The evolution of self-gravitating systems, and long-range interacting systems more generally, from initial configurations far from dynamical equilibrium is often described as a simple two phase process: a first phase of violent relaxation bringing it to a quasi-stationary state in a few dynamical times, followed by a slow adiabatic evolution driven by collisional processes. In this context the complex spatial structure evident, e.g., in spiral galaxies is understood either in terms of instabilities of quasi-stationary states, or a result of dissipative non-gravitational interactions. We illustrate here, using numerical simulations, that purely self-gravitating systems evolving from quite simple initial configurations can in fact give rise easily to structures of this kind of which the lifetime can be large compared to the dynamical characteristic time, but short compared to the collisional relaxation time scale. More specifically, for a broad range of non-spherical and non-uniform rotating initial conditions, gravitational relaxation gives rise quite generically to long-lived non-stationary structures of a rich variety, characterized by spiral-like arms, bars and even ring-like structures in special cases. These structures are a feature of the intrinsically out-of-equilibrium nature of the system's collapse, associated with a part of the system's mass while the bulk is well virialized. They are characterized by predominantly radial motions in their outermost parts, but also incorporate an extended flattened region which rotates coherently about a well virialized core of triaxial shape with an approximately isotropic velocity dispersion. We discuss the possible relevance of these simple toy models to the observed structure of real galaxies emphasizing the difference between dissipative and dissipationless disc formation.
Certain thermal non-equilibrium situations, outside of the astrophysical realm, suggest that entropy production extrema, instead of entropy extrema, are related to stationary states. In an effort to better understand the evolution of collisionless self-gravitating systems, we investigate the role of entropy production and develop expressions for the entropy production rate in two particular statistical families that describe self-gravitating systems. From these entropy production descriptions, we derive the requirements for extremizing the entropy production rate in terms of specific forms for…
The short--wave asymptotics (WKB) of spiral density waves in self-gravitating stellar discs is well suited for the study of the dynamics of tightly--wound wavepackets. But the textbook WKB theory is not well adapted to the study of the linear eigenmodes in a collisionless self-gravitating disc because of the transcendental nature of the dispersion relation. We present a modified WKB of spiral density waves, for collisionless discs in the epicyclic limit, in which the perturbed gravitational potential is related to the perturbed surface density by the Poisson integral in Kalnaj's logarithmic sp…
Building on the development of a Hermite-Legendre analysis of one-dimensional gravitating collisionless systems, we present a technique for determining the steady states of such systems. This provides an important component for understanding the physics involved in the relaxation of these kinds of systems. As the dark matter structures in the universe should have traits in common with these systems, insight into this relaxation can provide clues to larger astrophysical questions. For large perturbation strengths, we determine physically motivated parameter ranges for the simplest families of s…
The long-standing puzzle surrounding the statistical mechanics of self-gravitating systems has not yet been solved successfully. We formulate a systematic theoretical framework of entropy-based statistical mechanics for spherically symmetric collisionless self-gravitating systems. We use an approach that is very different from that of the conventional statistical mechanics of short-range interaction systems. We demonstrate that the equilibrium states of self-gravitating systems consist of both mechanical and statistical equilibria, with the former characterized by a series of velocity-moment e…
In an effort to better understand collisionless relaxation processes in gravitational systems, we investigate one-dimensional models. Taking advantage of a Hermite-Legendre expansion of relevant distribution functions, we present analytical and numerical behaviors of Maxwell-Boltzmann entropy. In particular, we modestly perturb systems about a separable-solution equilibrium and observe their collisionless evolution to a steady state. We verify the time-independence of fine-grained entropy in these systems before turning our attention to the behavior of coarse-grained entropy. We also verify th…
