Stochastic Dynamics Toward the Steady State of Self-Gravitating Systems

TL;DR

This paper develops a stochastic model using a Langevin equation with additive and multiplicative noise to explain the approach to steady states in self-gravitating systems like globular clusters and galaxies, supported by numerical simulations.

Contribution

It introduces a novel stochastic framework with a specialized Langevin equation to describe the dynamics of self-gravitating systems toward equilibrium, including non-Maxwellian distributions.

Findings

Derived non-Maxwellian density profiles around the core.

Showed the model matches observed density profiles by tuning parameters.

Applied the model to systems with heavier particles, like black holes.

Abstract

We will construct a theory which can explain the dynamics toward the steady state self-gravitating systems (SGSs) where many particles interact via the gravitational force. Real examples of SGS in the universe are globular clusters and galaxies. The idea is to represent an interaction by which a particle of the system is affected from the others by a special random force. That is, we will use a special Langevin equation, just as the normal Langevin equation can unveil the dynamics toward the steady state described by the Maxwell-Boltzmann distribution. However, we cannot introduce the randomness into the system without any evidence. Then, we must confirm that each orbit is random indeed. Of course, it is impossible to understand orbits of stars in globular clusters from observations. Thus we use numerical simulations. From the numerical simulations of SGS, grounds that we use the random noise become clear. The special Langevin equation includes the additive and the multiplicative noise. By using the random process, we derive the non-Maxwellian distribution of SGS especially around the core. 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 the density profiles around the core by adjusting the friction coefficient and the intensity of the multiplicative noise. Moreover, we also show that our model can be applied in the system which has a heavier particle, corresponding to the black hole in a globular cluster.