Feedback-controlled transport in an interacting colloidal system

Abstract

Abstract. -Based on dynamical density functional theory (DDFT) we consider a non-equilibrium system of interacting colloidal particles driven by a constant tilting force through a periodic, symmetric "washboard" potential. We demonstrate that, despite of pronounced spatio-temporal correlations, the particle current can be reversed by adding suitable feedback control terms to the DDFT equation of motion. We explore two distinct control protocols with time delay, focussing on either the particle positions or the density profile. Our study shows that the DDFT is an appropriate framework to implement time-delayed feedback control strategies widely used in other fields of nonlinear physics.Introduction. -Transport phenomena of Brownian particles in complex geometries are a topic receiving intense and continuos attention since decades [1][2][3]. A large number of studies has been devoted to transport in structured 1D systems such as colloids or biomolecules in microchannels [4], colloids in optical potentials [5,6], or cold atoms in optical lattices [7]. Theoretical studies of such systems have predicted spectacular effects such as ratchet mechanisms in systems with asymmetric spatial potential [1], giant diffusion [5,6,8,9] and dispersionless transport [10] in symmetric systems under constant external bias ("tilted washboards"), and the negative mobility effect [4,11]. Many of these effects have also been observed experimentally (see, e.g., [5,6,11,12]), often involving colloidal systems. A related topic is how these non-equilibrium phenomena can be manipulated by control forces [13]. Particularly promising are feedback control schemes which depend on the state of the system. A special case is the time-delayed feedback control method suggested by Pyragas [14], where the control term involves the difference between an output variable (the control target) at time t and its value at time t − τ , with τ being the delay time. This method is particularly suitable to stabilize certain, otherwise unstable (periodic) states. Moreover, a time delay naturally occurs in experiments involving feedback control due to the lag between the collection of information and the feedback. Indeed, time-delayed feedback control is nowadays used in a broad variety of non-linear systems such as lasers, neural dynamics, and excitable

Feedback-induced oscillations in one-dimensional colloidal transport

“…10, respectively). This is consistent with our earlier finding [27] that repulsive interactions support the particles in crossing the barrier, yielding an increase of the long-time diffusion coefficient. Moreover, for ε 0 = 15k B T we find stable (time-periodic) density oscillations with cycle time T = 2.315τ B .…”

“…Before proceeding, it is important to notice that the linear stability of the stationary distribution ρ s crucially depends on the choice of the coupling function f z 0 . Following [27] we use a linear, non-periodic coupling function f (z) = z. For further calculations, the origin of the z-axis is chosen in the maximum of the washboard potential U (z), implying that z 0 = 0.…”

“…Following an earlier study of two of us [27], we here employ a feedback control with delay where the control term involves the difference between an system variable (the control target) at time t and its value at time t − τ , with τ being the delay time. Such a time delay often occurs in experiments due to the time lag between measurement and feedback.…”

Novel structure formation of a phase separating colloidal fluid in a ratchet potential

“…Our study is based on DDFT [31][32][33], a generalized diffusion equation where the microscopic interactions enter via the (Helmholtz) free energy. In the last years, DDFT has been successfully applied to a variety of driven systems such as colloids in unstable traps [34], sedimenting colloids [35] and colloids in washboard potentials with feedback-control [36,37]. The present DDFT results demonstrate that, in combination with a ratchet potential, the attractive forces between the magnetic species in the driven mixture lead to a novel instability, that is, the formation of stripes perpendicular to the direction of the ratchet potential.…”

Feedback control of flow vorticity at low Reynolds numbers

“…Reynolds numbers to control the chaotic Taylor-Couette flow [21] and most recently in colloidal systems [22][23][24] and in liquid crystals [25]. Most often, in extended systems a local feedback scheme is used, which acts on each variable.…”