We present an effective evolution equation for a coarse-grained distribution function of a long-range-interacting system preserving the symplectic structure of the non-collisional Boltzmann, or Vlasov, equation. We first derive a general form of such an equation based on symmetry considerations only. Then, we explicitly derive the equation for one-dimensional systems, finding that it has the form predicted on general grounds. Finally, we use such an equation to predict the dependence of the damping times on the coarse-graining scale and numerically check it for some one-dimensional models, inc…
In this paper we study the kinetic theory of many-particle astrophysical systems and we present a consistent version of the collisionless Boltzmann equation in the 1PN approximation. We argue that the equation presented by Rezania and Sobouti in A&A 354 1110 (2000) is not the correct expression to describe the evolution of a collisionless self-gravitating gas. One of the reasons that account for the previous statement is that the energy of a free-falling test particle, obeying the 1PN equations of motion for static gravitational fields, is not a static solution of the mentioned equation. The s…
Self-gravitating Newtonian systems consisting of a very large number of particles have generally defied attempts to describe them using statistical mechanics. This is paradoxical since many astronomical systems, or simulations thereof, appear to have universal, equilibrium structures for which no physical basis exist. A decade ago we showed that extremizing the number of microstates with a given energy per unit mass, under the constraints of conserved total energy and mass, leads to the maximum entropy state, $n(E) \propto \exp (-β(E-Φ_0))-1$, known as DARKexp. This differential energy distrib…
In this paper we study the kinetic theory of many-particle astrophysical systems imposing axial symmetry and extending our previous analysis in Phys. Rev. D 83, 123007 (2011). Starting from a Newtonian model describing a collisionless self-gravitating gas, we develop a framework to include systematically the first general relativistic corrections to the matter distribution and gravitational potentials for general stationary systems. Then, we use our method to obtain particular solutions for the case of the Morgan & Morgan disks. The models obtained are fully analytical and correspond to the po…
The time irreversibility and fast relaxation of collapsing $N$-body gravitating systems (as opposed to the time reversibility of the equations of motion for individual stars or particles) are traditionally attributed to information loss due to coarse-graining in the observation. We show that this subjective element is not necessary once one takes into consideration the fundamental fact that these systems are discrete, i.e. composed of a finite number $N$ of stars or particles. We show that a connection can be made between entropy estimates for discrete systems and the Nyquist-Shannon sampling …
In the mean field limit, isolated gravitational systems often evolve towards a steady state through a violent relaxation phase. One question is to understand the nature of this relaxation phase, in particular the role of radial instabilities in the establishment/destruction of the steady profile. Here, through a detailed phase-space analysis based both on a spherical Vlasov solver, a shell code and a $N$-body code, we revisit the evolution of collisionless self-gravitating spherical systems with initial power-law density profiles $ρ(r) \propto r^n$, $0 \leq n \leq -1.5$, and Gaussian velocity …
We investigate the old problem of the fast relaxation of collisionless $N$-body systems which are collapsing or perturbed, emphasizing the importance of (non-collisional) discreteness effects. We integrate orbit ensembles in fixed external potentials, estimating the entropy of the ensemble to analyze the time evolution of the distribution function. We show that these estimates capture the correct physical behavior expected from the 2nd Law of Thermodynamics, without any spurious entropy production. For self-consistent (i.e. stationary) samples, the entropy is conserved, while for non-self-cons…
The collapse of a collisionless self-gravitating system, with the fast achievement of a quasi-stationary state, is driven by violent relaxation, with a typical particle interacting with the time-changing collective potential. It is traditionally assumed that this evolution is governed by the Vlasov-Poisson equation, in which case entropy must be conserved. We run N-body simulations of isolated self-gravitating systems, using three simulation codes: NBODY-6 (direct summation without softening), NBODY-2 (direct summation with softening) and GADGET-2 (tree code with softening), for different numb…
We study, using both theory and simulations, a system of self-gravitating sheets. A new statistical mechanics theory - free of any adjustable parameters - is derived to quantitatively predict the final stationary state achieved by this system after the process of collisionless relaxation is completed. The theory shows a very good agreement with the numerical simulations. The model sheds new light on the general mechanism of relaxation of self-gravitating systems and may help us to understand cold matter distribution in the Universe.
