Force-linearization closure for non-Markovian Langevin systems with time delay

Antipersistent random walks in time-delayed systems

We show that the occurrence of chaotic diffusion in a typical class of time-delayed systems with linear instantaneous and nonlinear delayed term can be well described by an antipersistent random walk. We numerically investigate the dependence of all relevant quantities characterizing the random walk on the strength of the nonlinearity and on the delay. With the help of analytical considerations, we show that for a decreasing nonlinearity parameter the resulting dependence of the diffusion coefficient is well described by Markov processes of increasing order.

Introducing Memory in Coarse-Grained Molecular Simulations

Preserving the correct dynamics at the coarse-grained (CG) level is a pressing problem in the development of systematic CG models in soft matter simulation. Starting from the seminal idea of simple time-scale mapping, there have been many efforts over the years toward establishing a meticulous connection between the CG and fine-grained (FG) dynamics based on fundamental statistical mechanics approaches. One of the most successful attempts in this context has been the development of CG models based on the Mori-Zwanzig (MZ) theory, where the resulting equation of motion has the form of a generalized Langevin equation (GLE) and closely preserves the underlying FG dynamics. In this Review, we describe some of the recent studies in this regard. We focus on the construction and simulation of dynamically consistent systematic CG models based on the GLE, both in the simple Markovian limit and the non-Markovian case. Some recent studies of physical effects of memory are also discussed. The Review is aimed at summarizing recent developments in the field while highlighting the major challenges and possible future directions.

Traveling band formation in feedback-driven colloids

Using simulation and theory we study the dynamics of a colloidal suspension in two dimensions subject to a time-delayed repulsive feedback that depends on the positions of the colloidal particles. The colloidal particles experience an additional potential that is a superposition of repulsive potential energies centered around the positions of all the particles a delay time ago. Here we show that such a feedback leads to self-organization of the particles into traveling bands. The width of the bands and their propagation speed can be tuned by the delay time and the range of the imposed repulsive potential. The emerging traveling band behavior is observed in Brownian dynamics computer simulations as well as microscopic dynamic density functional theory. Traveling band formation also persists in systems of finite size leading to rotating traveling waves in the case of circularly confined systems.

Heat flow due to time-delayed feedback

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.

Delayed feedback control of active particles: a controlled journey towards the destination

We explore theoretically the navigation of an active particle based on delayed feedback control. The delayed feedback enters in our expression for the particle orientation which, for an active particle, determines (up to noise) the direction of motion in the next time step. Here we estimate the orientation by comparing the delayed position of the particle with the actual one. This method does not require any real-time monitoring of the particle orientation and may thus be relevant also for controlling sub-micron sized particles, where the imaging process is not easily feasible. We apply the delayed feedback strategy to two experimentally relevant situations, namely, optical trapping and photon nudging. To investigate the performance of our strategy, we calculate the mean arrival time analytically (exploiting a small-delay approximation) and by simulations.