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We consider the spectrum of eigenmodes in a stellar system dominated bygravitational forces in the limit of zero collisions. We show analytically andnumerically using the Lenard-Bernstein collision operator that the Landaumodes, which are not true eigenmodes in a strictly collisionless system (exceptfor the Jeans unstable mode), become part of the true eigenmode spectrum in thelimit of zero collisions. Under these conditions, the continuous spectrum oftrue eigenmodes in the collisionless system, also known as the Case-van Kampenmodes, is eliminated. Furthermore, since the background distribution functionin a weakly collisional system can exhibit significant deviations from aMaxwellian distribution function over long times, we show that the spectrum ofLandau modes can change drastically even in the presence of slight deviationsfrom a Maxwellian, primarily through the appearance of weakly damped modes thatmay be otherwise heavily damped for a Maxwellian distribution. Our resultsprovide important insights for developing statistical theories to describethermal fluctuations in a stellar system, which are currently a subject ofgreat interest for N-body simulations as well as observations of gravitationalsystems.

Landau Modes are Eigenmodes of Stellar Systems in the Limit of Zero Collisions