New J. Phys. 22 (2020) 113025 https://doi.org/10.1088/1367-2630/abc91e
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PAPER
Effective Langevin equations for a polar tracer in an active bath
Miloˇ
sKneˇ
zevi´
c∗and Holger Stark
Institut für Theoretische Physik, Technische Universit¨
at Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany
∗Author to whom any correspondence should be addressed.
E-mail: knezev[email protected].de
Keywords: Langevin equations, tracer, active bath
Abstract
We study the motion of a polar tracer, having a concave surface, immersed in a two-dimensional
suspension of active particles. Using Brownian dynamics simulations, we measure the distributions
and auto-correlation functions of force and torque exerted by active particles on the tracer. The
tracer experiences a finite average force along its polar axis, while all the correlation functions show
exponential decay in time. Using these insights we construct the full coarse-grained Langevin
description for tracer position and orientation, where the active particles are subsumed into an
effective self-propulsion force and exponentially correlated noise for both translations and
rotations. The ensuing mesoscopic dynamics can be described in terms of five dimensionless
parameters. We perform a thorough parameter study of the mean squared displacement, which
illustrates how the different parameters influence the tracer dynamics, which crosses over from a
ballistic to diffusive motion. We also demonstrate that the distribution of tracer displacements
evolves from a non-Gaussian shape at early stages to a Gaussian behavior for sufficiently long
times. Finally, for a given set of microscopic parameters, we establish a procedure to estimate the
matching parameters of our effective model, and show that the resulting dynamics is in a very
good quantitative agreement with the one obtained in Brownian dynamics simulations.
1. Introduction
In recent years active motion has evolved into a thriving field combining different disciplines from physics
and chemistry to biology and engineering sciences [1–4]. Microorganisms swim in a fluid environment at
low Reynolds number, meaning that viscous forces dictate over inertial forces. Tremendous research
activities have been devoted to better understand their propulsion mechanisms [2,5,6], as well as to
construct artificial microswimmers [7–10] and to explore their fascinating patterns of collective motion
[11–15]. Artificial or biological microswimmers, which we simply term active particles, consume energy to
swim forward, and therefore are constantly driven out of equilibrium. Fascinating generic properties arise
in such nonequilibrium settings, as illustrated, for example, by active particles getting stuck at confining
walls [16–20], on which they exert a swim pressure [21–25].
Combining active motion with concepts from Brownian ratchets, one of the existing paradigms in
nonequilibrium statistical mechanics [26], provides new possibilities of rectified motion [27]. In the
direction of applications the following works are of interest: capturing active particles [28,29], sorting
active particles based on their velocity [30] or the mechanism how they reorient [31], effective interactions
between inclusions in active suspensions [32–37], cargo transport [38,39] and active assembly [40–42].
Active particles accumulate in corners, which causes directed transport through a wall of funnels [43,44], in
an asymmetric potential [45,46], or in a symmetric potential in combination with a position-dependent
swimming speed [47], in a corrugated channel [48], and in arrays of asymmetric obstacles [49].
When many active particles act on a mesoscopic object, they can be regarded as a nonequilibrium active
bath, which is strongly determined by fluctuations in the swimming directions of particles. Rotational and
translational ratchet motors can be constructed by placing asymmetric objects in active baths. Notably, a
wheel with sawtooth-like contour deposited in an active bath exhibits unidirectional rotation [50–52].
When passive mesoscopic objects, which do not self-propel, are suspended in such a bath, they are
© 2020 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft
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Figure 1. (a) Tracer geometry used in Brownian dynamics simulations. In the eigenframe of the tracer the orientation of its
symmetry axis, passing through the center of mass C, is determined by the unit vector ewhich makes the angle θwith the x-axis
of the lab frame. (b) A snapshot of the motion of the tracer immersed in a bath of active particles; we selected only a small region
of the simulation box centered around the tracer.
stochastically pushed around, and, more importantly, their motion can even be rectified if they have a polar
shape and a pronounced concave surface [53]. In what follows we shall refer to such a mesoscopic object as
a polar tracer. Well known examples include semicircle forms and wedge-like structures [53–55]. The
directed motion of the tracer can be explained by the fact that a portion of active particles, trapped within
some cavity of the object, exercise certain pressure on the surface of the cavity and thus they push the object
in the outward direction as illustrated in figure 1.Thereby,thepolartracersareendowedwithsubstantial
persistence of motion and can act as microshuttles [53–55]. Contrarily, spherical tracers in an active bath
display only enhanced diffusive motion [56–65].
