Extended fluctuation theorems for repeated measurements and feedback within Hamiltonian framework

• Extended Fluctuation Theorems under multiple measurements and feedback have been derived using Hamiltonian dynamics.
• We prove the theorems without using the notion of reverse trajectory.
• We show that the correction terms are not unique.
• The efficacy parameter is shown to have a unique physical meaning.

We derive the extended fluctuation theorems in presence of multiple measurements and feedback, when the system is governed by Hamiltonian dynamics. We use only the forward phase space trajectories in the derivation. However, to obtain an expression for the efficacy parameter, we must necessarily use the notion of reverse trajectory. Our results show that the correction term appearing in the exponent of the extended fluctuation theorems is non-unique, whereas the physical meaning of the efficacy parameter is unique.