Delay-induced transport in a rocking ratchet under feedback control

Heat production in a stochastic system with nonlinear time-delayed feedback

Using the framework of stochastic thermodynamics we study heat production related to the stochastic motion of a particle driven by repulsive, nonlinear, time-delayed feedback. Recently it has been shown that this type of feedback can lead to persistent motion above a threshold in parameter space [R. A. Kopp et al., Phys. Rev. E 107, 024611 (2023)2470-004510.1103/PhysRevE.107.024611]. Here we investigate, numerically and by analytical methods, the rate of heat production in the different regimes around the threshold to persistent motion. We find a nonzero average heat production rate, 〈q[over ̇]〉, already below the threshold, indicating the nonequilibrium character of the system even at small feedback. In this regime, we compare to analytical results for a corresponding linearized delayed system and a small-delay approximation which provides a reasonable description of 〈q[over ̇]〉 at small repulsion (or delay time). Beyond the threshold, the rate of heat production is much larger and shows a maximum as a function of the delay time. In this regime, 〈q[over ̇]〉 can be approximated by that of a system subject to a constant force stemming from the long-time velocity in the deterministic limit. The distribution of dissipated heat, however, is non-Gaussian, contrary to the constant-force case.

Efficient control protocols for an active Ornstein-Uhlenbeck particle

Designing a protocol to efficiently drive a stochastic system is an active field of research. Here we extend such control theory to an active Ornstein-Uhlenbeck particle (AOUP) in a bistable potential, driven by a harmonic trap. We find that protocols designed to minimize the excess work (up to linear response) perform better than naive protocols with constant velocity for a wide range of protocol durations.

Persistent motion of a Brownian particle subject to repulsive feedback with time delay

Based on analytical and numerical calculations we study the dynamics of an overdamped colloidal particle moving in two dimensions under time-delayed, nonlinear feedback control. Specifically, the particle is subject to a force derived from a repulsive Gaussian potential depending on the difference between its instantaneous position, r(t), and its earlier position r(t-τ), where τ is the delay time. Considering first the deterministic case, we provide analytical results for both the case of small displacements and the dynamics at long times. In particular, at appropriate values of the feedback parameters, the particle approaches a steady state with a constant, nonzero velocity whose direction is constant as well. In the presence of noise, the direction of motion becomes randomized at long times, but the (numerically obtained) velocity autocorrelation still reveals some persistence of motion. Moreover, the mean-squared displacement (MSD) reveals a mixed regime at intermediate times with contributions of both ballistic motion and diffusive translational motion, allowing us to extract an estimate for the effective propulsion velocity in presence of noise. We then analyze the data in terms of exact, known results for the MSD of active Brownian particles. The comparison indeed indicates a strong similarity between the dynamics of the particle under repulsive delayed feedback and active motion. This relation carries over to the behavior of the long-time diffusion coefficient D_{eff} which, similarly to active motion, is strongly enhanced compared to the free case. Finally, we show that, for small delays, D_{eff} can be estimated analytically.

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

Generalized Langevin equations (GLEs) can be systematically derived via dimensional reduction from high-dimensional microscopic systems. For linear models the derivation can either be based on projection operator techniques such as the Mori-Zwanzig (MZ) formalism or by 'integrating out' the bath degrees of freedom. Based on exact analytical results we show that both routes can lead to fundamentally different GLEs and that the origin of these differences is based inherently on the non-equilibrium nature of the microscopic stochastic model. The most important conceptional difference between the two routes is that the MZ result intrinsically fulfills the generalized second fluctuation-dissipation theorem while the integration result can lead to its violation. We supplement our theoretical findings with numerical and simulation results for two popular non-equilibrium systems: time-delayed feedback control and the active Ornstein-Uhlenbeck process.

