Process Of Violent Relaxation Research Articles

Early dynamical evolution of rotating star clusters in a tidal field

ABSTRACT In order to explore how the early internal rotational properties of star clusters are affected by the external potential of their host galaxies, we have run a suite of N-body simulations following the early dynamical evolution and violent relaxation of rotating star clusters embedded in a tidal field. Our study focuses on models for which the cluster’s rotation axis has a generic orientation relative to the torque of the tidal field. The interaction between the violent relaxation process, angular momentum of the cluster, and the external torque creates a complex kinematic structure within the cluster, most prominently a radial variation in the position of the rotation axis, along both the polar and azimuthal directions. We also examine the cluster’s velocity dispersion anisotropy and show that the projected anisotropy may be affected by the variation of the rotation axis directions within the cluster; the combination of projection effects and the complex kinematical features may result in the measurement of tangential anisotropy in the cluster’s inner regions. We also characterize the structural properties of our clusters as a function of their initial rotation and virial ratio and find that clusters may develop a triaxial morphology and a radial variation of the minor axis not necessarily aligned with the rotation axis. Finally, we examine the long-term evolution of these complex kinematic features.

Symplectic coarse graining approach to the dynamics of spherical self-gravitating systems

ABSTRACT We investigate the evolution of the phase–space distribution function around slightly perturbed stationary states and the process of violent relaxation in the context of the dissipationless collapse of an isolated spherical self-gravitating system. By means of the recently introduced symplectic coarse graining technique, we obtain an effective evolution equation that allows us to compute the scaling of the frequencies around a stationary state, as well as the damping times of Fourier modes of the distribution function, with the magnitude of the Fourier k −vectors themselves. We compare our analytical results with N-body simulations.

Gravity or turbulence V: star-forming regions undergoing violent relaxation

ABSTRACT Using numerical simulations of the formation and evolution of stellar clusters within molecular clouds (MCs), we show that the stars in clusters formed within collapsing MC clumps exhibit a constant velocity dispersion regardless of their mass, as expected in violent relaxation processes. In contrast, clusters formed in turbulence-dominated environments exhibit an inverse mass segregated velocity dispersion, where massive stars exhibit larger velocity dispersions than low-mass cores, consistent with massive stars formed in massive clumps, which, in turn, are formed through strong shocks. We furthermore use Gaia EDR3 to show that the stars in the Orion Nebula Cluster exhibit a constant velocity dispersion as a function of mass, suggesting that it has been formed by collapse within one free-fall time of its parental cloud, rather than in a turbulence-dominated environment during many free-fall times of a supported cloud. Additionally, we have addressed several of the criticisms of models of collapsing star-forming regions: namely, the age spread of the ONC, the comparison of the ages of the stars to the free-fall time of the gas that formed it, the star formation efficiency, and the mass densities of clouds versus the mass densities of stellar clusters, showing that observational and numerical data are consistent with clusters forming in clouds undergoing a process of global, hierarchical and chaotic collapse, rather than being supported by turbulence.

Theoretical Predictions of Colors and Metallicity of the Intracluster Light

Abstract We study the colors and metallicities of the brightest cluster galaxies (BCGs) and intracluster light (ICL) in galaxy groups and clusters, as predicted by a semi-analytic model of galaxy formation, coupled with a set of high-resolution N-body simulations. The model assumes stellar stripping and violent relaxation processes during galaxy mergers to be the main channels for the formation of the ICL. We find that BCGs are more metal-rich and redder than the ICL, at all redshifts once the ICL starts to form (z ∼ 1). In good agreement with several observed data, our model predicts negative radial metallicity and color gradients in the BCG+ICL system. By comparing the typical colors of the ICL with those of satellite galaxies, we find that the mass and metals in the ICL come from galaxies of different mass, depending on the redshift. Stripping of low-mass galaxies, , is the most important contributor in the early stages of ICL formation, but the bulk of the mass/metals contents are given by intermediate/massive galaxies, , at lower redshift. Our analysis supports the idea that stellar stripping is more important than galaxy mergers in building up the ICL, and highlights the importance of colors/metallicity measurements for understanding the formation and evolution of the ICL.

