Information thermodynamics for a multi-feedback process with time delay

TL;DR

This paper explores the thermodynamics of a multi-feedback process with time delay, deriving bounds on heat flow and analyzing stability and heating effects in a system with measurement and feedback delays.

Contribution

It introduces a thermodynamic framework for feedback with time delay, deriving tighter bounds on heat flow and analyzing stability in delayed feedback systems.

Findings

Tighter bounds on heat flow are derived considering memory states.

The long-time limit shows an instability transition and heating effects.

The process's stability depends on measurement errors, delay, and protocol strength.

Abstract

We investigate a measurement-feedback process of repeated operations with time delay. During a finite-time interval, measurement on the system is performed and the feedback protocol derived from the measurement outcome is applied with time delay. This protocol is maintained into the next interval until a new protocol from the next measurement is applied. Unlike a feedback process without delay, both memories associated with previous and present measurement outcomes are involved in the system dynamics, which naturally brings forth a joint system described by a system state and two memory states. The thermodynamic second law provides a lower bound for heat flow into a thermal reservoir by the (3-state) Shannon entropy change of the joint system. However, as the feedback protocol depends on memory states sequentially, we can deduce a tighter bound for heat flow by integrating out irrelevant memory states during dynamics. As a simple example, we consider the so-called cold damping feedback process where the velocity of a particle is measured and a dissipative feedback protocol is applied to decelerate the particle. We confirm that the heat flow is well above the tightest bound. We also examine the long-time limit of this feedback process, which turns out to exhibit an interesting instability transition as well as heating by controlling parameters such as measurement errors, time interval, protocol strength, and time delay length. We discuss the underlying mechanism for instability and heating, which might be unavoidable in reality.