Yasuhide Sota

Chaos in static axisymmetric spacetimes: I. Vacuum case

We study the motion of a test particle in static axisymmetric vacuum spacetimes and discuss two criteria for strong chaos to occur: (i) a local instability measured by the Weyl curvature, and (ii) a tangle of a homoclinic orbit, which is closely related to an unstable periodic orbit in general relativity. We analyse several static axisymmetric spacetimes and find that the first criterion is a sufficient condition for chaos, at least qualitatively. Although some test particles which do not satisfy the first criterion show chaotic behaviour in some spacetimes, these can be accounted for by the second criterion.We study the motion of test particle in static axisymmetric vacuum spacetimes and discuss two criteria for strong chaos to occur: (1) a local instability measured by the Weyl curvature, and (2) a tangle of a homoclinic orbit, which is closely related to an unstable periodic orbit in general relativity. We analyze several static axisymmetric spacetimes and find that the first criterion is a sufficient condition for chaos, at least qualitatively. Although some test particles which do not satisfy the first criterion show chaotic behavior in some spacetimes, these can be accounted for the second criterion.

Origin of scaling structure and non-Gaussian velocity distribution in a self-gravitating ring model

Fractal structures and non-Gaussian velocity distributions are characteristic properties commonly observed in virialized self-gravitating systems, such as galaxies and interstellar molecular clouds. We study the origin of these properties using a one-dimensional ring model that we propose in this paper. In this simple model, N particles are moving, on a circular ring fixed in three-dimensional space, with mutual interaction of gravity. This model is suitable for the accurate symplectic integration method by which we argue the phase transition in this system. Especially, in between the extended phase and the collapsed phase, we find an interesting phase (halo phase) that has negative specific heat at the intermediate energy scale. Moreover, in this phase, there appear scaling properties and nonthermal and non-Gaussian velocity distributions. In contrast, these peculiar properties are never observed in other gas and core phases. Particles in each phase have a typical time scale of motions determined by the cutoff length xi, the ring radius R, and the total energy E. Thus all relaxation patterns of the system are determined by these three time scales.

Universal non-Gaussian velocity distribution in violent gravitational processes.

We study the velocity distribution in spherical collapses and cluster-pair collisions by use of N -body simulations. Reflecting the violent gravitational processes, the velocity distribution of the resultant quasistationary state generally becomes non-Gaussian. Through the strong mixing of the violent process, there appears a universal non-Gaussian velocity distribution, which is a democratic (equal-weighted) superposition of many Gaussian distributions (DT distribution). This is deeply related with the local virial equilibrium and the linear mass-temperature relation which characterize the system. We show the robustness of this distribution function against various initial conditions which leads to the violent gravitational process. The DT distribution has a positive correlation with the energy fluctuation of the system. On the other hand, the coherent motion such as the radial motion in the spherical collapse and the rotation with the angular momentum suppress the appearance of the DT distribution.

Local virial relation for self-gravitating system.

We demonstrate that the quasi-equilibrium state in a self-gravitating N-body system after cold collapse is uniquely characterized by the local virial relation using numerical simulations. Conversely, assuming the constant local virial ratio and Jeans equation for a spherically steady-state system, we investigate the full solution space of the problem under the constant anisotropy parameter and obtain some relevant solutions. Specifically, the local virial relation always provides a solution which has a power-law density profile in both the asymptotic regions r --> 0 and infinity. This type of solution is commonly observed in many numerical simulations. Only the anisotropic velocity dispersion controls this asymptotic behavior of density profile.

Local virial relation and velocity anisotropy for collisionless self-gravitating systems

The collisionless quasi-equilibrium state realized after the cold collapse of self-gravitating systems has two remarkable characters. One of them is the linear temperature-mass (TM) relation, which yields a characteristic non-Gaussian velocity distribution. The other is the local virial (LV) relation, the virial relation which holds even locally in collisionless systems through phase mixing such as cold-collapse. A family of polytropes is examined from a view point of these two characters. The LV relation imposes a strong constraint on these models: only polytropes with index n ∼ 5 with a flat boundary condition at the center are compatible with the numerical results, except for the outer region. Using the analytic solutions based on the static and spherical Jeans equation, we show that this incompatibility in the outer region implies the important effect of anisotropy of velocity dispersion. Furthermore, the velocity anisotropy is essential in explaining various numerical results under the condition of the local virial relation.The collisionless quasi-equilibrium state realized after the cold collapse of self-gravitating systems has two remarkable characters. One of them is the linear temperature-mass (TM) relation, which yields a characteristic non-Gaussian velocity distribution. Another is the local virial (LV) relation, the virial relation which holds even locally in collisionless systems through phase mixing such as cold-collapse. A family of polytropes are examined from a view point of these two characters. The LV relation imposes a strong constraint on these models: only polytropes with index

Self-organized relaxation in a collisionless gravitating system.

n \sim 5

Renormalization group approach in Newtonian cosmology

with a flat boundary condition at the center are compatible with the numerical results, except for the outer region. Using the analytic solutions based on the static and spherical Jeans equation, we show that this incompatibility in the outer region implies the important effect of anisotropy of velocity dispersion. Furthermore, the velocity anisotropy is essential in explaining various numerical results under the condition of the local virial relation.

Local virial relation and velocity anisotropy in self-gravitating system

Abstract We propose the self-organized relaxation process which drives a collisionless self-gravitating system (SGS) 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 LV relation is attained if the LV ratio exceeds the crticaial 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 pre-saturated state. However, small phase-space perturbation can bring the inactive part into the LV critical state.

Renormalization Group Approach in Newtonian Cosmology

We apply the renormalization group (RG) method to examine the observable scaling properties in Newtonian cosmology. The original scaling properties of the equations of motion in our model are modified for averaged observables on constant time slices. In the RG flow diagram, we find three robust fixed points: Einstein\char21{}de Sitter, Milne, and quiescent fixed points. Their stability (or instability) property does not change under the effect of fluctuations. Inspired by the inflationary scenario in the early Universe, we set the Einstein\char21{}de Sitter fixed point with small fluctuations as the boundary condition at the horizon scale. Solving the RG equations under this boundary condition toward the smaller scales, we find a generic behavior of observables such that the density parameter

Statistical mechanics of self-gravitating system: Cluster expansion method

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