  The two-dimensional, periodic Lorentz gas, is the dynamical system
corresponding with the free motion of a point particle in a planar system of
fixed circular obstacles centered at the vertices of a square lattice in the
Euclidian plane. Assuming elastic collisions between the particle and the
obstacles, this dynamical system is studied in the Boltzmann-Grad limit,
assuming that the obstacle radius $r$ and the reciprocal mean free path are
asymptotically equivalent small quantities, and that the particle's
distribution function is slowly varying in the space variable. In this limit,
the periodic Lorentz gas cannot be described by a linear Boltzmann equation
(see [F. Golse, Ann. Fac. Sci. Toulouse 17 (2008), 735--749]), but involves an
integro-differential equation conjectured in [E. Caglioti, F. Golse, C.R. Acad.
Sci. S\'er. I Math. 346 (2008) 477--482] and proved in [J. Marklof, A.
Str\"ombergsson, preprint arXiv:0801.0612], set on a phase-space larger than
the usual single-particle phase-space. The main purpose of the present paper is
to study the dynamical properties of this integro-differential equation:
identifying its equilibrium states, proving a H Theorem and discussing the
speed of approach to equilibrium in the long time limit. In the first part of
the paper, we derive the explicit formula for a transition probability
appearing in that equation following the method sketched in [E. Caglioti, F.
Golse, loc. cit.].