Most theoretical studies [53–55] performed so far were based on methods of Brownian dynamics
simulations, replicating a collection of active particles interacting with the tracer. An alternative approach is
the extraction of effective Langevin equations from the microscopic many-particle dynamics, which is a
long term goal of theoreticians both in and out of thermal equilibrium. Specifically, for an active bath it
remains a challenge to describe the motion of a polar tracer or of even more complicated structures by
mesoscopic equations. Due to the nonequilibrium nature of the bath [66], it is clear that the standard
Langevin equation is not appropriate in this case. It has been shown, for instance, that the motion of a
spherical tracer in a bath of E. coli bacteria can be described by a Langevin equation containing
instantaneous friction kernel and colored noise [53,56,63,67]. Such noise can be generated by an auxiliary
Ornstein–Uhlenbeck process [68] and it brings the system outside of thermodynamic equilibrium.
In this article we develop an effective Langevin description for a polar tracer with a concave surface
immersed in an active bath (see figure 1). To determine the coarse-grained active noise resulting from the
impact of the active bath particles, we performed simulations based on Brownian dynamics equations by
extending previous studies. Our simulations support previous findings [53,55] concerning the existence of
a finite average force acting along tracer’s symmetry axis and the exponential time decay of relevant
correlation functions. In addition, we demonstrate that the cross-correlation function between the torque
and the force acting perpendicularly on the tracer’ssymmetryaxisisalwaysnegative,butdecays
exponentially with time as well. Based on these insights we propose a complete description of the tracer
motion with effective Langevin equations. More precisely, we show that the previous complex problem of
many-body Brownian dynamics can be reduced to a simple system of three stochastic equations of Langevin
type. Using this approach, we performed a detailed study of the tracer mean squared displacement (MSD)
and its displacement probability distributions as a function of time. We show for the first time that the
distribution of tracer displacements crosses over from a non-Gaussian at early stages of evolution to a
Gaussian behavior for sufficiently long times.
The article is organized as follows: after the introduction, section 2is devoted to the presentation of our
model and its description in the frameworks of Brownian dynamics and effective Langevin equations
approach. In section 3we present our results together with an extensive discussion of the mean squared
tracer displacement and associated probability distributions. Some concluding remarks and a summary of
the main results are given in section 4.Finally,inappendixAwe describe some technical details of our
Brownian dynamics simulations.
2. Model
To introduce the quantities of interest for the coarse-grained dynamics, we start with a description of the
problem within the framework of Brownian dynamics. Thus we begin with the general equations of tracer
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motion in the overdamped limit. In the lab frame, the vector r=(x,y) denotes the center of mass position
of the tracer and the direction of its symmetry axis eis characterized by an angle θ(see figure 1). In dyadic
notation the translational mobility matrix Mof the tracer in its eigenframe can be written as
M=μe⊗e+μ⊥(I−e⊗e), (1)
where μand μ⊥are translational scalar mobilities, and Iis the unit matrix. In the overdamped limit, where
inertial contributions are negligible, the equations of motion of the tracer are
V=MF,(2)
˙
θ=κT.(3)
Here, the tracer velocity Vand the force Facting on it, in the eigenframe take the form V=ve+v⊥e⊥
and F=Fe+F⊥e⊥, respectively. The quantity ˙
θand T=Tez=T(e×e⊥) are the angular velocity of
the tracer and the torque exerted by active particles on it, while κdenotes its rotational mobility. For future
convenience, we also introduce a typical length lof the tracer connecting the mobilities μ⊥and κthrough
the relation l2=μ⊥/κ.
