Direct Route to Thermodynamic Uncertainty Relations and Their Saturation

TL;DR

This paper provides a direct proof of thermodynamic uncertainty relations (TURs) from Langevin equations, extends TURs to time-dependent currents and densities, and introduces a sharper TUR for transient dynamics, enhancing thermodynamic inference.

Contribution

It offers the first direct proof of TURs from Langevin equations, extends TURs to transient and time-dependent cases, and derives a new sharpened TUR for transient dynamics.

Findings

Direct proof of TURs from Langevin equations.

Extension of TURs to time-dependent currents and densities.

Derivation of a sharpened TUR for transient dynamics.

Abstract

Thermodynamic uncertainty relations (TURs) bound the dissipation in non-equilibrium systems from below by fluctuations of an observed current. Contrasting the elaborate techniques employed in existing proofs, we here prove TURs directly from the Langevin equation. This establishes the TUR as an inherent property of overdamped stochastic equations of motion. In addition, we extend the transient TUR to currents and densities with explicit time-dependence. By including current-density correlations we, moreover, derive a new sharpened TUR for transient dynamics. Our arguably simplest and most direct proof, together with the new generalizations, allows us to systematically determine conditions under which the different TURs saturate and thus allows for a more accurate thermodynamic inference. Finally we outline the direct proof also for Markov jump dynamics.