### 5.2 Fokker-Planck

The computational limitations of -body simulations can be
sidestepped by describing the system in terms of distribution
functions with the number of stars of mass
at time in the range and given by .
This description requires that either the phase-space element be small enough to be infinitesimal yet large enough
to be statistically meaningful, or that be interpreted as
the probability distribution for finding a star of mass
at a location in phase space. The evolution of the cluster is then
described by the evolution of . The gravitational
interaction is provided by a smoothed gravitational potential , which is determined by
The effect of gravitational interactions is modeled by a collision
term (see [19, 117] for
specific descriptions of ). The dynamics of the
globular cluster are then governed by the Fokker-Planck equation:
In the Fokker-Planck approach, the mass spectrum
of stars is binned, with a separate for each bin.
Increasing the resolution of the mass spectrum requires increasing
the number of distribution functions and thus increasing the
complexity of the problem. The inclusion of additional physical
variables such as binaries adds further complexity. Methods for
numerically solving the Fokker-Planck equation use either an
orbit-averaged form of Equation (28) [25], or a Monte Carlo
approach [50, 55, 56, 89].

The two time scales involved in the evolution of
are (which governs changes in
position) and (which governs changes in energy). The
orbit-averaged form of Equation (28) derives from the
realization that changes in position are essentially periodic with
orbital period . Thus, one can average over the rapid changes in
position and retain the slow changes in the phase space coordinates
that occur over relaxation times. When one does this, the
Fokker-Planck equation is reduced to an equation involving three
phase-space variables and time. The most appropriate variables are
the energy and angular momentum. The orbit-averaged solutions of
the Fokker-Planck equation cannot easily handle the effect of
binaries and the binary interactions that occur during the
evolution of a globular cluster [52]. These effects are
usually inserted by hand using statistical methods. The advantages
of the orbit-averaged approach are that one can generalize it to
handle anisotropy in velocity, thus allowing study of the effects
of the galactic gravitational field and tidal stripping. One can
also include the rotation of the cluster [90].

Monte Carlo simulations do not actually solve the
Fokker-Planck equation, but rather rely on a representative sample
of test stars which are followed numerically with random
velocity perturbations applied as are appropriate to the diffusion
coefficients in . The advantage of the Monte Carlo
approach is that one can handle binary interactions and stellar
evolution directly (and then extrapolate the results of the
test stars to the actual stars in the cluster). However,
like Fokker-Planck, the Monte Carlo method currently requires the
cluster to be spherically symmetric, so gravitational effects of
the galaxy must be treated in an ad
hoc way.