One-dimensional Self-gravitating System Research Articles

Ergodicity and slow relaxation in the one-dimensional self-gravitating system.

We show that homogeneous and nonhomogeneous states of the one-dimensional self-gravitating sheets models have different ergodic properties. The former are nonergodic and the one-particle distribution function has a zero collision term if a proper limit is taken for the periodic boundary conditions. As a consequence, homogeneous states of the sheets model are nonergodic and do not relax to the equilibrium state, while nonhomogeneous states are ergodic in a time window of the order of the relaxation time to equilibrium, as similarly observed in other systems with a long range interaction.

Long-term relaxation of one-dimensional self-gravitating systems.

We investigate the long-term relaxation of one-dimensional (1D) self-gravitating systems, using both kinetic theory and N-body simulations. We consider thermal and Plummer equilibria, with and without collective effects. All combinations are found to be in clear agreement with respect to the Balescu-Lenard and Landau predictions for the diffusion coefficients. Interestingly, collective effects reduce the diffusion by a factor ∼10. The predicted flux for Plummer equilibrium matches the measured one, which is a remarkable validation of kinetic theory. We also report on a situation of quasikinetic blocking for the same equilibrium.

Dynamics to the universal structure of one-dimensional self-gravitating systems in the quasi-equilibrium state

We investigate the quasi-equilibrium state of one-dimensional self-gravitating systems. If the null virial condition is satisfied at initial time, it is found that the number density around the center of the system at the quasi-equilibrium state has the universality similar to two- and three-dimensional self-gravitating systems reported in Refs. [1,2]. The reason why the null virial condition is sufficient for the universality is unveiled by the envelope equation. We present a phenomenological model to describe the universal structure by using a special Langevin equation with a distinctive random noise to self-gravitating systems. Additionally, we unveil a mechanism which decides the radius of the system.

Predicting Collisionless Equlibria in Dark Matter Simulations

By using a transformation from phase-space variables to a set of orthogonal Hermite-Legendre polynomials, the dynamics of one-dimensional self-gravitating systems has been reduced to a linear set of equations. This transformation changes the long-range forces into local interactions in the new problem. This makes it possible to determine the steady-state solutions by projecting the initial conditions into the time-independent solutions.

Dynamics of One-dimensional Self-gravitating Systems Using Hermite-Legendre Polynomials

The current paradigm for understanding galaxy formation in the universe de- pends 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 inter- acting via long-range forces described by the Vlasov equation. As a step towards developing a clearer picture of collisionless self-gravitating relaxation, we analyze the linearized dynamics of isolated one-dimensional systems near thermal equi- librium by expanding their phase space distribution functions f(x,v) in terms of Hermite functions in the velocity variable, and Legendre functions involving the position variable. This approach produces a picture of phase-space evolution in terms of expansion coefficients, rather than spatial and velocity variables. We obtain equations of motion for the expansion coefficients for both test-particle dis- tributions and self-gravitating linear perturbations of thermal equilibrium. This development presents the opportunity to avoid time-consuming N-body simula- tions that are limited by statistical uncertainty and provides a powerful analysis tool for understanding the relaxation to equilibrium.Â

Scaling Quasistationary States in Long-Range Systems with Dissipation

Hamiltonian systems with long-range interactions give rise to long-lived out-of-equilibrium macroscopic states, so-called quasistationary states. We show here that, in a suitably generalized form, this result remains valid for many such systems in the presence of dissipation. Using an appropriate mean-field kinetic description, we show that models with dissipation due to a viscous damping or due to inelastic collisions admit "scaling quasistationary states," i.e., states that are quasistationary in rescaled variables. A numerical study of one-dimensional self-gravitating systems confirms the relevance of these solutions and gives indications of their regime of validity in line with theoretical predictions. We underline that the velocity distributions never show any tendency to evolve towards a Maxwell-Boltzmann form.

Dynamics of one-dimensional self-gravitating systems using Hermite–Legendre polynomials

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 collisionless self-gravitating relaxation, we analyze the linearized dynamics of isolated one-dimensional systems near thermal equilibrium by expanding their phase space distribution functions f(x,v) in terms of Hermite functions in the velocity variable, and Legendre functions involving the position variable. This approach produces a picture of phase-space evolution in terms of expansion coefficients, rather than spatial and velocity variables. We obtain equations of motion for the expansion coefficients for both test-particle distributions and self-gravitating linear perturbations of thermal equilibrium. N-body simulations of perturbed equilibria are performed and found to be in excellent agreement with the expansion coefficient approach over a time duration that depends on the size of the expansion series used.

