Stochastic Dynamics Toward the Steady State of Self-Gravitating Systems
  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.
