Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay

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.

Non-Markovian systems out of equilibrium: exact results for two routes of coarse graining

“…The first one is obtained by setting the coupling constants b i = k i which can lead to time-delayed friction kernels with a maximum at t > 0. Very different from the usual kernels this implies that the system will exhibit a time-delayed response to internal or external influences and thus causing strongly non-equilibrium phenomena, as has recently been discussed in the context of stochastic thermodynamics [35][36][37][38]. Another model is found by setting b i = 0 for which the SDE becomes equivalent to a stochastic model suggested by Wu and Libchaber describing passive colloids in active bacteria baths [39].…”

Non-Markovian systems out of equilibrium: Exact results for two routes of coarse graining

“…[33]. [38]. Different from the memory kernel K I (t) we find that the autocorrelation function of the noise, C I η (t), decays monotonically and thus the 2FDT is violated.…”

“…The first one is given by setting the coupling constants b i = k i , which can lead to time-delayed friction kernels with a maximum at t > 0. Such systems have recently attracted attention in the context of stochastic thermodynamics [35][36][37][38]. The second system is defined by choosing b i = 0, which is equivalent to the stochastic model suggested by Wu and Libchaber to describe passive colloids in active bacteria baths [39], and is also known as active Ornstein-Uhlenbeck process [40].…”

“…We will present results for four different models: The first corresponds to equilibrium dissipative dynamics, the second to a non-equilibrium system where both solutions fulfill the 2FDT. The third model has similarities to a non-equilibrium system with time-delayed feedback control [35][36][37][38]. All these models consist of a colloid coupling to two bath particles, N = 2.…”

“…The immediate question to ask would be: Is it possible to formulate a projection operator P, which leads to the linear GLE (8) with the memory kernel K I (t)? A related question comprises the entropy production in such coarse-grained systems and whether it is possible to connect thermodynamic quantities such as entropy [37,38,50] as discussed intensively in the framework of stochastic thermodynamics [51][52][53] to dynamic coarse-graining, for example via projection operator techniques? We believe that answering these questions would pave the way towards systematic dynamic coarse-graining of non-equilibrium soft matter systems with applications to active microrheology [25,54], active matter [27,51,55], but also non-stationary situations such as crystallization [56], colloid self-assembly [57] or at phase transitions [58] using a non-stationary GLE [22,59].…”