The 4-body problem in a (1+1)-dimensional self-gravitating system

We report on the results of a study of the motion of a four particle non-relativistic one-dimensional self-gravitating system. We show that the system can be visualized in terms of a single particle moving within a potential whose equipotential surfaces are shaped like a box of pyramid-shaped sides. As such this is the largest N-body system that can be visualized in this way. We describe how to classify possible states of motion in terms of Braid Group operators, generalizing this to N bodies. We find that the structure of the phase space of each of these systems yields a large variety of interesting dynamics, containing regions of quasiperiodicity and chaos. Lyapunov exponents are calculated for many trajectories to measure stochasticity and previously unseen phenomena in the Lyapunov graphs are observed.

Velocity Correlation Function of the One-Dimensional Self-Gravitating System

We develop a kinetic theory for the one-dimensional self-gravitating system (1d-SG system). It is known that the collision term in usual kinetic theory for one-dimensional system is identically zero, when all particles have the same mass. Here we choose a phenomenological approach. Assuming that the system is near equilibrium, we use an approximation that the system is in a thermal bath, called the thermal bath approximation (TBA). With this approximation, velocity correlation function of the 1d-SG system is investigated. The direct numerical simulation shows ∼ t -α for small N (α= 2.0–1.4), where N is the number of sheets. On the other hand, the TBA shows ∼ t -2 for small N . For large N , the velocity correlation function exhibits slow relaxation. The reason of slow relaxation is explained in terms of the stochastic differential equation derived from the TBA.

Thermodynamics and dynamics of a 1-D gravitational system

We describe a one-dimensional self-gravitating system derived from the problem of large-scale structure formation in cosmology. Considering small times so that the expansion can be neglected we present a thermodynamical analysis of this system. We find a second-order phase transition at $T_{c1}$ from an homogeneous equilibrium at high temperature to a clustered phase (with a density peak at one of the boundaries of the system) at low temperature. There also exists an infinite series of unstable equilibria which appear at lower temperatures $T_{cn}$, reflecting the scale-free nature of the gravitational interaction and the usual Jeans instability. We find that, as for the similar HMF (Hamiltonian mean field) model, all three micro-canonical, canonical and grand-canonical ensembles agree with each other, as well as with the stability properties associated with a hydrodynamical approach. On the other hand, the collisionless dynamics governed by the Vlasov equation yields the same results except that at low $T$ the equilibrium associated with two density peaks (one at each boundary) becomes stable.

Dynamics and thermodynamics of a simple model similar to self-gravitating systems: the HMF model

We discuss the dynamics and thermodynamics of the Hamiltonian Mean Field model (HMF) which is a prototypical system with long-range interactions. The HMF model can be seen as the one Fourier component of a one-dimensional self-gravitating system. Interestingly, it exhibits many features of real self-gravitating systems (violent relaxation, persistence of metaequilibrium states, slow collisional dynamics, phase transitions,...) while avoiding complicated problems posed by the singularity of the gravitational potential at short distances and by the absence of a large-scale confinement. We stress the deep analogy between the HMF model and self-gravitating systems by developing a complete parallel between these two systems. This allows us to apply many technics introduced in plasma physics and astrophysics to a new problem and to see how the results depend on the dimension of space and on the form of the potential of interaction. This comparative study brings new light in the statistical mechanics of self-gravitating systems. We also mention simple astrophysical applications of the HMF model in relation with the formation of bars in spiral galaxies.

Three-body dynamics in a (1+1) -dimensional relativistic self-gravitating system.

