much more comprehensive and easy-to-use set of data here, which uses as entries dynamical age and flattening of observed cluster and then offers a limited range of applicable models in full detail. The method, data structure and some exemplary comparison with observations are presented. Future work will improve modelling and data base to take a central black hole, a mass spectrum and stellar evolution into account.

On the law of increase of entropy for non-equilibrium systems

Under the assumption of a smooth full phase-space distribution function we prove that the non-equilibriumentropy S which is considered as a functional of the distribution vector for anN-bodysystem possesses a lower bound and therefore cannot decrease. We also compute the rate of changeof S, ∂S/∂t, showing that this is non-negative and having a global minimum at equilibrium. As anapplication we obtain a generalization of the BGK (Bhatnagar–Gross–Krook) relaxationmodel.

A New Mass Modelling Trick

The estimation of the distribution of the total (luminous and dark) mass in early type systems is hard! Even for the lucky few systems for which kinematic information is available, its implementation is mired in problems, given uncertainties about the assumptions that enter the calculations; the most critical of such assumptions involve considerations of the system geometry and the shape of its velocity ellipsoid. This work offers an independent means of getting to the mass distributions of early type galaxies, without relying directly on the phase space distribution function. The methodology is based upon the well established idea that in elliptical galaxies, the largest variations in normalised velocity dispersion profiles occur typically at R < 0.5Re (Re≡ half-light radius) and at R ≥ 2Re.

Structure and adiabatic compression of dark matter halos: Simple analytic model

A simple analytical model for describing inner parts of dark matter halo is considered. It is assumed that dark matter density is power-law. The model deals with dark matter distribution function in phase space of adiabatic invariants (radial action and angular momentum). Two variants are considered for the angular part of the distribution function: narrow and broad distribution. The model allows to describe explicitly the process of adiabatic contraction of halo due to change of gravitational potential caused by condensation of baryonic matter in the centre. The modification of dark matter density in the centre is calculated, and is it shown that the standard algorithm of adiabatic contraction calculation overestimates the compressed halo density, especially in the case of strong radial anisotropy.

Angular Momentum Transfer in Dark Matter Halos: Erasing the Cusp

We propose that angular momentum transfer from the baryons to the Dark Matter (DM) during the early stages of galaxy formation can flatten the halo inner density profile and modify the halo dynamics. We compute the phase-space distribution function of DM halos, that corresponds to the density and anisotropy profiles obtained from N-body simulations in the concordance cosmology. We then describe an injection of angular momentum into the halo by modifying the distribution function, and show that the system evolves into a new equilibrium configuration; the latter features a constant central density and a tangentially-dominated anisotropy profile in the inner regions, while the structure is nearly unchanged beyond 10% of the virial radius. Then we propose a toy model to account for such a halo evolution, based on the angular momentum exchange due to dynamical friction; at the epoch of galaxy formation this is efficiently exerted by the DM onto the gas clouds spiralling down the potential well. The comparison between the angular momentum profile gained by the halo through dynamical friction and that provided by the perturbed distribution function reveals a surprising similarity, hinting at the reliability of the process.

Fine structure in the phase space distribution of nearby subdwarfs

We analysed the fine structure of the phase space distribution function of nearby subdwarfs using data extracted from various catalogues. Applying a new search strategy based on Dekker's theory of galactic orbits, we found four overdensely populated regions in phase space. Three of them were correlated with previously known star streams: the Hyades-Pleiades and Hercules streams in the thin disk of the Milky Way and the Arcturus stream in the thick disk. In addition we find evidence for another stream in the thick disk, which resembles closely the Arcturus stream and probably has the same extragalactic origin.

