Persistent motion of a Brownian particle subject to repulsive feedback with time delay Robin A. Koppand Sabine H. L. Klapp Institut f ur Theoretische Physik

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

Robin A. Kopp∗and Sabine H. L. Klapp†

Institut f¨ur Theoretische Physik,

Technische Universit¨at Berlin,

Hardenbergstr. 36, D-10623 Berlin, Germany

(Dated: March 7, 2025)

Based on analytical and numerical calculations we study the dynamics of an overdamped colloidal

particle moving in two dimensions under time-delayed, non-linear 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 τis 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, non-zero 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 Deﬀ which, similarly to active motion, is strongly enhanced compared to the free case.

Finally we show that, for small delays, Deﬀ can be estimated analytically.

I. INTRODUCTION

Within the last years, feedback (closed-loop) control

of colloidal systems, that is, nano- to micron-sized parti-

cles in a thermally ﬂuctuating bath of solvent particles,

has become a focus of growing interest. Conceptually,

feedback control implies that the dynamics of a system

is subject to a protocol depending on an internal vari-

able, i.e., internal information from the system. This

concept is already widely used in other ﬁelds of physics

(and related sciences) and on many diﬀerent length- and

time scales [1, 2], examples ranging from the stabiliza-

tion of exotic quantum states and quantum computation

over control of transport of (passive) colloids to switching

processes in neurosystems, applications in robotics and

cars, and chaos control of satellites. In contrast, the use

and design of feedback control in colloidal or, more gen-

erally, soft matter systems is a rather new development.

Recent experimental applications include control of DNA

molecules by feedback-driven temperature ﬁelds [3, 4] or

optical traps [5], magnetic feedback control of cells [6],

electrophoretic feedback control of nanoparticles [7], col-

loids in a electrokinetic feedback trap [8], in optical line

traps [9], and feedback cooling of nanoparticles [10].

A new, exciting ﬁeld of application is feedback con-

trol of active (self-propelled) colloids which move au-

tonomously due to an intrinsic source of energy. Con-

trol of the translational or rotational motion of active

∗r.kopp@tu-berlin.de

†sabine.klapp@tu-berlin.de

colloids can be realized, e.g., by photon nudging and by

adaptive light ﬁelds [11–20]. Interest in this topic is trig-

gered by the immense body of work on active particles

showing intriguing collective behaviors such as swarming

and clustering. Feedback control of such systems oﬀers

a way of modelling living matter involving not only au-

tonomous motion, but also communication, sensing and

thereby new types of self-organization. In addition to

these eﬀorts, feedback control of colloids is interesting, on

a fundamental level, to study its interplay with thermo-

dynamics and information exchange in small stochastic

systems [8, 21–25] on the basis of stochastic thermody-

namics [26].

In any realistic setup of feedback control (and simi-

larly in biological systems), there is some time lag (or

time delay) between the reception of information (e.g.,

via a camera) and the actual control. Similarly, biological

living systems with feedback mechanisms often exhibit

some degree of sensorial delay [17], communication delay

[27], or, more generally, memory eﬀects due to viscoelas-

tic environments [28]. Thus, the idea of instantaneous

feedback is often an idealization. Moreover, there is now

increasing awareness that this time delay is not per se an

annoyance, but can rather be an important ingredient to

observe and stabilize certain dynamical behavior. The

constructive role of time delay has already been shown in

other contexts [1, 2, 29] including chaos control [30, 31],

[32], laser systems [33, 34], chemical oscillatory systems

[35] and reaction networks [36]. Recently is has been

shown that time delay has also important consequences

for the thermodynamics of feedback-controlled systems

[24, 37, 38].

In colloidal systems, time-delayed feedback has already

arXiv:2210.03182v3 [cond-mat.soft] 6 Mar 2025

been shown to create intriguing eﬀects. Examples in the

ﬁeld of transport of (one-dimensional) passive colloidal

systems include current reversals [39] and enhancements

[9, 40, 41] in ratchet and tilted washboard systems, but

also band formation in two-dimensional interacting col-

loidal systems with collective time-delayed feedback [42].