The results of our study of the motion of a three particle, self-gravitating system in general relativistic lineal gravity is presented for an arbitrary ratio of the particle masses. We derive a canonical expression for the Hamiltonian of the system and discuss the numerical solution of the resulting equations of motion. This solution is compared to the corresponding nonrelativistic and post-Newtonian approximation solutions so that the dynamics of the fully relativistic system can be interpreted as a correction to the one-dimensional Newtonian self-gravitating system. We find that the structure of the phase space of each of these systems yields a large variety of interesting dynamics that can be divided into three distinct regions: annulus, pretzel, and chaotic; the first two being regions of quasiperiodicity while the latter is a region of chaos. By changing the relative masses of the three particles we find that the relative sizes of these three phase space regions changes, and that this deformation can be interpreted physically in terms of the gravitational interactions of the particles. Furthermore, we find that many of the interesting characteristics found in the case where all of the particles share the same mass also appear in our more general study. We find that there are additional regions of chaos in the unequal mass system which are not present in the equal mass case. We compare these results to those found in similar systems.

Chaos in an exact relativistic three-body self-gravitating system.

We consider the problem of three-body motion for a relativistic one-dimensional self-gravitating system. After describing the canonical decomposition of the action, we find an exact expression for the three-body Hamiltonian, implicitly determined in terms of the four coordinates and momentum degrees of freedom in the system. Nonrelativistically these degrees of freedom can be rewritten in terms of a single particle moving in a two-dimensional hexagonal well. We find the exact relativistic generalization of this potential, along with its post-Newtonian approximation. We then specialize to the equal-mass case and numerically solve the equations of motion that follow from the Hamiltonian. Working in hexagonal-well coordinates, we obtain orbits in both the hexagonal and three-body representations of the system, and plot the Poincaré sections as a function of the relativistic energy parameter eta. We find two broad categories of periodic and quasiperiodic motions that we refer to as the annulus and pretzel patterns, as well as a set of chaotic motions that appear in the region of phase space between these two types. Despite the high degree of nonlinearity in the relativistic system, we find that the global structure of its phase space remains qualitatively the same as its nonrelativistic counterpart for all values of eta that we could study. However, the relativistic system has a weaker symmetry and so its Poincare section develops an asymmetric distortion that increases with increasing eta. For the post-Newtonian system we find that it experiences a chaotic transition in the interval 0.21<theta<0.26, above which some of the near-integrable regions degenerate into chaos.

Incomplete relaxation in a two-mass one-dimensional self-gravitating system.

Due to the apparent ease with which they can be numerically simulated, one-dimensional gravitational systems were first introduced by astronomers to explore different modes of gravitational evolution. These include violent relaxation and the approach to thermal equilibrium. Careful work by dynamicists and statistical physicists has shown that several claims made by astronomers regarding these models were incorrect. Unusual features of the evolution include the development of long lasting structures on large scales, which can be thought of as one-dimensional analogs of Jupiter's red spot or a galactic spiral density wave or bar. The existence of these structures demonstrates that in gravitational systems evolution is not entirely dominated by the second law of thermodynamics and also appears to contradict the Arnold diffusion ansatz. Thus it is correct to assert that the one-dimensional planar sheet gravitational system is the nonextensive analog of the Fermi-Pasta-Ulam model of dynamical systems. This paper is an extension of a preliminary study where we conclusively showed mass segregation and equipartition of kinetic energy in a two-mass planar sheet system for the first time. Here we employ both mean-field theory and dynamical simulation to more thoroughly probe the statistical and ergodic properties of these systems. Valuable information is obtained from local and global time averaging, and temporal and spatial correlation functions. Using these tools we show that the system appears to approach the equilibrium distribution on very long time scales, but the relaxation is incomplete.

Long-time behavior and relaxation of power-law correlation in one-dimensional self-gravitating system

Long-time behavior of spatial power-law correlation in one-dimensional self-gravitating system is numerically investigated. The power-law structure persists even after the system is virialized. The structure gradually disappears with energy exchange among particles. Lifetime of the power-law structure is estimated to be proportional to the system size.

Statistical mechanics of relativistic one-dimensional self-gravitating systems.

We consider the statistical mechanics of a general relativistic one-dimensional self-gravitating system. The system consists of N particles coupled to lineal gravity and can be considered as a model of N relativistically interacting sheets of uniform mass. The partition function and one-particle distribution functions are computed to leading order in 1/c where c is the speed of light; as c --> infinity results for the nonrelativistic one-dimensional self-gravitating system are recovered. We find that relativistic effects generally cause both position and momentum distribution functions to become more sharply peaked, and that the temperature of a relativistic gas is smaller than its nonrelativistic counterpart at the same fixed energy. We consider the large-N limit of our results and compare this to the nonrelativistic case.