Feedback-controlled dynamics of neuronal cells on directional surfaces

The formation of neuronal networks is a complex phenomenon of fundamental importance for understanding the development of the nervous system. The basic process underlying the network formation is axonal growth, a process involving the extension of axons from the cell body and axonal navigation toward target neurons. Axonal growth is guided by the interactions between the tip of the axon (growth cone) and its extracellular environmental cues, which include intercellular interactions, the biochemical landscape around the neuron, and the mechanical and geometrical features of the growth substrate. Here, we present a comprehensive experimental and theoretical analysis of axonal growth for neurons cultured on micropatterned polydimethylsiloxane (PDMS) surfaces. We demonstrate that closed-loop feedback is an essential component of axonal dynamics on these surfaces: the growth cone continuously measures environmental cues and adjusts its motion in response to external geometrical features. We show that this model captures all the characteristics of axonal dynamics on PDMS surfaces for both untreated and chemically modified neurons. We combine experimental data with theoretical analysis to measure key parameters that describe axonal dynamics: diffusion (cell motility) coefficients, speed and angular distributions, and cell-substrate interactions. The experiments performed on neurons treated with Taxol (inhibitor of microtubule dynamics) and Y-27632 (disruptor of actin filaments) indicate that the internal dynamics of microtubules and actin filaments plays a critical role for the proper function of the feedback mechanism. Our results demonstrate that axons follow geometrical patterns through a contact-guidance mechanism, in which high-curvature geometrical features impart high traction forces to the growth cone. These results have important implications for our fundamental understanding of axonal growth as well as for bioengineering novel substrate to guide neuronal growth and promote nerve repair.

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.

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

This paper is concerned with the Fokker-Planck (FP) description of classical stochastic systems with discrete time delay. The non-Markovian character of the corresponding Langevin dynamics naturally leads to a coupled infinite hierarchy of FP equations for the various n-time joint distribution functions. Here, we present an approach to close the hierarchy at the one-time level based on a linearization of the deterministic forces in all members of the hierarchy starting from the second one. This leads to a closed equation for the one-time probability density in the steady state. Considering two generic nonlinear systems, a colloidal particle in a sinusoidal or bistable potential supplemented by a linear delay force, we demonstrate that our approach yields a very accurate representation of the density as compared to quasiexact numerical results from direct solution of the Langevin equation. Moreover, the results are significantly improved against those from a small-delay approximation and a perturbation-theoretical approach. We also discuss the possibility of accessing transport-related quantities, such as escape times, based on an additional Kramers approximation. Our approach applies to a wide class of models with nonlinear deterministic forces.

Efficiency at maximum power of a discrete feedback ratchet

Efficiency at maximum power is found to be of the same order for a feedback ratchet and for its open-loop counterpart. However, feedback increases the output power up to a factor of five. This increase in output power is due to the increase in energy input and the effective entropy reduction obtained as a consequence of feedback. Optimal efficiency at maximum power is reached for time intervals between feedback actions two orders of magnitude smaller than the characteristic time of diffusion over a ratchet period length. The efficiency is computed consistently taking into account the correlation between the control actions. We consider a feedback control protocol for a discrete feedback flashing ratchet, which works against an external load. We maximize the power output optimizing the parameters of the ratchet, the controller, and the external load. The maximum power output is found to be upper bounded, so the attainable extracted power is limited. After, we compute an upper bound for the efficiency of this isothermal feedback ratchet at maximum power output. We make this computation applying recent developments of the thermodynamics of feedback-controlled systems, which give an equation to compute the entropy reduction due to information. However, this equation requires the computation of the probability of each of the possible sequences of the controller's actions. This computation becomes involved when the sequence of the controller's actions is non-Markovian, as is the case in most feedback ratchets. We here introduce an alternative procedure to set strong bounds to the entropy reduction in order to compute its value. In this procedure the bounds are evaluated in a quasi-Markovian limit, which emerge when there are big differences between the stationary probabilities of the system states. These big differences are an effect of the potential strength, which minimizes the departures from the Markovianicity of the sequence of control actions, allowing also to minimize the departures from the optimal performance of the system. This procedure can be applied to other feedback ratchets and, more in general, to other control systems.

Enhancement of mobility in an interacting colloidal system under feedback control

Feedback control schemes are a promising way to manipulate transport properties of driven colloidal suspensions. In the present article, we suggest a feedback scheme to enhance the collective transport of colloidal particles with repulsive interactions through a one-dimensional tilted washboard potential. The control is modeled by a harmonic confining potential, mimicking an optical "trap," with the center of this trap moving with the (instantaneous) mean particle position. Our theoretical analysis is based on the Smoluchowski equation combined with dynamical density functional theory for systems with hard-core or ultrasoft (Gaussian) interactions. For either type of interaction, we find that the feedback control can lead to an enhancement of the mobility by several orders of magnitude relative to the uncontrolled case. The largest effects occur for intermediate stiffness of the trap and large particle numbers. Moreover, in some regions of the parameter space the feedback control induces oscillations of the mean velocity. Finally, we show that the enhancement of mobility is robust against a small time delay in implementing the feedback control.

