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, we consider this dynamical system in the Boltzmann-Grad limit, i.e. 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. While it is known that the particle's distribution function in that limit cannot be governed by a linear Boltzmann equation [F.Golse, Ann. Fac. Sci. Toulouse 17 (2008), 735--749], we propose a limiting theory involving an extended phase space larger than the usual phase space of a classical point particle in which the limiting particle's distribution function evolves under an integro-differential equation somewhat analogous to a linear Boltzmann equation. The particle's distribution function in the classical phase space is obtained by averaging out the additional variables involved in the extended phase space. We derive explicit formulas for the coefficients appearing in that integro-differential equation, and study its basic dynamical properties --- especially, we establish an analogue of Boltzmann's H theorem, provide a complete description of the equilibrium distribution functions and investigate the convergence to these equilibrium distribution functions in the long time limit. This article provides in particular complete proofs of the results announced in [E. Caglioti, F. Golse, C.R. Acad. Sci. Ser. I Math. 346 (2008) 477--482].