We will now use equation (21) to evaluate the fluctuation energy at different spatial scales assuming that individual particles are uncorrelated. The particle wakes do in fact give rise to correlations but this is of higher order in 1/N in the BBGKY expansion (cf. Gilbert 1969) than the lowest-order effect we will consider here.
This leaves us with individual particles reacting coherently to the
effect of their own wakes. Because the particles are uncorrelated,
the number density of particles at at time 0
and at at time t is
where is the equilibrium particle distribution with
Direct substitution demonstrates that equation (23) solves the Liouville equation with the initial condition and at t=0. Similarly, integrating equation (23) over all coordinates gives N.
For a given harmonic lm, the fluctuation energy is then
where the expectation value of some quantity is defined by
Applying equations (21) and
(26) to equation (25) gives
Gathering terms, this can be simplified as follows:
Note that each term in the fluctuation energy, equation (28), is negative definite as expected. The contribution for each triple in the angle expansion, , and each term in the basis expansion k may be tabulated separately.
We may compute the fluctuation energy in the absence of gravity by
returning to equation (25) and evaluating without any dynamics. For N
particles, the sample value for is Using
the expectation defined by equation (26) one finds:
This is identical to equation (28) with out the particle dressing: .