Heat production in a stochastic system with nonlinear time-delayed feedback

TL;DR

This study investigates heat production in a stochastic system with nonlinear, time-delayed feedback, revealing nonzero heat dissipation below the threshold for persistent motion and complex behavior above it, including non-Gaussian heat distribution.

Contribution

It provides analytical and numerical analysis of heat production in a nonlinear delayed feedback system, highlighting nonequilibrium effects and the impact of delay on heat dissipation.

Findings

Nonzero heat production rate below the threshold indicates nonequilibrium behavior.

Heat production peaks at an optimal delay time above the threshold.

Heat distribution is non-Gaussian, especially beyond the threshold.

Abstract

Using the framework of stochastic thermodynamics we study heat production related to the stochastic motion of a particle driven by repulsive, nonlinear, time-delayed feedback. Recently it has been shown that this type of feedback can lead to persistent motion above a threshold in parameter space [Physical Review E 107, 024611 (2023)]. Here we investigate, numerically and by analytical methods, the rate of heat production in the different regimes around the threshold to persistent motion. We find a nonzero average heat production rate, $\langle \dot{q}\rangle$, already below the threshold, indicating the nonequilibrium character of the system even at small feedback. In this regime, we compare to analytical results for a corresponding linearized delayed system and a small-delay approximation which provides a reasonable description of $\langle \dot{q}\rangle$ at small repulsion (or delay time). Beyond the threshold, the rate of heat production is much larger and shows a maximum as function of the delay time. In this regime, $\langle \dot{q}\rangle$ can be approximated by that of a system subject to a constant force stemming from the long-time velocity in the deterministic limit. The distribution of dissipated heat, however, is non-Gaussian, contrary to the constant-force case.