Moreover, recent studies of (time-delayed) feedback con-

trol in active systems [13–15] have revealed new eﬀects

such as delay-induced clustering and swarming [17, 43–

45]. Despite the increasing awareness of the relevance of

delay and feedback eﬀects in colloidal systems, theoreti-

cal studies of speciﬁc model systems are still rare. This

is mainly due to the non-Markovianity of the underlying

stochastic equations of motion, making explicit calcula-

tions challenging. Indeed, exact results only exist for lin-

ear models [46–49]. However, in real colloidal systems,

the interactions and forces involved are typically nonlin-

ear. Motivated by these developments, we here present

a theoretical study of a particular nonlinear model of a

colloidal particle under time-delayed feedback. Speciﬁ-

cally, we consider the two-dimensional translational mo-

tion of a single, spherical colloid subject to a repulsive

force generated by a Gaussian potential centered around

the particle position at an earlier time t−τ. Thus, the

position acts as a stochastic variable on which the control

is performed. Our model diﬀers in several aspects from

earlier ones (see, e.g. [40, 41]). We here consider repul-

sive feedback, rather than the often studied case of trap-

ping a particle by an attractive potential (stemming, e.g.,

from an optical tweezer). We note that similar nudging

mechanisms have already been used in the context of the

“photo-nudging” method for Janus particles [11–15, 50].

A second diﬀerence is that the force is nonlinear in the

particle position and ﬁnite in range. Thus, although our

model is not directly designed to describe a speciﬁc ex-

periment and is, in this sense, hypothetical, it allows us

to focus exclusively on several important features of a

control scheme for Brownian particles: time-delay, repul-

sion, and nonlinearity.

Explicit results are obtained based on analytical con-

siderations in some limiting cases, as well as on numer-

ical solution of the corresponding (overdamped) non-

Markovian Langevin equation. The ﬁrst main result of

our study is that for appropriate values of delay time and

strength of repulsion, and in the absence of noise, the par-

ticle develops a stationary state characterized by a con-

stant velocity vector. Second, thermal ﬂuctuations lead

to a randomization of the direction of motion, but the

particle still possesses a persistence of motion and a con-

stant magnitude of speed (after noise-averaging). This

is reminiscent of the stochastic behavior of self-propelled

particles, and indeed we identify close similarities. In

particular, we discuss relations to the prominent model

of active Brownian particles (ABP) [51]. For this model,

which is widely used also to explain experimental data

(see, e.g. [52, 53]), there exists a closed analytical expres-

sion for the mean-squared displacement which we utilize

to ﬁt and interpret our numerical data. Further, the ABP

model is one of the best studied systems concerning the

collective behavior induced by activity [53, 54]. Thus, es-

tablishing a link to the ABP model will provide us with

a basis for future investigation of the collective behavior

of our model. By investigating, in the present study, the

single-particle relations between the diﬀerent models we

make a ﬁrst step to establishing a link between feed-back

controlled and active matter.

The rest of the paper is organized as follows: In the

following section II we introduce our model and the cor-

responding overdamped Langevin equation governing the

dynamics of the system. Subsequently we study, ﬁrst, the

deterministic limit in Sec. III, using a combination of

analytical and numerical methods. We then present our

results for the full, nonlinear stochastic system, based on

Brownian dynamics simulations, in Sec. IV. Our conclu-

sions are summarized in Sec. V.

II. MODEL

We consider the two-dimensional motion of a Brow-

nian particle in the x-yplane. In addition to thermal

ﬂuctuations due to a coupled heat bath at temperature

T, the particle is subject to a time-delayed feedback

force Fdepending on both, its instantaneous position

r(t)=(x(t), y(t))Tand its position at an earlier time,

r(t−τ), where τis the (discrete) delay time. The parti-

cle’s motion at times t > 0 is governed by the overdamped

Langevin equation

γdr

dt =F(r(t),r(t−τ)) + ξ(t),(1)

where γis the friction coeﬃcient, and the position at ear-

lier times t∈[−τ, 0] is determined by the history func-

tion Φ(t). Furthermore, ξrepresents a two-dimensional,

Gaussian white noise with zero mean and correlation

function ⟨ξα(t)ξβ(t′)⟩= 2γkBT δαβδ(t−t′) where α,β

are the cartesian components of the noise vector ξ, and

kBT(with kBbeing the Boltzmann constant) is the ther-

mal energy. The diﬀusion constant of the free particle

motion (F=0) follows from the Stokes-Einstein relation

D=kBT/γ.

Within our model, the feedback force Fis derived from

a Gaussian feedback potential involving the displacement

r(t)−r(t−τ), that is,

V(r(t),r(t−τ)) = Aexp −(r(t)−r(t−τ))2

2b2!,(2)

yielding

F=−∇rV(r(t),r(t−τ))

b2(r(t)−r(t−τ)) exp −(r(t)−r(t−τ))2

2b2!.(3)

We choose the feedback strength A > 0, representing re-

pulsive feedback whose range is determined by the width

r(t−τ)

r(t)

V(r(t),r(t−τ))

FIG. 1. Schematic of the repulsive Gaussian delayed feedback

potential V. At each time t, the Gaussian is centered around

the earlier position r(t−τ), resulting in a “pushing” force (red

arrow) directed along the diﬀerence vector r(t)−r(t−τ).

of the Gaussian, b(b > 0). The width of the feed-

back potential serves as the length scale in our system.