To characterize the fluctuating force and torque, which result from the active particles hitting the polar
tracer, we performed Brownian dynamics simulations of a semicircle tracer immersed in a bath of active
Brownian particles [69]. The technical details of simulations are given in appendix A. As one can infer from
figure 2(top row, left) the probability distribution of force Fhas a Gaussian profile centered at a finite
mean (similar results were reported in [53], where the motion of a wedge shaped tracer in a bath of active
rods has been studied). One can also see that the auto-correlation function C(t)=F(t0)F(t0+t)
−F2in the stationary regime (top row, right) decays with time following an exponential law; similar
behavior has been observed in [53]. As the graph shows, the characteristic time of this decay is of the order
of the reorientation time τRof an individual active bath particle. This hints that the force Fcan be
modeled as F=F+ξ,whereFis a net drift force acting along the tracer’s polar axis, and ξis a
random noise term, with a zero mean, exponentially correlated in time. In contrast to the case of F,
Brownian dynamics simulations show that the average value of the perpendicular component of the force is
equaltozero,F⊥=0, which suggests that the perpendicular force can be taken in the simple form
F⊥=ξ⊥,withξ⊥=0. As before, the corresponding correlation function C⊥(t) appears to follow an
exponential decay in time (see figure 2, middle row). Furthermore, the simulations also reveal that the
different components of the random force are not mutually correlated, ξ(t)ξ⊥(t)=0, which is also clear
by the polar symmetry of the tracer. All these findings can be summarized by relations
ξα=0, ξα(t)ξβ(t)=δαβ
1
μ2
α
DA
α
1
τα
e−|t−t|
τα,(4)
where the indices αand βtake values and ⊥,whileDA
αare the diffusion constants along the principal
directions and ταrepresents the corresponding persistence time of the active noise.
We have seen that the active particles affect the motion of the tracer through the force F.Theseparticles
also produce a random torque leading to a rotation of the tracer, which causes a gradual degradation in
directed motion. Brownian dynamics simulations demonstrate that the average value of torque is equal to
zero, and that its auto-correlation function CT(t) decays exponentially in time (see figure 2(bottom row)
and [53]). Since the fluctuating torque results from the fluctuating perpendicular force component, we also
measured the cross-correlation function CF⊥T(t)=F⊥(t0)T(t0+t).Asonecanseefromfigure3,itis
always negative and displays an exponential behavior. In our coarse-grained model the presence of these
anti-correlations is taken into account in the following way
T(t)=−μ⊥
κlT
ξ⊥(t), (5)
where the characteristic length μ⊥/(κlT)linkingTand ξ⊥can be deduced using dimensional analysis; here
we have introduced the new length lT=DA
⊥/DA
Rwith DA
Rbeing the tracer’s rotational diffusion constant.
Let us mention in passing that, in contrast to the case of F⊥,theforceFand the torque Tare not mutually
correlated.
Taking into account the above considerations, after transforming the equation (2)intothelabframe,we
obtain the following system of three stochastic equations for the polar tracer
˙
x=μF+ξcos θ−μ⊥ξ⊥sin θ,(6)
˙
y=μF+ξsin θ+μ⊥ξ⊥cos θ,(7)
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Figure 2. The probability distribution Pof the force Facting on the tracer is presented on the left side of the top row; the force
is measured in units of kBT/σ,whereTis the temperature of the bath and σis the characteristic length of the interaction
potential between the active particles. The solid black line is the best fit of Pto a Gaussian distribution; note that F>0,
which means that it has a positive projection on the polar axis eof figure 1. On the right side of the top row the scaled
auto-correlation function C(t)=C(t)/C(0) for F(t) is shown (light gray symbols), together with its fit to the exponential
form e−t/τ(black solid line); note that the time is measured in units of the persistence time, τR, of an active particle, discussed in
the appendix A. The middle row presents the corresponding data for P⊥(F⊥)andC⊥(t). Finally in the bottom row we presented
the behavior of PTand CT. Note that the mean values of the perpendicular force and torque acting on the tracer are vanishing,
F⊥=0andT=0; the torque is measured in units of thermal energy kBT. According to our observations the values of all
three correlation times τ,τ⊥,andτTare approximately 0.45τR. All results presented in this panel correspond to an active bath
having the area packing fraction φ≈0.08 and the persistence number Per =80/3 of active particles (see appendix Afor
definitions of φand Per and more details).
