Landau Modes are Eigenmodes of Stellar Systems in the Limit of Zero
  Collisions
  We consider the spectrum of eigenmodes in a stellar system dominated by
gravitational forces in the limit of zero collisions. We show analytically and
numerically using the Lenard-Bernstein collision operator that the Landau
modes, which are not true eigenmodes in a strictly collisionless system (except
for the Jeans unstable mode), become part of the true eigenmode spectrum in the
limit of zero collisions. Under these conditions, the continuous spectrum of
true eigenmodes in the collisionless system, also known as the Case-van Kampen
modes, is eliminated. Furthermore, since the background distribution function
in a weakly collisional system can exhibit significant deviations from a
Maxwellian distribution function over long times, we show that the spectrum of
Landau modes can change drastically even in the presence of slight deviations
from a Maxwellian, primarily through the appearance of weakly damped modes that
may be otherwise heavily damped for a Maxwellian distribution. Our results
provide important insights for developing statistical theories to describe
thermal fluctuations in a stellar system, which are currently a subject of
great interest for N-body simulations as well as observations of gravitational
systems.
