  In order to study analytically the nature of the size segregation in granular
mixtures, we introduce a mean field theory in the framework of a statistical
mechanics approach, based on Edwards' original ideas. For simplicity we apply
the theory to a lattice model for hard sphere binary mixture under gravity, and
we find a new purely thermodynamic mechanism which gives rise to the size
segregation phenomenon. By varying the number of small grains and the mass
ratio, we find a crossover from Brasil nut to reverse Brasil nut effect, which
becomes a true phase transition when the number of small grains is larger then
a critical value. We suggest that this transition is induced by the effective
attraction between large grains due to the presence of small ones (depletion
force). Finally the theoretical results are confirmed by numerical simulations
of the 3d system under taps.
