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5.2 Fokker–Planck

The computational limitations of N-body simulations can be sidestepped by describing the system in terms of distribution functions f (x, v,t) m with the number of stars of mass m at time t in the range 3 (x, x + d x) and 3 (v, v + d v) given by 3 3 dN = fm d x d v. This description requires that either the phase-space element 3 3 d xd v be small enough to be infinitesimal yet large enough to be statistically meaningful, or that fm be interpreted as the probability distribution for finding a star of mass m at a location in phase space. The evolution of the cluster is then described by the evolution of fm. The gravitational interaction is provided by a smoothed gravitational potential φ, which is determined by
∑ [ ∫ ] ∇2φ = 4π mi fm (x,v, t) d3v . (28 ) i i
The effect of gravitational interactions is modeled by a collision term Γ [f] (see [24172] for specific descriptions of Γ). The dynamics of the globular cluster are then governed by the Fokker–Planck equation:
∂f ∂f ∂t-+ v ⋅ ∇f − ∇ φ ⋅∂v- = Γ [f]. (29 )

In the Fokker–Planck approach, the mass spectrum of stars is binned, with a separate f m 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. Consequently, Fokker–Planck codes can handle at most a few dozen different fm. The inclusion of additional physical variables such as binaries adds sufficient further complexity that the codes are taxed beyond their capacity. Methods for numerically solving the Fokker–Planck equation use either an orbit-averaged form of Equation (29View Equation[33], or a Monte Carlo approach [69Jump To The Next Citation Point77Jump To The Next Citation Point78126Jump To The Next Citation Point63].

The two time scales involved in the evolution of fm are tcross (which governs changes in position) and trelax (which governs changes in energy). The orbit-averaged form of Equation (29View Equation) derives from the realization that changes in position are essentially periodic with orbital period T ∼ t ≪ t cross relax. 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. Given suitable assumptions on the symmetry of the potential and the velocity distribution, when one does this, the Fokker–Planck equation is reduced to an equation involving the energy and the magnitude of the 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 [73]. 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 [129].

The more recent Monte Carlo simulations [77126Jump To The Next Citation Point69] do not actually deal with the distribution functions, but rather treat the cluster as a collection of particles that represent a spherical shell of similar stars. Based on the pioneering work of Hénon [103102], they are able to represent an arbitrary number of species and can follow binary evolution and other effects. The underlying treatment of relaxation throughout the simulation is done in the Fokker–Planck approximation, but the interactions and evolution of the stars are handled on a particle by particle basis. Consequently, these codes are significantly more robust in their ability to handle realistic populations of stars. A nearly continuous mass spectrum can be used, and stellar evolution and binarity can be included with relative ease. In addition, both stellar collisions and large-angle scatterings can also be tracked. The primary disadvantages of these Monte Carlo codes are that they require spherical symmetry and that they suffer from statistical noise despite the large number of particles being tracked. For an excellent overview of the implementations and history of the Monte Carlo methods based on Hénon’s work, see Marc Freitag’s link on the Working Group 3 page at the MODEST website [151].

Another approach to solving the Fokker–Planck equation makes use of the analogy between a globular cluster and a self-gravitating gaseous sphere [14580]. The most effective use of the gaseous models are in a hybrid code that treats the single stars in a gaseous model while treating the relaxation of binary, three-, and four-body interactions using a Monte Carlo code [81Jump To The Next Citation Point82Jump To The Next Citation Point]. This approach shows promise for its flexibility in adding new physics.


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