A unifying picture of generalized thermodynamic uncertainty relations

The thermodynamic uncertainty relation is a universal trade-off relation connecting the precision of a current with the average dissipation at large times. For continuous time Markov chains (also called Markov jump processes) this relation is valid in the time-homogeneous case, while it fails in the time-periodic case. The latter is relevant for the study of several small thermodynamic systems. We consider here a time-periodic Markov chain with continuous time and a broad class of functionals of stochastic trajectories, which are general linear combinations of the empirical flow and the empirical density. Inspired by the analysis done in our previous work Barato et al (2018 New J. Phys. 20 103023), we provide general methods to get local quadratic bounds for large deviations, which lead to universal lower bounds on the ratio of the diffusion coefficient to the squared average value in terms of suitable universal rates, independent of the empirical functional. These bounds are called 'generalized thermodynamic uncertainty relations' (GTUR's), being generalized versions of the thermodynamic uncertainty relation to the time-periodic case and to functionals which are more general than currents. Previously, GTUR's in the time-periodic case have been obtained in Barato et al (2018 New J. Phys. 20 103023); Koyuk et al (2019 J. Phys. A: Math. Theor. 52 02LT02); Proesmans and Van den Broeck (2017 Europhys. Lett. 119 20001). Here we recover the GTUR's in Barato et al (2018 New J. Phys. 20 103023); Koyuk et al (2019 J. Phys. A: Math. Theor. 52 02LT02) and produce new ones, leading to even stronger bounds and also to new trade-off relations for time-homogeneous systems. Moreover, we generalize to arbitrary protocols the GTUR obtained in Proesmans and Van den Broeck (2017 Europhys. Lett. 119 20001) for time-symmetric protocols. We also generalize to the time-periodic case the GTUR obtained in Garrahan (2017 Phys. Rev. E 95 032134) for the so called dynamical activity, and provide a new GTUR which, in the time-homogeneous case, is stronger than the one in Garrahan (2017 Phys. Rev. E 95 032134). The unifying picture is completed with a comprehensive comparison between the different GTUR's.