  We discuss the dynamics and thermodynamics of the Hamiltonian Mean Field
model (HMF) which is a prototypical system with long-range interactions. The
HMF model can be seen as the one Fourier component of a one-dimensional
self-gravitating system. Interestingly, it exhibits many features of real
self-gravitating systems (violent relaxation, persistence of metaequilibrium
states, slow collisional dynamics, phase transitions,...) while avoiding
complicated problems posed by the singularity of the gravitational potential at
short distances and by the absence of a large-scale confinement. We stress the
deep analogy between the HMF model and self-gravitating systems by developing a
complete parallel between these two systems. This allows us to apply many
technics introduced in plasma physics and astrophysics to a new problem and to
see how the results depend on the dimension of space and on the form of the
potential of interaction. This comparative study brings new light in the
statistical mechanics of self-gravitating systems. We also mention simple
astrophysical applications of the HMF model in relation with the formation of
bars in spiral galaxies.