An illustration of the “bump”-like potential is given in

Fig. 1. Physically, the potential (2) can be interpreted

as a source of a “nudge” following the particle with some

delay. Such a potential could, in principle, be created

by optical forces [15, 19, 50, 55–57]. We note that, dif-

ferent to the often used harmonic potentials leading to

linear feedback forces [58], the potential (2) and the force

(3) decay to zero when the displacement within one de-

lay time, r(t)−r(t−τ), increases to large values. The

ﬁnite range is indeed advantageous when using Eq. (2)

in a many-particle system with periodic boundary con-

ditions. We also note, from the perspective of control

theory, the feedback force has “Pyragas” form [30, 31]

since it only depends on the diﬀerence between r(t) and

r(t−τ). Thus, the force vanishes trivially if τ= 0; in

this case, our model reduces to that of a free Brownian

particle. Note, however, that the feedback force also van-

ishes when the particle is at rest, that is, r(t−τ) = r(t),

or when it moves in a cyclic fashion with period τ.

Inserting the expression for the force (3) into Eq. (1),

the complete equation of motion reads

γdr

dt =A

b2(r(t)−r(t−τ)) exp −(r(t)−r(t−τ))2

2b2!

+ξ(t).(4)

Due to the presence of time delay, the particle’s dynam-

ics deﬁned by Eq. (4) is strongly non-Markovian (where

“strongly” refers to the fact that the kernel emerging

when we write the right side as a convolution over past

times is a delta peak located at a ﬁnite time τ[59]). As a

consequence, several tools well established for Markovian

Brownian systems (such as the link between Langevin

and Fokker Planck descriptions) do not straightforwardly

apply. More mathematically spoken, Eq. (4) represents

a stochastic delay diﬀerential equation (SDDE) which is,

furthermore, non-linear due to the Gaussian shape of the

underlying potential. Even in the absence of noise, the

resulting diﬀerential equation (DDE) formally becomes

inﬁnite-dimensional due to the presence of the continuous

history function Φ(t). In recent years, a number of re-

sults have been obtained for deterministic DDEs [60–62],

as well as for linear SDDEs see, e.g., [46–49]. However,

here we are dealing with a nonlinear SDDE for which, to

our knowledge, no full analytical solution exists. Still, we

can analytically consider some limiting cases, which we

will discuss in the subsequent Sections III and IV. In ad-

dition, we study the particle dynamics given by Eq. (4)

and its deterministic limit numerically using Brownian

Dynamics (BD) simulations. Some technical details of

the numerical calculations are given in Appendix A.

III. DETERMINISTIC LIMIT

We start by considering the deterministic limit of

Eq. (4) deﬁned by ξ=0, which may be realized by

setting the temperature Tto zero. In this way we can

explore the role of delay alone, thereby providing a use-

ful starting point for the later investigation of the noisy

case.

A. Linear behavior

Some ﬁrst insights can already be obtained by investi-

gating the behavior at small displacements |r(t)−r(t−

τ)| ≪ √2b. In this case, we can expand the force (3)

up to ﬁrst order in the displacement, yielding the linear

equation

γ˙

r=A

b2(r(t)−r(t−τ)) .(5)

In deriving Eq. (5) we have used that the Hessian ma-

trix H(r) with elements Hαβ (r) = ∂2V /∂xα∂xβ(with

V(r) being the feedback potential) is diagonal at r= 0,

and Hαα(0) = −A/b2. As a consequence of the diago-

nality, each component xαof rcan be considered sepa-

rately. The resulting scalar, linear DDEs are indeed well

studied. For example, it is well known [63–65] that the

(ﬁxed) point r=r0with xα=const is marginally stable

in the sense that a perturbation ∝exp[µt] (with µbe-

ing a complex number) remains constant, that is, µ= 0.

Physically, this expresses the metastability of a particle

on a “parabolic mountain” [58]. Moreover, explicit solu-

tions of the linear equation (5) can be constructed in a

piecewise manner by using the method of steps [60, 61].

