Generalized uncertainty relations for semi-Markov processes

Abstract

The thermodynamic and kinetic uncertainty relations provide finite-time bounds on the observable fluctuation in Markov processes. Herein, we generalize these bounds for semi-Markov processes. Specifically, we prove that, unlike in the Markovian case, the fluctuation of time-antisymmetric observables is bounded not only by entropy production but also by a memory term. For generic observables, we analogously show that the fluctuation is bounded by both dynamical activity and a memory term. Our results indicate that memory plays an important role in the bounds. Interestingly, with a proper form of the waiting-time distribution, the memory can decrease the observable fluctuation. When the waiting-time distribution is Poissonian (i.e., the process becomes Markov), the memory terms vanish, and the derived bounds reduce to the conventional bounds.