Abstract
Many natural and artificial systems are subject to some sort of delay, which can be in the form of a single discrete delay or distributed over a range of times. Here, we discuss the impact of this distribution on (thermo-)dynamical properties of time-delayed stochastic systems. To this end, we study a simple classical model with white and colored noise, and focus on the class of Gamma-distributed delays which includes a variety of distinct delay distributions typical for feedback experiments and biological systems. A physical application is a colloid subject to time-delayed feedback control, which is, in principle, experimentally realizable by co-moving optical traps. We uncover several unexpected phenomena in regard to the system’s linear stability and its thermodynamic properties. First, increasing the mean delay time can destabilize or stabilize the process, depending on the distribution of the delay. Second, for all considered distributions, the heat dissipated by the controlled system (e.g., the colloidal particle) can become negative, which implies that the delay force extracts energy and entropy of the bath. As we show here, this refrigerating effect is particularly pronounced for exponential delay. For a specific non-reciprocal realization of a control device, we find that the entropic costs, measured by the total entropy production of the system plus controller, are the lowest for exponential delay. The exponential delay further yields the largest stable parameter regions. In this sense, exponential delay represents the most effective and robust type of delayed feedback.
Highlights
In addition to the omnipresent noise, many complex systems are governed by dynamical equations that involve some sort of memory or time delay
We study a simple classical model with white and colored noise, and focus on the class of Gamma-distributed delays which includes a variety of distinct delay distributions typical for feedback experiments and biological systems
For a specific non-reciprocal realization of a control device, we find that the entropic costs, measured by the total entropy production of the system plus controller, are the lowest for exponential delay
Summary
In addition to the omnipresent noise, many complex systems are governed by dynamical equations that involve some sort of memory or time delay. Recent experiments on small-scale, fluctuating systems successfully apply feedback control involving video microscopy and co-moving laser traps to create virtual potentials [23,24] or realize arbitrary interactions between colloidal particles [25,26] Such experiments give rise to a stochastic motion with time delay of the feedback-controlled colloidal particles [2,23,24,25,26,27,28,29]. We discuss the total entropy production and the thermodynamic efficiency for a specific realization of a feedback controller involving non-reciprocal interactions between some linear, stochastic internal degrees of freedom (which we interpret as the “memory cells”) and gives rise to a delay force with Gamma-distributed delay.
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