A classical result of Bunimovich, Sinai, and Chernov is that the sum of the displacements between successive collisions in the periodic Lorentz gas satisfies a central limit theorem. We study the fluctuations of order 1 for such averages and prove that they satisfy a large deviation principle. We will also discuss some applications to nonequilibrium statistical mechanics (thermostatted Lorentz gas). This is joint work with Lai-Sang Young (Courant Institute).