Thermodynamic uncertainty relation for underdamped dynamics driven by time-dependent protocols

Chulan Kwon,Hyun Keun Lee

doi:10.1088/1367-2630/ac2c8f

Chulan Kwon, Hyun Keun Lee

Open Access

https://doi.org/10.1088/1367-2630/ac2c8f

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Abstract

The thermodynamic uncertainty relation (TUR) for overdamped dynamics has recently been derived for general time-dependent protocols. However, its precedent TUR for underdamped dynamics has not been known yet. Even for the case of steady states, a proper way to match underdamped and overdamped TURs has not been found. We derive the TUR for underdamped systems subject to general time-dependent protocols, that covers steady states, by using the celebrating Cramér–Rao inequality. We show the resultant TUR to give rise to the inequality for the product of the variance and entropy production. We prove it to approach to the known overdamped result for large viscosity limit. It is the eventual step to pursue the TUR for Langevin systems. We present three examples to confirm our rigorous result.

Highlights

Non-equilibrium fluctuation in timeaccumulated observables such as work, heat, and entropy production (EP) has been an important subject since the discovery of the fluctuation theorem (FT) for deterministic systems about two decades ago [1–3]
We investigate the thermodynamic uncertainty relation (TUR) for underdamped dynamics subject to general time-dependent protocols which goes to Eq (2) for large viscosity
Out of many choices for the adjoint dynamics, we propose one to give rise to the TUR which relates the variance of Φ and the total entropy production (EP), and goes to that for overdamped dynamics for large γ recently derived by Koyuk and Seifert [34]

Summary

Introduction

Non-equilibrium fluctuation in timeaccumulated observables such as work, heat, and entropy production (EP) has been an important subject since the discovery of the fluctuation theorem (FT) for deterministic systems about two decades ago [1–3]. Thermodynamic uncertainty relation for underdamped dynamics driven by time-dependent protocols We derive the TUR for underdamped systems subject to general time-dependent protocols, that covers steady states, by using the Cramér-Rao inequality.

Results

Conclusion