Brownian Dynamics around the Core of Self-Gravitating Systems

Abstract

We derive the non-Maxwellian distribution of self-gravitating N-body systems around the core by a model based on the random process with the additive and the multiplicative noise. The number density can be obtained through the steady state solution of the Fokker-Planck equation corresponding to the random process. We exhibit that the number density becomes equal to that of the King model around the core by adjusting the friction coefficient and the intensity of the multiplicative noise. We also show that our model can be applied in the system which has a heavier particle. Moreover, we confirm the validity of our model by comparing with our numerical simulation.How relevant is the equilibrium statistical mechanics when we describe the steady state of a self-gravitating system (SGS) where many particles interact via the gravitational force? Let's assume that the state of the SGS with equal mass m becomes isothermal with temperature T and the particles of the system will be distributed spherically symmetrically. Then, the structure in the phase space can be determined by the Maxwell-Boltzmann distribution. For example, the number density at a radial distance r in the real space iswhere Φ(r) is the mean gravitational potential per mass generated by this whole system at r and k B is the Boltzmann constant. This potential should satisfy a relation with the number density by the Poisson equation △Φ(r) = 4πGmn MB (r) where G is the gravitational constant. A special solution of eq.(1) and this Poisson equation is n MB (r) = k B T /2πGm 2 r 2 known as the singular isothermal sphere. 1) This solution has *

Stochastic Dynamics Toward the Steady State of Self-Gravitating Systems

“…Here, note that we have reported similar results in our previous letter [11]. In this paper, however, we demonstrate how we executed our numerical simulations.…”

Universal structure of two- and three-dimensional self-gravitating systems in the quasiequilibrium state

“…(1) with κ ∼ 3/2 [4][5][6]. Furthermore, we have reported that the same density profile is obtained through N-body simulations of 3DSGS [2,3]. Therefore, we can conclude that the SGSs without boundary have the universal density profile depicted by Eq.…”

“…QES of Hamiltonian mean field model which is a toy model of systems with long range interactions varies from a ferromagnetic state to a paramagnetic (homogeneous) state with increase of the initial total energy [1]. We have obtained numerically the result that the number density N of three dimensional self-gravitating system (3DSGS) in the real space can be approximated by the following representation when the initial virial ratio V (≡ 2K/Ω) is 0 where K and Ω are the total kinetic and gravitational energy respectively [2,3]:…”

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

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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 [1,2]. The reason why the null virial condition is sufficient for the universality is unveiled by the envelope equation.

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“…QES of Hamiltonian mean field model which is a toy model of systems with long range interactions varies from a ferromagnetic state to a paramagnetic (homogeneous) state with increase of the initial total energy [1]. We have obtained numerically the result that the number density N of three dimensional self-gravitating system (3DSGS) in the real space can be approximated by the following representation when the initial virial ratio V (≡ 2K/Ω) is 0 where K and Ω are the total kinetic and gravitational energy respectively [2,3]:…”