Consider a closed system which is described by a Hamiltonian H ( X ) that depends on some control parameters X = ( X 1 , X 2 , X 3 ) . If X 3 is an Aharonov Bohm magnetic flux through the ring, then by Faraday law − − X 3 ˙ ˙ is the electro motive force. If linear response theory applies we have the proportionality I = − − G 33 X ˙ ˙ 3 , where G 33 is the called the Ohmic conductance. In complete analogy if we change X 1 the current is I = − − G 31 X ˙ ˙ 1 , and if we change X 2 the current is I = − − G 32 X ˙ ˙ 2 , where G 31 and G 32 are elements of a conductance matrix. Accordingly, for a full pumping cycle: