Stochastic thermodynamics of Langevin systems under time-delayed feedback control: I. Second-law-like inequalities

TL;DR

This paper develops second-law-like inequalities for small Langevin systems with time-delayed feedback, revealing how measurement and non-Markovian effects influence entropy production and work extraction in nonequilibrium steady states.

Contribution

It introduces a theoretical framework for analyzing thermodynamics of Langevin systems with time delays, including new inequalities and modifications to fluctuation relations.

Findings

Derived bounds on work extraction in delayed feedback systems.

Identified entropy reduction contributions from measurement processes.

Validated the formalism with analytical and numerical studies of a harmonic oscillator.

Abstract

Response lags are generic to almost any physical system and often play a crucial role in the feedback loops present in artificial nanodevices and biological molecular machines. In this paper, we perform a comprehensive study of small stochastic systems governed by an underdamped Langevin equation and driven out of equilibrium by a time-delayed continuous feedback control. In their normal operating regime, these systems settle in a nonequilibrium steady state in which work is permanently extracted from the surrounding heat bath. By using the Fokker-Planck representation of the dynamics, we derive a set of second-law-like inequalities that provide bounds to the rate of extracted work. These inequalities involve additional contributions characterizing the reduction of entropy production due to the continuous measurement process. We also show that the non- Markovian nature of the dynamics requires a modification of the basic relation between dissipation and time-reversal symmetry breaking. The new relation includes a contribution arising from the acausal character of the reverse process. This in turn leads to another second-law-like inequality. We illustrate the general formalism by a detailed analytical and numerical study of a harmonic oscillator driven by a linear feedback, which describes actual experimental setups.