Entropy production and Vlasov equation for self-gravitating systems

The evolution of a self-gravitating system to a non-equilibrium steady state occurs through a process of violent relaxation. In the thermodynamic limit the dynamics of a many body system should be governed by the Vlasov equation. Recently, however, a question was raised regarding the validity of Vlasov equation during the process or violent relaxation. In this paper we will explore the entropy production during the relaxation process using N-body molecular dynamics simulations. We will show that the entropy production time grows as , with and in the limit , entropy will remain constant, consistent with the Vlasov equation. Furthermore, we will show that the mean field dynamics constructed on the basis of the Vlasov equation is in excellent agreement with the full molecular dynamics simulations, justifying the applicability of Vlasov equation during the violent relaxation phase of evolution.

Derivation of a generalized Schrödinger equation for dark matter halos from the theory of scale relativity

Abstract Using Nottale’s theory of scale relativity, we derive a generalized Schrodinger equation applying to dark matter halos. This equation involves a logarithmic nonlinearity associated with an effective temperature and a source of dissipation. Fundamentally, this wave equation arises from the nondifferentiability of the trajectories of the dark matter particles whose origin may be due to ordinary quantum mechanics, classical ergodic (or almost ergodic) chaos, or to the fractal nature of spacetime at the cosmic scale. The generalized Schrodinger equation involves a coefficient D , possibly different from ħ ∕ 2 m (where ħ is the Planck constant and m the mass of the particles), whose value for dark matter halos is D = 1 . 02 × 1 0 23 m2/s. This model is similar to the Bose–Einstein condensate dark matter model except that it does not require the dark matter particle to be ultralight. It can accommodate any type of particles provided that they have nondifferentiable trajectories. We suggest that the cold dark matter crisis may be solved by the fractal (nondifferentiable) structure of spacetime at the cosmic scale, or by the chaotic motion of the particles on a very long timescale, instead of ordinary quantum mechanics. The equilibrium states of the generalized Schrodinger equation correspond to configurations with a core-halo structure. The quantumlike potential generates a solitonic core that solves the cusp problem of the classical cold dark matter model. The logarithmic nonlinearity accounts for the presence of an isothermal halo that leads to flat rotation curves (it also accounts for the isothermal core of large dark matter halos). The damping term ensures that the system relaxes towards an equilibrium state. This property is guaranteed by an H -theorem satisfied by a Boltzmann-like free energy functional. In our approach, the temperature and the friction arise from a single formalism. They correspond to the real and imaginary parts of the complex friction coefficient present in the scale covariant equation of dynamics that is at the basis of Nottale’s theory of scale relativity. They may be the manifestation of a cosmic aether or the consequence of a process of violent relaxation and gravitational cooling on a coarse-grained scale.

Kinematical fingerprints of star cluster early dynamical evolution

Abstract We study the effects of the external tidal field on the violent relaxation phase of star clusters dynamical evolution, with particular attention to the kinematical properties of the equilibrium configurations emerging at the end of this phase. We show that star clusters undergoing the process of violent relaxation in the tidal field of their host galaxy can acquire significant internal differential rotation and are characterized by a distinctive radial variation of the velocity anisotropy. These kinematical properties are the result of the symmetry breaking introduced by the external tidal field in the collapse phase and of the action of the Coriolis force on the orbit of the stars. The resulting equilibrium configurations are characterized by differential rotation, with a peak located between one and two half-mass radii. As for the anisotropy, similar to clusters evolving in isolation, the systems explored in this Letter are characterized by an inner isotropic core, followed by a region of increasing radial anisotropy. However, for systems evolving in an external tidal field, the degree of radial anisotropy reaches a maximum in the cluster intermediate regions and then progressively decreases, with the cluster outermost regions being characterized by isotropy or a mild tangential anisotropy. Young or old but less-relaxed dynamically young star clusters may keep memory of these kinematical fingerprints of their early dynamical evolution.