˙
θ=−μ⊥
lT
ξ⊥.(8)
In these equations we neglected the usual thermal noise because we confined ourselves to the physically
most interesting case of large speed of active particles. To generate the exponentially correlated noises ξ
and ξ⊥of equation (4) with exactly the same parameters, we use two auxiliary Ornstein–Uhlenbeck
processes [68]
˙
ξα=−1
ταξα+2DA
α
μα
ηαwith α=,⊥,(9)
where ηαare Gaussian white noises of zero mean and unit variance: ηα=0, ηα(t)ηβ(t)=δαβδ(t−t).
We use the typical extent of the tracer las the unit of length, persistence time τof the noise as the unit
of time, and we measure forces in units of the effective self-propulsion force F. Now, keeping the same
notation, the equations (6)–(9) can be rewritten in the dimensionless form:
˙
x=P(1 +ξ)cos θ−ξ⊥sin θ, (10)
˙
y=P(1 +ξ)sin θ+ξ⊥cos θ, (11)
˙
θ=−P
bξ⊥, (12)
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Figure 3. The scaled cross-correlation function between the perpendicular force and torque acting on the tracer,
CF⊥T(t)=CF⊥T(t)/|CF⊥T(0)|, versus time measured in units of τR(light gray symbols); note that F⊥and Tare anti-correlated,
CF⊥T<0. The solid black curve represents the best fit of CF⊥Tdata to the exponential form −e−t/τc,whereτcis the characteristic
cross-correlation time. The simulation data were obtained for the same values of bath parameters as those in figure 2.
˙
ξ=−ξ+√2Q
Pη, (13)
˙
ξ⊥=−wξ⊥+w
P2Q
dη⊥, (14)
where we introduced five independent dimensionless parameters:
P=μFτ
l,Q=DA
τ
l2,d=DA
DA
⊥
,w=τ
τ⊥
,b=lT
l.(15)
Here, the persistence number Pquantifies the effective persistence length μFτ, which is the distance the
tracer traverses in roughly the same direction. The parameter Qis the ratio of two timescales: the
persistence time τand the time l2/DA
it takes the tracer to diffuse its own length ldue to active noise. The
parameters dand wdescribe the ratios of diffusion constants and persistence times of the active noise along
and ⊥directions, respectively. Finally, bdenotes the characteristic length lTthat we introduced earlier
measured in units of l.Itisusefultonotethatallthesedimensionless parameters depend on the geometry
of the tracer and the active bath properties. Let us add yet that in writing the above dimensionless equations
we removed the parameter μ/μ⊥by absorbing it into the definition of the perpendicular component of
noise: ξ⊥μ⊥/μ→ξ⊥.
The stochastic equations (10)–(14) are integrated using a simple Euler scheme with a time step of
δt/τ=10−5. The simulation time goes up to t/τ=2000, and the results are averaged over 100
independent simulation runs for each parameter set.
3. Results
One of the most important characteristics of tracer movement is the behavior of its MSD, which will be
presented in section 3.1. Of course more detailed characterization of tracer’s motion is provided by
probability distributions of its displacements. We explore them in section 3.2. Finally, in section 3.3 we
present a procedure which allows us to estimate the parameters of our effective Langevin model from direct
Brownian dynamics simulations. We then demonstrate that the time evolution of probability distribution of
displacements obtained in our model is in a very good quantitative agreement with the one acquired from
Brownian dynamics simulations.
3.1. Mean squared displacement
We analyze the motion of the tracer by computing its MSD: Δr2(t)=[r(t+T)−r(T)]2T,wherethe
averaging is performed over different initial times Tand over 100 independent simulation runs. Our results
span over several decades in time. In the following we evaluate how the MSD changes with varying each of
the above dimensionless parameters (15).
The MSD obtained for different values of the persistence number Pisshowninfigure4.Asonecan
infer from figure 4the MSD displays a ballistic behavior, Δr2(t)∼t2, for short times (t/τ1, for our
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Figure 4. The MSD, measured in units of l2, as a function of time, measured in units of τ, for three selected persistence number
values P. All other dimensionless parameters are set to 1. The solid black lines are guides to the eye.
