“…Examples include systems with arbitrary initial states [16], systems with time-dependent driving [17], discrete-time processes [16,18,19], systems without time-reversal symmetry [11,20,21] including underdamped Langevin systems [10,[22][23][24], waiting-time statistics [25][26][27][28][29], and processes with unidirectional transitions [30]. Another direction is to replace the entropy production with other quantities such as activity [27,29,31] and probability current [30], which provide a new type of TUR-type inequalities. The short-time limit of the TUR [32][33][34][35] has also been discussed in the context of inference of entropy production [34,35].…”