Abstract
Many stochastic systems in biology, physics and technology involve discrete time delays in the underlying equations of motion, stemming, e. g., from finite signal transmission times, or a time lag between signal detection and adaption of an apparatus. From a mathematical perspective, delayed systems represent a special class of non-Markovian processes with delta-peaked memory kernels. It is well established that delays can induce intriguing behaviour, such as spontaneous oscillations, or resonance phenomena resulting from the interplay between delay and noise. However, the thermodynamics of delayed stochastic systems is still widely unexplored. This is especially true for continuous systems governed by nonlinear forces, which are omnipresent in realistic situations. We here present an analytical approach for the net steady-state heat rate in classical overdamped systems subject to time-delayed feedback. We show that the feedback inevitably leads to a finite heat flow even for vanishingly small delay times, and detect the nontrivial interplay of noise and delay as the underlying reason. To illustrate this point, and to provide an understanding of the heat flow at small delay times below the velocity-relaxation timescale, we compare with the case of underdamped motion where the phenomenon of “entropy pumping” has already been established. Application to an exemplary (overdamped) bistable system reveals that the feedback induces heating as well as cooling regimes and leads to a maximum of the medium entropy production at coherence resonance conditions. These observations are, in principle, measurable in experiments involving colloidal suspensions.
Highlights
A finite heat flow is a generic feature of systems out of thermal equilibrium
The second law does not impose nonnegativity on ΔSm alone27–29. While these statements are generic, explicit expressions for the thermodynamic quantities are, so far, only available for systems governed by linear forces27–30, excluding wide classes of physically interesting processes which exclusively arise in nonlinear systems
We have shown that the heat rate can be evaluated based on positional moments, implying that, despite the inherent memory, no temporal correlations are needed
Summary
A finite heat flow is a generic feature of systems out of thermal equilibrium. In the last decades, special interest has been devoted to heat exchange and other thermodynamic properties of small (mesoscopic) systems coupled to a bath, which are noisy per se. By considering Q = S m rather than S tot, we avoid the above-mentioned problem induced by acausality, and at the same time consider a key thermodynamic quantity and nontrivial part of the total entropy production, which already provides important physical insight into the thermodynamics of delayed systems. This strategy enables us to address several fundamental questions: What is the impact of time-delayed feedback on heat exchange and entropy production? Plugging the LE [1] into Eqs [3, 4] results in the NESS ensemble averages
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