Figure 5. The MSD as a function of time for three selected values of parameter Q. All other dimensionless parameters are set to
1. The solid black lines are guides to the eye.
choice of parameters). The practically pure ballistic motion is due to the persistence in random force acting
on the tracer (there are no thermal fluctuations in our model). The higher the persistence number, the
more space is explored by the tracer. From the equations (10)and(11), it is easy to see that the MSD should
scale as Δr2∼P2t2in the ballistic regime, which is supported by the numerical results in figure 2.Onthe
other hand, for long times (t/τ100) the tracer motion is eventually randomized for all Pso that the
normal diffusion sets in, Δr2(t)∼t. One can notice that a larger value of Pgives rise to an enhanced
effective value of the diffusion coefficient.
Varying parameter Q=DA
τ/l2yields a nontrivial change of the MSD, see figure 5.BychangingQone
essentially alters the active diffusion constants DA
and DA
⊥=DA
/d.ComparedtothecaseQ=1, for
Q=10 the tracer is subjected to a larger value of correlated noise, which also affects short-time ballistic
motion leading to a larger effective speed. However, the tracer is also exposed to a greater active diffusion
constant DA
⊥or correlated noise along its lateral direction, causing a destruction of its ordered motion at
earlier times if compared to the case Q=1. As a consequence of this, the effective diffusion constant at long
times is not markedly distinct between these two cases. On the other hand, for Q=0.1thelateralrandom
force exerted on the tracer is sufficiently small, such that for a chosen P=1, one obtains a pronounced
ballistic regime spanning up to times t/τ≈10. Consequently, the effective diffusion constant at long times
is noticeably larger with respect to the previous two cases.
The MSD for persistence number P=10 and for diverse values of parameter d, quantifying the ratio of
active diffusion constants along the main and lateral axis of the tracer, is shown in figure 6. The effect of
changing dis straightforward. Increasing dabove the reference value d=1, corresponding to DA
=DA
⊥,the
tracer exhibits longer ballistic movement due to elevated diffusion constant of the persistent active noise
along its symmetry axis. In contrast, d<1 signifies less persistent ballistic motion.
In figure 7we show the MSD for P=10 and two values of the parameter w=τ/τ⊥. Note that the
measurements of time auto-correlations of forces Fand F⊥in Brownian dynamics simulations suggest that
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Figure 6. The MSD as a function of time for P=10 and three selected values of parameter d. All other dimensionless
parameters are set to 1. The solid black lines are guides to the eye.
Figure 7. The MSD as a function of time for P=10 and w=1, 10. All other dimensionless parameters are set to 1. The solid
black lines are guides to the eye.
Figure 8. The MSD as a function of time for P=10 and four selected values of parameter b. All other dimensionless parameters
are set to 1. The solid black lines are guides to the eye.
τ⩾τ⊥with w1. Thus, figure 7indicates that in the physically relevant region of parameter space
1⩽w<10 the MSD is not appreciably sensitive to variations of w. This implies that in most practical
cases one can set w=1.
Finally, the effect of changing the parameter b=lT/lon the MSD is depicted in figure 8. As can be seen
from the equation (12) larger values of bcorrespond to a slower variation of tracer’s angular velocity, and
thus to a longer persistence of motion. As before for early times we obtain a ballistic regime, while for
longer times diffusive motion takes place. One notes that the duration of the ballistic regime grows with b.
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Figure 9. The probability distribution Pxof displacements Δx,measuredinunitsofl, for four characteristic time values t/τ.
Here we chose the same values of parameters as those used to obtain the cyan line in figure 6,P=d=10 and Q=w=b=1.
For t/τ=200 the solid black line provides the best fit of Pxto a Gaussian form.
Figure 10. The probability distribution Prof displacements Δr,measuredinunitsofl, for four characteristic time values t/τ.
Here we chose the same values of parameters as those used to obtain the cyan line in figure 6,P=d=10 and Q=w=b=1.
For t/τ=1 the solid black line represents the best fit of Prto a Gaussian form. For t/τ=200 the solid black line is the best fit
of Prto the form Pr=(Δr/σ2
r)e−Δr2/(2σ2
r),whereσris a fit parameter.
