We study the seasonal changes in the thickness distribution of Arctic sea ice, g(h), under climate forcing. Our analytical and numerical approach is based on a Fokker-Planck equation for g(h) (Toppaladoddi and Wettlaufer in Phys Rev Lett 115(14): 148501, 2015), in which the thermodynamic growth rates are determined using observed climatology. In particular, the Fokker-Planck equation is coupled to the observationally consistent thermodynamic model of Eisenman and Wettlaufer (Proc Natl Acad Sci USA 106: 28-32, 2009). We find that due to the combined effects of thermodynamics and mechanics, g(h) spreads during winter and contracts during summer. This behavior is in agreement with recent satellite observations from CryoSat-2 (Kwok and Cunningham in Philos Trans R Soc A 373(2045): 20140157, 2015). Because g(h) is a probability density function, we quantify all of the key moments (e. g., mean thickness, fraction of thin/thick ice, mean albedo, relaxation time scales) as greenhouse-gas radiative forcing, Delta F-0, increases. The mean ice thickness decays exponentially with Delta F-0, but much slower than do solely thermodynamic models. This exhibits the crucial role that ice mechanics plays in maintaining the ice cover, by redistributing thin ice to thick ice-far more rapidly than can thermal growth alone.