Stochastic thermodynamics provides a consistent description of a wide class of Langevin systems, but the Markov assumption is often essential. While some non-Markovian systems have been studied in great detail, the important case of feedback control with a discrete delay is still insufficiently understood [1,2]. The thermodynamical description of nonlinear delayed systems turns out to be particularly challenging.
In this talk, I will consider the paradigmatic example of a Brownian particle in a doublewell potential subject to time-delayed feedback in its non-equilibrium steady state. We use different strategies to tackle the technical challenges arising due to the delay, such as closure schemes for the infinite Fokker-Planck hierarchy [3,4] and a Markovian embedding technique [4,5].
We find an unavoidable heat flow induced by the control (even in the absence of any additional driving), i.e., the feedback inevitably cools down or heats up the system [1]. Interestingly, the heat flow is enhanced in the limit of small delay times, which is a phenomenon related to entropy pumping. We further compute the total entropy production of the super system, consisting of the particle plus the memory of the controller. By varying the information capacity of the memory device, we can consider different Gamma-distributed delays, including the limit of an infinitely sharp (delta-peaked) kernel, i. e., discrete delay. We identify an entropic contribution due to the memory, which is a consequence of the unidirectional information flow induced by the feedback-controller. This contribution diverges in the case of error-free measurement, as well as in the case of a perfectly sharp (delta-peaked) time delay.
 
[1] Loos & Klapp, Sci. Rep. 9, 2491 (2019)
[2] Rosinberg, Munakata, Tarjus, PRE 91, 042114 (2015) [3] Loos & Klapp, PRE 96, 012106 (2017)
[4] Loos & Klapp, ArXiv:1903.02322 (2019)
[5] Puglisi & Villamaina, EPL 88, 30004 (2009)