3.2. Probability distribution of displacements
The time evolution of probability distribution Pxof tracer displacement Δx=x−x0,withrespecttosome
initial position x0, obtained for some representative values of relevant parameters, is shown in figure 9.As
one can infer from this figure, at early times, when the tracer displays ballistic motion, the probability
distribution Pxis bimodal with two peaks located at Δx/l≈±P. As the time progresses, the height of these
peaks decreases until the end of the ballistic regime. After a characteristic time t/τ(in our case t/τ=30)
they completely disappear, and Pxexhibits a plateau. Later in time (see the figure corresponding to
t/τ=50) the shoulders of Pxsubside, and with further increase in time Pxcrosses over to a purely
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Figure 11. The probability distribution Pxof tracer displacements Δx,measuredinunitsofσ, for four characteristic time
values t/τR. The results obtained in Brownian dynamics simulations are shown as gray histograms; here the persistence number
of an active particle was chosen to be Per =80/3 and the bath packing fraction was set to φ≈0.08. The solid black lines show Px
for matching parameters of the coarse-grained model P≈0.4, Q≈0.1, d≈1.5, w≈1andb≈1.7, without any free fitting
parameter.
Gaussian form for sufficiently long times. The width of this Gaussian is directly related to the MSD
presented in figure 6.
For the same parameter choice, the corresponding time evolution of the probability distribution Prof
radial displacement Δr=Δx2+Δy2is shown in figure 10. In this representation the initial two peak
structure of Px,presentedinfigure9, maps onto a Gaussian centered at Δr/l≈P. Later in time the peak of
Prpropagates to higher values of Δr/l, and develops a shoulder for smaller displacements Δr/l(see
figure 10 for t/τ=30). For even longer times (in our case t/τ=50) the shoulder becomes more
pronounced and eventually the probability distribution attains the expected form Pr=(Δr/σr)e−Δr2/(2σ2
r),
which is typical for the diffusive regime.
3.3. Comparison between the models
In Brownian dynamics simulations the tracer motion depends on its geometry (through the radius Rand
the mobilities μ,μ⊥and κ) and on the properties of the surrounding active bath (characterized by the
packing fraction of active particles φand their persistence number Per). For a given set of parameters of this
system, we would like to find the matching parameters (15)intheeffectiveLangevindescription.Toachieve
this goal, we use the insights from Brownian dynamics simulations to numerically compute the physical
quantities entering (15). We illustrate this mapping procedure for the example of a tracer of radius R/σ =5
immersed in an active bath described by Per =80/3andφ≈0.08. One can extract the effective
self-propulsion force Facting on the tracer, and the persistence times of the active noise by analyzing the
statistic of force and torque (figure 2). This gives F≈75kBT/σ,andτ≈τ⊥≈τT≈0.45τR.
The translational active diffusivities DA
and DA
⊥follow directly from the MSDs along the polar axis e
and the axis e⊥perpendicular to it, Δr2
(t)and Δr2
⊥(t),where˙
r=P(1 +ξ)and˙
r⊥=Pξ⊥are
equations analogous to (10)–(11) but written in the eigenframe of the tracer (do not confuse Δr2
(t)and
Δr2
⊥(t)with the previously introduced quantity Δr2(t)=Δx2(t)+Δy2(t)referring to the lab
system xOy). Although the colored noise (4) enters these equations, due to their simplicity, they can be
solved analytically
Δr2
(t)=P2t2+2Qt−(1 −e−t), (16)
Δr2
⊥(t)=2Q
dt−1
w(1 −e−wt).(17)
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On the other hand, the same quantities can be measured in Brownian dynamics simulations. Our
simulation results are presented in figure A1.Fromequations(16)and(17), in the limit of large times, one
gets Δr2
(t)→P2t2and Δr2
⊥(t)→2Qt/d. Converting these expressions back to the dimensional form,
and fitting the data of figure A1 to them, we obtain F≈75kBT/σ, in perfect agreement with the above
estimate from force statistics, and DA
⊥≈5σ2/τR. On the other hand, in the limit of small times, one obtains
Δr2
(t)→(P2+Q)t2. Fitting the Δr2
(t)data corresponding to this regime allows us to extract the
diffusion constant DA
≈7.5σ2/τR,seeinsetoffigureA1.