Caloric curves fitted by polytropic distributions in the HMF model

We perform direct numerical simulations of the Hamiltonian mean field (HMF) model starting from non-magnetized initial conditions with a velocity distribution that is (i) Gaussian; (ii) semi-elliptical, and (iii) waterbag. Below a critical energy E c , depending on the initial condition, this distribution is Vlasov dynamically unstable. The system undergoes a process of violent relaxation and quickly reaches a quasi-stationary state (QSS). We find that the distribution function of this QSS can be conveniently fitted by a polytrope with index (i) n = 2; (ii) n = 1; and (iii) n = 1/2. Using the values of these indices, we are able to determine the physical caloric curve T kin (E) and explain the negative kinetic specific heat region C kin = dE/d T kin < 0 observed in the numerical simulations. At low energies, we find that the system has a “core-halo” structure. The core corresponds to the pure polytrope discussed above but it is now surrounded by a halo of particles. In case (iii), we recover the “uniform” core-halo structure previously found by Pakter and Levin [Phys. Rev. Lett. 106, 200603 (2011)]. We also consider unsteady initial conditions with magnetization M 0 = 1 and isotropic waterbag velocity distribution and report the complex dynamics of the system creating phase space holes and dense filaments. We show that the kinetic caloric curve is approximately constant, corresponding to a polytrope with index n 0 ≃ 3.56 (we also mention the presence of an unexpected hump). Finally, we consider the collisional evolution of an initially Vlasov stable distribution, and show that the time-evolving distribution function f(θ,v,t) can be fitted by a sequence of polytropic distributions with a time-dependent index n(t) both in the non-magnetized and magnetized regimes. These numerical results show that polytropic distributions (also called Tsallis distributions) provide in many cases a good fit of the QSSs. They may even be the rule rather than the exception. However, in order to moderate our message, we also report a case where the Lynden-Bell theory (which assumes ergodicity or efficient mixing) provides an excellent prediction of an inhomogeneous QSS. We therefore conclude that both Lynden-Bell and Tsallis distributions may be useful to describe QSSs depending on the efficiency of mixing.

Self-organized relaxation in a collisionless gravitating system

We propose the self-organized relaxation process which drives a collisionless self-gravitating system to the equilibrium state satisfying local virial (LV) relation. During the violent relaxation process, particles can move widely within the time interval as short as a few free-fall times, because of the effective potential oscillations. Since such particle movement causes further potential oscillations, it is expected that the system approaches the critical state where such particle activities, which we call gravitational fugacity, is independent of the local position as much as possible. Here we demonstrate that gravitational fugacity can be described as the functional of the LV ratio, which means that the LV ratio is a key ingredient estimating the particle activities against gravitational potential. We also demonstrate that the LV relation is attained if the LV ratio exceeds the critical value b=1 everywhere in the bound region during the violent relaxation process. The local region which does not meet this criterion can be trapped into the presaturated state. However, small phase-space perturbation can bring the inactive part into the LV critical state.

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

We discuss the kinetic theory of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system described by the Liouville equation from the canonical description of a stochastically forced Brownian system described by the Fokker–Planck equation. We show that the mean-field approximation is exact in a proper thermodynamic limit. For N → + ∞ , a Hamiltonian system is described by the Vlasov equation. In this collisionless regime, coherent structures can emerge from a process of violent relaxation. These metaequilibrium states, or quasi-stationary states (QSS), are stable stationary solutions of the Vlasov equation. To order 1 / N , the collision term of a homogeneous system has the form of the Lenard–Balescu operator. It reduces to the Landau operator when collective effects are neglected. The statistical equilibrium state (Boltzmann) is obtained on a collisional timescale of order N or larger (when the Lenard–Balescu operator cancels out). We also consider the stochastic motion of a test particle in a bath of field particles and derive the general form of the Fokker–Planck equation describing the evolution of the velocity distribution of the test particle. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. For Brownian systems, in the N → + ∞ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit ξ → + ∞ , or for large times t ⪢ ξ - 1 , it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, 2D vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker–Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.

Inconsistency in theories of violent relaxation

We examine an inconsistency in theories of violent-relaxation by Lynden-Bell and Nakamura. The inconsistency arises from the non-transitive nature of these theories: a system that under goes a violent-relaxation, relaxes and then upon an addition of energy, undergoes violent-relaxation once again would settle in an equilibrium state that is different from the one that is predicted had the system would go directly from the initial to the final states. We conclude that a proper description of the violent-relaxation process cannot be achieved by equilibrium statistical mechanics approach, but instead a dynamical theory for the coarse-grained phase-space density is needed.

Statistical theory of high-gain free-electron laser saturation.

We propose an approach, based on statistical mechanics, to predict the saturated state of a single-pass, high-gain free-electron laser. In analogy with the violent relaxation process in self-gravitating systems and in the Euler equation of two-dimensional turbulence, the initial relaxation of the laser can be described by the statistical mechanics of an associated Vlasov equation. The laser field intensity and the electron bunching parameter reach a quasistationary value which is well fitted by a Vlasov stationary state if the number of electrons N is sufficiently large. Finite N effects (granularity) finally drive the system to Boltzmann-Gibbs statistical equilibrium, but this occurs on times that are unphysical (i.e., excessively long undulators). All theoretical predictions are successfully tested by means of finite- N numerical experiments.

