Thermodynamic Uncertainty Relation for Arbitrary Initial States

TL;DR

This paper derives a finite-time thermodynamic uncertainty relation applicable to arbitrary initial states, expanding the scope of TURs beyond steady or specific initial conditions, with implications for feedback control and discrete-time systems.

Contribution

The authors introduce a new finite-time TUR valid for arbitrary initial states, derived from the Cramér-Rao inequality, and applicable to both continuous and discrete-time Markov processes.

Findings

The variance of accumulated current is bounded by the instantaneous current at the final time.

The new TUR explains recent experimental violations involving feedback control.

The bound exponentially improves existing bounds in discrete-time regimes.

Abstract

The thermodynamic uncertainty relation (TUR) describes a trade-off relation between nonequilibrium currents and entropy production and serves as a fundamental principle of nonequilibrium thermodynamics. However, currently known TURs presuppose either specific initial states or an infinite-time average, which severely limits the range of applicability. Here we derive a finite-time TUR valid for arbitrary initial states from the Cram\'er-Rao inequality. We find that the variance of an accumulated current is bounded by the instantaneous current at the final time, which suggests that ``the boundary is constrained by the bulk". We apply our results to feedback-controlled processes and successfully explain a recent experiment which reports a violation of a modified TUR with feedback control. We also derive a TUR that is linear in the total entropy production and valid for discrete-time Markov chains with non-steady initial states. The obtained bound exponentially improves the existing bounds in a discrete-time regime.