Friction ofN-bead macromolecules in solution: Effects of the bead-solvent interaction

The role of the bead-solvent interaction has been studied for its influence on the dynamics of an N-bead macromolecule which is immersed into a solution. Using a Fokker-Planck equation for the phase-space distribution function of the macromolecule, we show that all the effects of the solution can be treated entirely in terms of the friction tensors which are assigned to each pair of interacting beads in the chain. For the high-density as well as for the critical solvent, the properties of these tensors are discussed in detail and are calculated by using several (realistic) choices of the bead-solvent potential. From the friction tensors, moreover, an expression for the center-of-mass friction coefficient of a (N-bead) chain macromolecule is derived. Numerical data for this coefficient for "truncated" Lennard-Jones bead-solvent potential are compared with results from molecular dynamic simulations and from the phenomenological theoretical data as found in the literature.

MACHOs in M 31? Absence of evidence but not evidence of absence

We present results of a microlensing survey toward the Andromeda Galaxy (M31) carried out during four observing seasons at the Isaac Newton Telescope (INT). This survey is part of the larger microlensing survey toward M31 performed by the Microlensing Exploration of the Galaxy and Andromeda (MEGA) collaboration. Using a fully automated search algorithm, we indentify 14 candidate microlensing events, three of which are reported here for the first time. Observations obtained at the Mayall telescope are combined with the INT data to produce composite lightcurves for these candidates. The results from the survey are compared with theoretical predictions for the number and distribution of events. These predictions are based on a Monte Carlo calculation of the detection efficiency and disk-bulge-halo models for M31. The models provide the full phase-space distribution functions for the lens and source populations and are motivated by dynamical and observational considerations. They include differential extinction and span a wide range of parameter space characterised primarily by the mass-to-light ratios for the disk and bulge. For most models, the observed event rate is consistent with the rate predicted for self-lensing -- a MACHO halo fraction of 30% or higher can be ruled at the 95% confidence level. The event distribution does show a large near-far asymmetry hinting at a halo contribution to the microlensing signal. Two candidate events are located at particularly large projected radii on the far side of the disk. These events are difficult to explain by self lensing and only somewhat easier to explain by MACHO lensing. A possibility is that one of these is due to a lens in a giant stellar stream.

Weyl ordering, normally ordering of Husimi operator as the squeezed coherent state projector and its applications

By introducing a new kind of squeezed coherent states | p , q 〉 κ we propose so-called Husimi operator Δ h ( q , p , κ ) corresponding to the Husimi distribution functions in phase space. Remarkably, we show Δ h ( q , p , κ ) = | p , q 〉 k 〈 p , q | k , i.e. the Husimi operator actually is a squeezed coherent state projector. The Weyl ordered form and the normally ordered form of Δ h ( q , p , κ ) are derived that provides us with an operator version to examine various properties of the Husimi distribution, which is concise and neat. In so doing, a new operator-representation theory which underlies the Husimi distribution is established. Throughout the Letter we have fully employed the technique of integration within an ordered product of operators.

Interaction between a galactic disk and a live dark halo with an anisotropic velocity distribution

We have extended previous analytical studies of the interaction of dark halos with galactic disks by introducing for the halo particles anisotropic distribution functions in phase space. For this purpose we have employed the shearing sheet model of a patch of a galactic disk embedded in a homogeneous halo. We find that velocity anisotropy increases considerably the maximum growth factor of perturbations in the disk.

Molecular theory of irreversibility

A generalization of the Gibbs entropy postulate is proposed based on the Bogolyubov-Born-Green-Kirkwood-Yvon hierarchy of equations as the nonequilibrium entropy for a system of N interacting particles. This entropy satisfies the basic principles of thermodynamics in the sense that it reaches its maximum at equilibrium and is coherent with the second law. By using a generalization of the Liouville equation describing the evolution of the distribution vector, it is demonstrated that the entropy production is a non-negative quantity. Moreover, following the procedure of nonequilibrium thermodynamics a transport matrix is introduced and a microscopic expression for this is derived. This framework allows one to perform the thermodynamic analysis of nonequilibrium steady states with smooth phase-space distribution functions which, as proven here, constitute the states of minimum entropy production when one considers small departures from stationarity.