The current paradigm for understanding galaxy formation in the universe depends on the existence of self-gravitating collisionless dark matter. Modeling such dark matter systems has been a major focus of astrophysicists, with much of that effort directed at computational techniques. Not surprisingly, a comprehensive understanding of the evolution of these self-gravitating systems still eludes us, since it involves the collective nonlinear dynamics of many-particle systems interacting via long-range forces described by the Vlasov equation. As a step towards developing a clearer picture of colli…
This paper studies the scale and redshift variation of density and velocity distributions in self-gravitating collisionless dark matter flow by a halo-based non-projection approach. All particles are divided into halo and out-of-halo particles for redshift variation of distributions. Without projecting particle fields onto a structured grid, the scale variation is analyzed by identifying all particle pairs on different scales $r$. We demonstrate that: i) Delaunay tessellation can be used to reconstruct the density field. The density correlation, spectrum, and dispersion functions were obtained…
The relation between properties of galaxies and dark matter halos they reside in can be valuable for structure formation and evolution. This paper focus on the baryonic-to-halo mass ratio (BHMR) and its evolution. We first review unique properties of self-gravitating collisionless dark matter flow (SG-CFD), followed by their application to derive BHMR. To maximize system entropy, the long-range interaction requires a broad size of halos to be formed. These halos facilitate inverse mass and energy cascade from small to large scales with a constant rate of energy cascade $\varepsilon_u$. In addi…
In this Letter, we investigate the stability of the statistical equilibrium of spherically symmetric collisionless self-gravitating systems. By calculating the second variation of the entropy, we find that perturbations of the relevant physical quantities should be classified as long- and short-range perturbations, which correspond to the long- and short-range relaxation mechanisms, respectively. We show that the statistical equilibrium states of self-gravitating systems are neither maximum nor minimum, but complex saddle-point entropy states, and hence differ greatly from the case of ideal ga…
The equilibria formed by the self-gravitating, collisionless collapse of simple initial conditions have been investigated for decades. We present the results of our attempts to describe the equilibria formed in $N$-body simulations using thermodynamically-motivated models. Previous work has suggested that it is possible to define distribution functions for such systems that describe maximum entropy states. These distribution functions are used to create radial density and velocity distributions for comparison to those from simulations. A wide variety of $N$-body code conditions are used to red…
Halo-mediated mass and energy cascades are key to understand dark matter flow. Both cascades origin from the mass exchange between halo and out-of-halo sub-systems. Kinetic energy can be from the motion of halos and particle motion in halos. Similarly, potential energy can be due to the inter- and intra-halo interactions. Intra-halo virial equilibrium is established much faster than inter-halo. Change of energy of entire system comes from virilization in halos. At statistically steady state, continuous mass exchange is required to sustain growth of total halo mass $M_h\propto a^{1/2}$ and ener…
We describe the dynamics and thermodynamics of collisionless particle disks orbiting a massive central body, in the case where the disk mass is small compared to the central mass, the self-gravity of the disk dominates the non-Keplerian force, and the spread in semi-major axes is small. We show that with plausible approximations such disks have logarithmic two-body interactions and a compact phase space, and therefore exhibit thermodynamics that are simpler than most other gravitating systems, which require a confining box and artificial softening of the potential at small scales to be thermod…
Systems with long range interactions present generically the formation of quasi-stationary long-lived non-equilibrium states. These states relax to Boltzmann equilibrium following a dynamics which is not well understood. In this paper we study this process in two-dimensional inhomogeneous self-gravitating systems. Using the Chandrasekhar -- or local -- approximation we write a simple approximate kinetic equation for the relaxation process, obtaining a Fokker -- Planck equation for the velocity distribution with explicit analytical diffusion coefficients. Performing molecular dynamics simulatio…
MOdified Gravity (MOG) is a covariant modification of Einstein's general relativity. This theory is one of the current alternatives to dark matter models. We describe dynamics of collisionless self-gravitating systems in the context of MOG. By studying the weak field approximation of this theory, we derive the equations governing the dynamics of the self-gravitating systems. More specifically, we consider the Jeans instability for self-gravitating fluid and stellar systems, and derive new Jeans mass limit $\tilde{M}_J$ and wave-number $\tilde{k}_J$. Furthermore, considering the gravitational i…
