In the mean-field regime, the evolution of a gas of
interacting particles is governed in first approximation by a Vlasov type
equation with a self-induced force field. This equation is conservative and
describes return to equilibrium only in the very weak sense of Landau
damping. However, the first correction to this approximation is given by the
Lenard–Balescu operator, which dissipates entropy on the very long timescale
. We
show how one can derive rigorously this correction on intermediate timescales (of
order
for
)
close to equilibrium.