Heat flow due to time-delayed feedback

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.

Entropy Production in Exactly Solvable Systems

“…This implies, at least for Markovian systems, the existence of steady-state probability currents in the state space, which change sign under time-reversal. When a thermodynamically consistent description is available, the average rate of entropy production can be related to the rate of energy or information exchange between the system, the heat bath(s) it is connected to, and any other thermodynamic entity involved in the dynamics, such as a measuring device [ 15 , 16 , 17 ]. Whilst the rate of energy dissipation is of immediate interest since it captures how ‘costly’ it is to sustain specific dynamics (e.g., the metabolism sustaining the development of an organism [ 18 , 19 ]), entropy production has also been found to relate non-trivially to the efficiency and precision of the corresponding process via uncertainty relations [ 3 , 20 ].…”

Multimode cold-damping optomechanics with delayed feedback

“…From Eq. (39) we see that next to the term expressing the singlemode solution, additional terms describing mode-to-mode coupling emerge, which describe mutual heating effects. For independent modes (such that |ω 1 − ω 2 | → ∞) the mutual heating vanishes.…”

“…We provide here a more in-depth analytical treatment of simultaneous cold damping of many mechanical resonances [33][34][35] and address a crucial aspect extremely relevant in experiments, i.e., the inherent time delay τ inh that characterizes any electronic feedback loop. It is generally agreed that the delayed action of the feedback loop can lead to unwanted heating eventually leading to an instability [36][37][38][39]. Extending the analytical approach that we have previously introduced in Ref.…”

Explicit Solution of the Generalised Langevin Equation

“…It is important to underline that one can not explicitly calculate the underdamped susceptibilities and take the massless limit m → 0 afterwards because this would lead to inconsistencies, as it can be seen in [22]. However, the direct solution of the overdamped dynamics (63) can be found (dropping the "over" superscript):…”