We consider a self-gravitating collisionless gas as described by the Vlasov-Poisson or Einstein-Vlasov system or a self-gravitating fluid ball as described by the Euler-Poisson or Einstein-Euler system. We give a simple proof for the finite extension of spherically symmetric equilibria, which covers all these models simultaneously. In the Vlasov case the equilibria are characterized by a local growth condition on the microscopic equation of state, i.e., on the dependence of the particle distribution on the particle energy, at the cut-off energy E_0, and in the Euler case by the corresponding g…
Spiral structure in disk galaxies is modeled with nine collisionless N-body simulations including live disks, halos, and bulges with a range of masses. Two of these simulations make long-lasting and strong two-arm spiral wave modes that last for $\sim5$ Gyr with constant pattern speed. These two had a light stellar disk and the largest values of the Toomre $Q$ parameter in the inner region at the time the spirals formed, suggesting the presence of a Q-barrier to wave propagation resulting from the bulge. The relative bulge mass in these cases is about 10\%. Models with weak two-arm spirals had…
N-body simulations are essential for understanding the formation and evolution of structure in the Universe. However, the discrete nature of these simulations affects their accuracy when modelling collisionless systems. We introduce a new approach to simulate the gravitational evolution of cold collisionless fluids by solving the Vlasov-Poisson equations in terms of adaptively refineable "Lagrangian phase space elements". These geometrical elements are piecewise smooth maps between Lagrangian space and Eulerian phase space and approximate the continuum structure of the distribution function. T…
It has long been known that no static, spherically symmetric, asymptotically flat Klein-Gordon scalar field configuration surrounding a nonrotating black hole can exist in general relativity. In a series of previous papers we proved that, at the effective level, this no-hair theorem can be circumvented by relaxing the staticity assumption: for appropriate model parameters there are quasi-bound scalar field configurations living on a fixed Schwarzschild background which, although not being strictly static, have a larger lifetime than the age of the universe. This situation arises when the mass …
The secular thickening of a self-gravitating stellar galactic disc is investigated using the dressed collisionless Fokker-Planck equation and the inhomogeneous multicomponent Balescu-Lenard equation. The thick WKB limits for the diffusion fluxes are found using the epicyclic approximation, while assuming that only radially tightly wound transient spirals are sustained by the disc. This yields simple quadratures for the drift and diffusion coefficients, providing a clear understanding of the positions of maximum vertical orbital diffusion within the disc, induced by fluctuations either external…
Initially far out-of-equilibrium self-gravitating systems form, through a collisionless relaxation dynamics, quasi-stationary states (QSS). These may arise from a bottom-up aggregation of structures or in a top-down frame; their quasi-equilibrium properties are well described by the Jeans equation and are not universal, i.e. they depend on initial conditions. To understand the origin of such dependence, we present results of numerical experiments of initially cold and spherical systems characterized by various choices of the spectrum of initial density fluctuations. The amplitude of such fluct…
Using $N$-body simulations ($N\sim 10^6 - 10^7$), we examine how a non-axisymmetric dark halo affects the dynamical evolution of the structure in collisionless (stellar) discs. We demonstrate how the model parameters such as mass of the halo, initial conditions in the disc and the halo axes ratio affect morphology and kinematics of the stellar discs. We show that a non-axisymmetric halo can generate a large-scale spiral density pattern in the embedded stellar disc. The pattern is observed in the disc for many periods of its revolution, even if the disc is gravitationally over-stable. The growt…
The collapse of an isolated, uniform and spherical cloud of self-gravitating particles represents a paradigmatic example of a relaxation process leading to the formation of a quasi-stationary state in virial equilibrium. We consider several N-body simulations of such a system, with the initial velocity dispersion as a free parameter. We show that there is a clear difference between structures formed when the initial virial ratio is b_0 =2K_0/W_0 < b_0^c ~ -1/2 and b_0> b_0^c. These two sets of initial conditions give rise respectively to a mild and violent relaxation occurring in about the sam…
