Thermodynamic interpretation to stochastic Fisher information and single-trajectory speed limits

The Fisher information (FI) metric is a Riemannian metric that allows a geometric treatment of stochastic thermodynamics, introducing the possibility of computing thermodynamic lengths and deviations from equilibrium. At the trajectory level, a related quantity can be introduced, the stochastic Fisher information (SFI), which, on average, is equivalent to the FI. In this work, we discuss two fundamental questions regarding the SFI; namely, (i) what is the thermodynamic interpretation of the SFI, and (ii) are there any trajectory-level thermodynamic bounds. We find that, contrary to previous results in the literature for the FI, the thermodynamic interpretation of the SFI depends only on the entropy produced by the system and on the thermodynamic force. Moreover, we find that the SFI allows one to derive single-trajectory speed limits, which we demonstrate to hold for a Brownian particle under a saturating drive force and a Brownian particle under a decreasing drive force. From the ensemble of single-trajectory bounds, one can derive a hierarchy of average speed limits that are always less tight than the one derived from the FI. We test our results for speed limits on the adopted models and find that the hierarchy of average speed limits is respected and that the single-trajectory speed limits behave qualitatively similar to the average and stochastic speed limits, with some trajectories achieving velocities higher than the tightest average bound whenever it does not saturate. Our results open avenues for the exploration of uncertainty relations at the trajectory level.