3 - The dynamics of random impulsive forces

Summary

The slings and arrows of outrageous fortune.

Shakespeare

Previous sections have made it plausible that an object in a gravitating system near equilibrium can be considered to be immersed in a bath of fluctuating forces, along with an average mean field force. We now consider a simple mathematical model for the time evolution of orbits. We use this intuitive physical picture to try to capture the essence of the problem in a fairly simple way. An advantage of this procedure is that it readily suggests modifications of the description for an improved physical picture. The results can always be checked against N-body computer experiments, and we will discuss their more exact derivation in Section 10.

At first sight, the simplest model might seem to represent the motion of each star by Newton's equation of motion with a stochastic force β(t) which fluctuates in time, i.e.,. But this turns out to be too simple. It makes the velocity undergo Brownian motion (for a Gaussian distribution of fluctuations) with an everincreasing root mean square value v_rms ∝t^½. Correspondingly, the root mean square position of an average star also departs monotonically from its initial value. These two properties are inconsistent with conservation of total energy, for the increase in kinetic energy must be compensated by a contraction of the system to decrease the potential energy. But the Brownian increase of every star's root mean square position from its initial value prevents the system from becoming very small.