In this paper we suggest the existence in the central regions of spiral galaxies of collisionless, scale-free, rigidly rotating, self-gravitating discs with spiral symmetry. Such discs must be truncated at a finite radius, and they must be stabilized and rendered self-similar by a suitable halo. The halo and the rotating disc share the self-similar class and must form together to arrive at the suggested state. We make comparisons with the well-known rigidly rotating, Kalnajs discs; one of which is axi-symmetric and finite while the other is infinite and decomposed into spiral modes. We find th…
Self-gravitating gaseous filaments exist on many astrophysical scales, from sub-pc filaments in the interstellar medium to Mpc scale streams feeding galaxies from the cosmic web. These filaments are often subject to Kelvin-Helmotz Instability (KHI) due to shearing against a confining background medium. We study the nonlinear evolution of KHI in pressure-confined self-gravitating gas streams initially in hydrostatic equilibrium, using analytic models and hydrodynamic simulations, not including radiative cooling. We derive a critical line-mass as a function of the stream Mach number and density …
Global m=1 slow mode in near-Keplerian self-gravitating torus: applications to stellar nuclear disks and AGN molecular tori
INAF - Astronomical Observatory of Capodimonte（意大利国家天体物理研究所卡波迪蒙特天文台）；Institute of Radio Astronomy, National Academy of Sciences of Ukraine（乌克兰国家科学院射电天文研究所）；INAF–Osservatorio Astrofisico di Torino（意大利国家天体物理研究所都灵天文台）；Main Astronomical Observatory, National Academy of Sciences of Ukraine（乌克兰国家科学院主天文台）；V.N. Karazin Kharkiv National University（V.N. 卡拉津哈尔科夫国立大学）；National University of Life and Environmental Sciences of Ukraine（乌克兰国家生命与环境科学大学）
We present a theory in three parts, of the secular dynamics of a (Keplerian) stellar system of mass $M$ orbiting a black hole of mass $M_\bullet \gg M$. Here we describe the collisionless dynamics; Papers II and III are on the (collisional) theory of Resonant Relaxation. The mass ratio, $\varepsilon = M/M_\bullet \ll 1$, is a natural small parameter implying a separation of time scales between the short Kepler orbital periods and the longer orbital precessional periods. The collisionless Boltzmann equation (CBE) for the stellar distribution function (DF) is averaged over the fast Kepler orbita…
Many barred galaxies harbor small-scale secondary bars in the center. The evolution of such double-barred galaxies is still not well understood, partly because of a lack of realistic N-body models with which to study them. Here we report the generation of such systems in the presence of rotating pseudobulges. We demonstrate with high mass and force resolution collisionless N-body simulations that long-lived secondary bars can form spontaneously without requiring gas, contrary to previous claims. We find that secondary bars rotate faster than primary ones. The rotation is not rigid: the seconda…
In the present work we apply virial analysis to the model of self-gravitating turbulent cloud ensembles introduced by Donkov \& Stefanov in two previous papers, clarifying some aspects of turbulence and extending the model to account not only for supersonic flows but for trans- and subsonic ones as well. Make use of the Eulerian virial theorem at an arbitrary scale, far from the cloud core, we derive an equation for the density profile and solve it in approximate way. The result confirms the solution $\varrho(\ell)=\ell^{-2}$ found in the previous papers. This solution corresponds to three pos…
The problem of modeling the distribution of dark matter in galaxies in terms of equilibrium configurations of collisionless self-gravitating quantum particles is considered. We first summarize the pioneering model of a Newtonian self-gravitating Fermi gas in thermodynamic equilibrium developed by Ruffini and Stella (1983), which is shown to be the generalization of the King model for fermions. We further review the extension of the former model developed by Gao, Merafina and Ruffini (1990), done for any degree of fermion degeneracy at the center ($θ_0$), within general relativity. Finally, we …
Noble gravitational atoms: Self-gravitating black hole scalar wigs with angular momentum number
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México（墨西哥国立自治大学核科学研究所）；Departamento de Física, División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato（瓜纳华托大学莱昂校区科学与工程分院物理系）；Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理科学研究所）；Instituto de Física Enrique Gaviola, CONICET（恩里克·加维亚拉物理研究所，阿根廷国家科学研究委员会）；Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo（圣尼古拉斯伊达尔戈米却肯大学物理与数学研究所）
We revisit the integer lattice (IL) method to numerically solve the Vlasov-Poisson equations, and show that a slight variant of the method is a very easy, viable, and efficient numerical approach to study the dynamics of self-gravitating, collisionless systems. The distribution function lives in a discretized lattice phase-space, and each time-step in the simulation corresponds to a simple permutation of the lattice sites. Hence, the method is Lagrangian, conservative, and fully time-reversible. IL complements other existing methods, such as N-body/particle mesh (computationally efficient, but…
Efficient numerical evolution of self-gravitating systems with dark-matter-baryon scattering
Johns Hopkins University（约翰斯·霍普金斯大学）；University of Southern California（南加州大学）；Institute for Advanced Study（高等研究院）；Flatiron Institute（平顿研究所）；University of British Columbia（不列颠哥伦比亚大学）