On the Origin of the Inner Structure of Halos

We calculate by means of the Press-Schechter formalism the density profile developed by dark matter halos during accretion, i.e., the continuous aggregation of small clumps. We find that the shape of the predicted profile is similar to that shown by halos in high-resolution cosmological simulations. Furthermore, the mass-concentration relation is correctly reproduced at any redshift in all the hierarchical cosmologies analyzed except for very large halo masses. The role of major mergers, which can cause the rearrangement of the halo structure through violent relaxation, is also investigated. We show that, as a result of the boundary conditions imposed by the matter continuously infalling into the halo during the violent relaxation process, the shape of the density profile emerging from major mergers is essentially identical to the shape the halo would have developed through pure accretion. This result explains why, according to high-resolution cosmological simulations, relaxed halos of a given mass have the same density profile regardless of whether they have had a recent merger and why both spherical infall and hierarchical assembly lead to very similar density profiles. Finally, we demonstrate that the density profile of relaxed halos is not affected by the capture of clumps of intermediate mass either.

The Internal Structure and Formation of Early‐Type Galaxies: The Gravitational Lens System MG 2016+112 atz= 1.004

We combine our recent measurements of the velocity dispersion and the surface brightness profile of the lens galaxy D in the system MG 2016+112 (z = 1.004) with constraints from gravitational lensing to study its internal mass distribution. We find the following: (1) dark matter accounts for more than 50% of the total mass within the Einstein radius (99% confidence limit [CL]), whereas ~75% is the more likely contribution. In particular, we can exclude at the 8 σ level that mass follows light inside the Einstein radius with a constant mass-to-light ratio (M/L). (2) The total mass distribution inside the Einstein radius is well described by a density profile ∝ r, with an effective slope γ' = 2.0 ± 0.1 ± 0.1, including random and systematic uncertainties. (3) The offset of galaxy D from the local fundamental plane independently constrains the stellar M/L and matches the range derived from our models, leading to a more stringent lower limit of more than 60% on the fraction of dark matter within the Einstein radius (99% CL). Under the assumption of adiabatic contraction, we show that the inner slope of the dark matter halo before the baryons collapsed to form the lens galaxy is γi < 1.4 (68% CL), only marginally consistent with the highest resolution cold dark matter simulations that indicate γi ~ 1.5. This might indicate either that adiabatic contraction is a poor description of early-type galaxy formation or that additional processes play a role as well. Indeed, the apparently isothermal density distribution inside the Einstein radius is not a natural outcome of adiabatic contraction models, where it appears to be a mere coincidence. By contrast, we argue that isothermality might be the result of a stronger coupling between luminous and dark matter, possibly the result of (incomplete) violent relaxation processes during the formation of the innermost regions of the galaxy. Hence, we conclude that galaxy D appears already relaxed ~8 Gyr ago. We briefly discuss the importance of our results for lens statistics and the determination of the Hubble constant from gravitational lens time delays.

On the gravitational collapse of stellar systems

We study the (maximum entropy) variational problem associated with the violent relaxation process in stellar dynamics. We prove that for confined systems, in the general case (not necessarily radially symmetric), this problem always has a solution. Thus the gravitational collapse, which occurs for classical systems, does not occur in the case of violent relaxation.

Galaxy clusters and microwave background anisotropy

Previous estimates of the microwave background anisotropies produced by freely falling spherical clusters are discussed. These estimates are based on the Swiss-Cheese and Tolman-Bondi models. It is proved that these models give only upper limits to the anisotropies produced by the observed galaxy clusters. By using spherically symmetric codes including pressureless matter and a hot baryonic gas, new upper limits are obtained. The contributions of the hot gas and the pressureless component to the total anisotropy are compared. The effects produced by the pressure are proved to be negligible; hence, estimations of the cluster anisotropies based on N-body simulations are hereafter justified. After the phenomenon of violent relaxation, any realistic rich cluster can only produce small anisotropies with amplitudes of order $10^{-7}$. During the rapid process of violent relaxation, the anisotropies produced by nonlinear clusters are expected to range in the interval $(10^{-6},10^{-5})$. The angular scales of these anisotropies are discussed.