This expression agrees with the expression given by Zwanzig, except that Zwanzig subsumes H ( Γ Γ ) in the slow variables. The Zwanzig projection operator fulfills P G ( A ( Γ Γ ) ) = G ( A ( Γ Γ ) ) and P 2 = P . The fast part of f ( Γ Γ ) is ( 1 − − P ) f ( Γ Γ ) . Functions of slow variables and in particular products of slow variables are slow variables. The space of slow variables thus is an algebra. The algebra in general is not closed under the Poisson bracket, including the Poisson bracket with the Hamiltonian.