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

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.