Critical Dynamics under the Vlassov‐Poisson Equations: Critical Exponents and Scaling of the Distribution Function near the Point of a Marginal Stability

Using direct integration of the Vlassov equation in configurational space, nonlinear dynamics of a model one-dimensional periodic self-gravitating system are investigated near the point of a marginal stability. The critical velocity dispersion σ, corresponding to a marginal stability of the Jeans mode with the least k, is assumed as the critical point. The peak amplitude of the Jeans mode computed in a run is assumed as the order parameter. In the neighborhood of the critical point, the dynamics can be described by a set of the power laws typical for a system undergoing a second-order phase transition. For the order parameter, this is A ∝ -θβ, where β = 1.907 ± 0.006 and θ = (σ2 - σ)/σ 0. Under this accuracy, these critical exponents satisfy the equality γ± = β(δ - 1). For a gravitating system, its existence is a direct consequence of scaling invariance of the distribution function at |θ| 1; i.e., f(λt, x, λv, λθ, λA0, λF1) = λf(t, x, v, θ, A0, F1). These critical exponents indicate a wide critical area where critical phenomena may determine macroscopic dynamics. The random external drive field of a small amplitude causes anomalous growth of the marginally stable Jeans perturbation into a spatially coherent structure.

Emergence of power-law correlation in 1-dimensional self-gravitating system

A new phase of temporal evolution of the one-dimensional self-gravitating system is numerically discovered. Fractal structure is dynamically created from non-fractal initial conditions. Implication to astrophysics and mathematical physics is discussed.

Equipartition and Mass Segregation in a One-Dimensional Self-Gravitating System

The one-dimensional self-gravitating system (OGS) has been used as a simple model to study relaxation in N-body gravitational systems for several decades. In the OGS the gravitational field is uniform, and therefore simple algebraic equations which can be easily and rapidly solved on a computer govern the motion of the particles. Current computer technology allows for very long time dynamical simulations of reasonably large systems with little loss of numerical accuracy. Because the phase space of the OGS is compact, these systems do not suffer from some of the difficulties encountered in three dimensions (e.g., singularities, evaporation). However, a consequence of this is a somewhat tenuous connection with real galactic systems. Computer simulations of the OGS show that they tend to progress through various quasiequilibrium states as they evolve from arbitrary initial conditions. These quasiequilibrium states often last for very long times, and are approximately stationary. Fluctuations caused by changes in the mean field potential within the initial nonstationary distribution rapidly decay and the system reaches a state of microscopic relaxation, distinguished from the longer macroscopic time scale for relaxation to thermal equilibrium. Early predictions and dynamical simulations put thermal equilibrium on a time scale proportional to N 2 tc, where N is the number of particles and tc is the typical time for a particle to traverse the system [1,2]. Later work tended to refute these earlier claims by showing that the system was still far from equilibrium after 2N 2 tc [3,4].

Ergodic properties and equilibrium of one-dimensional self-gravitating systems

Recent studies of one-dimensional self-gravitating systems have raised new questions about their ergodic properties, what defines equilibrium for these systems, and their ability to reach a state of thermal equilibrium in a finite time. Earlier studies of small-$N$ systems $(N&lt;11)$ using Lyapunov exponents have shown that stable regions exist in the phase space which prevent these systems from thermalizing. Here we investigate several small-$N$ systems with specific initial states in an attempt to answer some of the questions of ergodicity and relaxation toward equilibrium which have been sparked by recent large-$N$ $(N=64)$ simulations. Using time averages of the specific particle energy deviations from equipartition, we see similar peaks occurring in the data for small-$N$ simulations as have been reported for large $N.$ Instead of being an indication of the onset of equilibrium, these peaks may indicate regions of the phase space where the system resides for extremely long periods of time. The existence of sticky regions in the phase space in both small- and large-$N$ systems raises questions about the structure of the phase space, relaxation, and the appropriateness of various tests of equilibrium. Here we show that equipartition is not sufficient to remove fundamental doubts concerning the system's ergodic properties.