Feedback between interacting transport channels

A model of particle transport through a large number of channels is introduced. Interactions among the particles can lead to a strong suppression of fluctuations in the particle number statistics. Within a mean-field-type limit, this becomes equivalent to a time-dependent (nonautonomous) collective feedback control mechanism. The dynamics can be interpreted as a diffusive spreading of a feedback signal across the channels that displays scaling, can be quantified via the flow of information, and becomes visible, e.g., in the spectral function of the particle noise.

Stochastic thermodynamics of Langevin systems under time-delayed feedback control: Second-law-like inequalities

Response lags are generic to almost any physical system and often play a crucial role in the feedback loops present in artificial nanodevices and biological molecular machines. In this paper, we perform a comprehensive study of small stochastic systems governed by an underdamped Langevin equation and driven out of equilibrium by a time-delayed continuous feedback control. In their normal operating regime, these systems settle in a nonequilibrium steady state in which work is permanently extracted from the surrounding heat bath. By using the Fokker-Planck representation of the dynamics, we derive a set of second-law-like inequalities that provide bounds to the rate of extracted work. These inequalities involve additional contributions characterizing the reduction of entropy production due to the continuous measurement process. We also show that the non-Markovian nature of the dynamics requires a modification of the basic relation linking dissipation to the breaking of time-reversal symmetry at the level of trajectories. The modified relation includes a contribution arising from the acausal character of the reverse process. This, in turn, leads to another second-law-like inequality. We illustrate the general formalism with a detailed analytical and numerical study of a harmonic oscillator driven by a linear feedback, which describes actual experimental setups.

Feedback control of flow vorticity at low Reynolds numbers

Our aim is to explore strategies of feedback control to design and stabilize novel dynamic flow patterns in model systems of complex fluids. To introduce the control strategies, we investigate the simple Newtonian fluid at low Reynolds number in a circular geometry. Then, the fluid vorticity satisfies a diffusion equation. We determine the mean vorticity in the sensing area and use two control strategies to feed it back into the system by controlling the angular velocity of the circular boundary. Hysteretic feedback control generates self-regulated stable oscillations in time, the frequency of which can be adjusted over several orders of magnitude by tuning the relevant feedback parameters. Time-delayed feedback control initiates unstable vorticity modes for sufficiently large feedback strength. For increasing delay time, we first observe oscillations with beats and then regular trains of narrow pulses. Close to the transition line between the resting fluid and the unstable modes, these patterns are relatively stable over long times.

Waiting time distribution for continuous stochastic systems

The waiting time distribution (WTD) is a common tool for analyzing discrete stochastic processes in classical and quantum systems. However, there are many physical examples where the dynamics is continuous and only approximately discrete, or where it is favourable to discuss the dynamics on a discretized and a continuous level in parallel. An example is the hindered motion of particles through potential landscapes with barriers. In the present paper we propose a consistent generalization of the WTD from the discrete case to situations where the particles perform continuous barrier crossing characterized by a finite duration. To this end, we introduce a recipe to calculate the WTD from the Fokker-Planck (Smoluchowski) equation. In contrast to the closely related first passage time distribution (FPTD), which is frequently used to describe continuous processes, the WTD contains information about the direction of motion. As an application, we consider the paradigmatic example of an overdamped particle diffusing through a washboard potential. To verify the approach and to elucidate its numerical implications, we compare the WTD defined via the Smoluchowski equation with data from direct simulation of the underlying Langevin equation and find full consistency provided that the jumps in the Langevin approach are defined properly. Moreover, for sufficiently large energy barriers, the WTD defined via the Smoluchowski equation becomes consistent with that resulting from the analytical solution of a (two-state) master equation model for the short-time dynamics developed previously by us [Phys. Rev. E 86, 061135 (2012)]. Thus, our approach "interpolates" between these two types of stochastic motion. We illustrate our approach for both symmetric systems and systems under constant force.