Hamiltonian Derivations of the Generalized Jarzynski Equalities under Feedback Control

In the presence of feedback control by "Maxwell's demon," the second law of thermodynamics and the nonequilibrium equalities such as the Jarzynski equality need to be generalized. In this paper, we derive the generalized Jarzynski equalities for classical Hamiltonian dynamics based on the Liouville's theorem, which is the same approach as the original proof of the Jarzynski equality [Phys. Rev. Lett. 78, 2690 (1997)]. The obtained equalities lead to the generalizations of the second law of thermodynamics for the Hamiltonian systems in the presence of feedback control.