  The relaxation of a weakly collisional plasma is described by the
Boltzmann-Poisson equations with the Lenard-Bernstein collision operator. We
perform a perturbative analysis of these equations, and obtain, for the first
time, exact analytic solutions, enabling definitive resolutions to
long-standing controversies regarding the impact of weak collisions on
continuous spectra and Landau eigenmodes. Unlike some previous studies, we
retain both damping and diffusion terms in the collision operator. We find that
the linear response is a temporal convolution of a continuum that depends on
the continuous velocities of particles, and discrete normal modes that
encapsulate coherent oscillations. The normal modes are exponentially damped
over time due to collective effects (Landau damping) as well as collisional
dissipation. The continuum is also damped by collisions but somewhat
differently. Up to a collision time, which is the inverse of the collision
frequency $\nu_{\mathrm{c}}$, the continuum decay is driven by velocity
diffusion and occurs super-exponentially over a timescale $\sim
\nu^{-1/3}_{\mathrm{c}}$. After a collision time, however, the continuum decay
is driven by the collisional damping of particle velocities and diffusion of
their positions, and occurs exponentially over a timescale $\sim
\nu_{\mathrm{c}}$. This hitherto unknown, slow exponential decay causes
perturbations to damp the most on scales comparable to the mean free path, but
very slowly on larger scales, which establishes the local thermal equilibrium,
the essence of the fluid limit. The long-term decay of the response is driven
by the normal modes on scales smaller than the mean free path, but, on larger
scales, is governed by the slowly decaying continuum and the least damped
normal mode. Our analysis firmly establishes a long-sought connection between
the collisionless and fluid limits of weakly collisional plasmas.
