  We show that small homogeneous solutions to the Einstein-Boltzmann-scalar
field system exist globally towards the future and tend to the de Sitter
solution in a suitable sense. More specifically, we assume that the spacetime
is of Bianchi type I--VIII, that the matter is described by Israel particles
and that there exists a scalar field with a potential which has a positive
lower bound. This represents a generalization of the work [19], where a
cosmological constant was considered, and a generalization of [16], where a
spatially flat FLRW spacetime was considered. We obtain the global existence
and asymptotic behavior of classical solutions to the Einstein-Boltzmann-scalar
field system for small initial data.
