  Systems with long-range (LR) forces, for which the interaction potential
decays with the interparticle distance with an exponent smaller than the
dimensionality of the embedding space, remain an outstanding challenge to
statistical physics. The internal energy of such systems lacks extensivity and
additivity. Although the extensivity can be restored by scaling the interaction
potential with the number of particles, the non-additivity still remains. Lack
of additivity leads to inequivalence of statistical ensembles. Before relaxing
to thermodynamic equilibrium, isolated systems with LR forces become trapped in
out-of-equilibrium quasi-stationary state (qSS), the lifetime of which diverges
with the number of particles. Therefore, in thermodynamic limit LR systems will
not relax to equilibrium. The qSSs are attained through the process of
collisionless relaxation. Density oscillations lead to particle-wave
interactions and excitation of parametric resonances. The resonant particles
escape from the main cluster to form a tenuous halo. Simultaneously, this cools
down the core of the distribution and dampens out the oscillations. When all
the oscillations die out the ergodicity is broken and a qSS is born. In this
report, we will review a theory which allows us to quantitatively predict the
particle distribution in the qSS. The theory is applied to various LR
interacting systems, ranging from plasmas to self-gravitating clusters and
kinetic spin models.