For a given history function Φ(t) with components Φα(t)

(deﬁned in the interval t∈[−τ, 0]), the solution in the

ﬁrst interval t∈[0, τ] follows as

xα= exp At

γb2(Φα(0)

−A

γb2Zt

dt′exp A(−t′)

γb2Φα(t′−τ)(6)

which shows directly the dependency on the history. In

particular, if the particle was at rest at some ﬁxed po-

sition (Φ=r0), it stays where it is (as expected from

the marginal stability) for all values of the prefactor

A/b2. This then also holds for the later time intervals

[nτ, (n+ 1)τ] (with n= 1,2, . . .). A more interesting

situation for the present study occurs if the particle had

moved with a constant velocity v0, that is, Φ=v0t,

t∈[−τ, 0]. The corresponding explicit solution for t > 0

up to time 2τis given in the Appendix B. It turns out

that the behavior of xα(t) (and the corresponding veloc-

ity) crucially depends on the value of the dimensionless

parameter Aτ/γb2. If this parameter is smaller than one,

the particle just approaches a constant position and the

velocity dies oﬀ to zero. In contrast, if Aτ /γb2>1,

the position and the corresponding velocity continue to

increase unboundly. Finally, at the “threshold” value

Aτ/γb2= 1, the behavior just remains identical to that

given by the history, that is, the particle continues to

move with the velocity v0. An illustration of these be-

haviors is given in Fig. 2 (dashed lines), where the history

is characterized by a one-dimensional velocity (generat-

ing one-dimensional motion), v0=v0ˆ

x,t∈[−τ, 0].

B. Nonlinear behavior and steady state

We now consider the full, nonlinear (deterministic) sys-

tem. The corresponding DDE (4) with ξ=0cannot

be solved any more by the method of steps, such that

we utilize a numerical solution to obtain the full trajec-

tory r(t) in dependence of the history Φ(t). Examples

are shown in Fig. 2 (solid lines) where we focus (again)

on an initial velocity along the x-axis. At small times

t/τ > 0, the solution agrees with that obtained for the

(linear) case describing small displacements, as it should

do. The further time dependence of the nonlinear system

again crucially depends on the dimensionless parameter

Aτ/γb2. In particular, for Aτ /γb2≤1 the nonlinear sys-

tem comes to rest in the sense that the position settles

and the velocity vanishes. Note that this is consistent

with the linear case for Aτ/γb2<1, but not for the

“threshold” value Aτ /γb2= 1, where the linear system

is fully governed by the history. Moreover, a crucial dif-

ference between the systems emerges when Aτ/γb2>1.

Here, the nonlinear system governed by the full, Gaussian

feedback develops a stationary state characterized by a

constant velocity v∞. This velocity can be determined

analytically as follows.

Let us assume that such a state develops. We then

have ˙

r=v∞and r(t)−r(t−τ) = v∞τ. Inserting this

−1 0 1 2

t/τ

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

x(t)/b

(a)

nonlinear linear

Aτ/γb2= 0.25

Aτ/γb2= 1.0

Aτ/γb2= 1.75

Aτ/γb2= 4.0

Aτ/γb2= 0.25

Aτ/γb2= 1.0

Aτ/γb2= 1.75

Aτ/γb2= 4.0

02468

t/τ

˙x(t)τ /b

v∞τ/b

(b)

nonlinear linear

Aτ/γb2= 0.25

Aτ/γb2= 1.0

Aτ/γb2= 1.75

Aτ/γb2= 4.0

Aτ/γb2= 0.25

Aτ/γb2= 1.0

Aτ/γb2= 1.75

Aτ/γb2= 4.0

FIG. 2. Numerical results for the time dependence of the

position (a) and velocity (b) of the particle subject to nonlin-

ear, Gaussian feedback in the deterministic limit (for a his-

tory with constant (one-dimensional) velocity v0=v0ˆ

x, i.e.,

Φ=v0t). Included are the analytical results for the linearized

case (small displacements [see Eq. (5)]). The curves are la-

belled according to the value of the dimensionless parameter

Aτ/γb2.

into the deterministic version of Eq. (4) we obtain an

implicit equation for the long-time velocity,

v∞=Aτ

γb2v∞exp −(v∞τ)2

2b2!,(7)

yielding

|v∞|=±√2b

τs−ln γb2

Aτ .(8)

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摘要：

PersistentmotionofaBrownianparticlesubjecttorepulsivefeedbackwithtimedelayRobinA.Kopp∗andSabineH.L.Klapp†Institutf¨urTheoretischePhysik,TechnischeUniversit¨atBerlin,Hardenbergstr.36,D-10623Berlin,Germany(Dated:March7,2025)Basedonanalyticalandnumericalcalculationswestudythedynamicsofanoverdampedcollo...

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Persistent motion of a Brownian particle subject to repulsive feedback with time delay Robin A. Koppand Sabine H. L. Klapp Institut f ur Theoretische Physik

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