where I is the current operator, and F j = − − ∂ ∂ H / ∂ ∂ X j is the generalized force that is associated with the control parameter X j . Though this formula is written using quantum mechanical notations it holds also classically if the commutator is replaced by Poisson brackets. In general G can be written as a sum of two terms: one has to do with dissipation, while the other, denoted as B has to do with geometry. The dissipative part vanishes in the strict quantum adiabatic limit, while the geometrical part B might be non-zero. It turns out that in the strict adiabatic limit B is the " Berry curvature " (mathematically known as ``two-form"). Using the notations B 1 = − − G 32 and B 2 = G 31 we can rewrite the formula for the amount of pumped particles as