Figure A2 shows the orientational auto-correlation function Co(t)=e(t+t0)e(t0)t0of the tracer
measured in Brownian dynamics simulations; here e(t)=cos θ(t)ex+sin θ(t)eyis the tracer’s orientation
vector, and the averaging is performed over different initial times t0and over 15 independent simulation
runs. The obtained data can be nicely fitted to a simple exponential form Co(t)=e−t/τo,withτo≈20τR
being the orientational correlation time. We can argue that τoshould be close to 1/DA
R.Indeed,from
equation (12) it follows that [θ(t)−θ(0)]2≈2DA
Rt,tτ⊥. On the other hand, since τoτ⊥,the
auto-correlation function can be written as Co(t)=cos[θ(t)−θ(0)]≈1−1
2[θ(t)−θ(0)]2=1−DA
Rt,
which can be compared to Co(t)≈1−t/τo. This allows us to estimate the rotational diffusion constant of
the coarse-grained model, DA
R≈0.05τ−1
R.
Using the above findings and the values of tracer mobilities μ,μ⊥and κfrom the appendix A,onecan
calculate the sought matching parameters of the effective Langevin model. This gives: P≈0.4, Q≈0.1,
d≈1.5, w≈1andb≈1.7. In figure 11 we show a comparison between the time evolution of probability
distribution Pxof tracer displacements Δxobtained in Brownian dynamics simulations (gray histograms)
and in the effective Langevin model (black solid curves). We achieve a very good quantitative agreement,
confirming that our effective model correctly describes the tracer motion in an active bath.
4. Conclusion
We have studied the dynamics of a polar tracer with a concave surface in a bath consisting of active
particles. By investigating the non-equilibrium statistics of the force and torque with which the active
particles push against the tracer, we were able to fully determine a set of three effective Langevin equations
for the tracer position and orientation. Thus, this procedure enabled us to reduce the complexity of the
problem, by going from an involved many-body dynamics approach to a coarse-grained description of the
bath, which appears in the tracer dynamics as a force drift and an exponentially correlated noise. Our
effective Langevin equations contain five independent dimensionless parameters, which depend on the
geometry of the tracer and the properties of active particles constituting the bath. For a given set of
parameters of the original Brownian dynamics approach, we managed to construct a numerical mapping to
obtain the matching parameters of the coarse-grained model without any free parameters. We
demonstrated a very good quantitative agreement between the time evolution of the probability distribution
of displacements obtained in Brownian dynamics simulations and in the effective Langevin model. In this
way we have been able to reduce the computational efforts by several orders of magnitude. For example, to
perform one run of a Brownian dynamics simulation up to time t/τR=1000 required about 12 h on 20
CPUs, while one run of the effective Langevin simulation for the same time t/τRtook only a few minutes
on a single CPU. Further work is needed in order to establish an analytical connection between the
parameters in our coarse-grained model and the parameters of the full many-body system in the Brownian
dynamics simulations.
Polar tracers can harness energy from the noisy non-equilibrium environment of an active bath and
thereby generate directed motion. Our work provides a complete effective description for the coupled
translational and rotational tracer motion. It will help to further explore the capabilities of active baths for
fueling directed transport, for example, with micro shuttles. An extension of this idea is to endow the polar
tracers with some intrinsic information processing system so that they can sense their environment and act
accordingly. Such smart micro shuttles can then use reinforcement learning to learn to perform some
prescribed task. For example, in [70] it was demonstrated how smart active particles learn to optimize their
travel time in a potential landscape.
Acknowledgments
We thank Benjamin Lindner for initiating discussions. MK gratefully acknowledges financial support from
the Alexander von Humboldt Foundation through a postdoctoral research fellowship and from SFB 910
funded by Deutsche Forschungsgemeinschaft.
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Appendix A. Brownian dynamics simulations
We consider a system of Ninteracting active Brownian particles in two dimensions, which self-propel with a
constant speed vand have a mobility μ. Their dynamics is described by overdamped stochastic
equations [37]
˙
ri=vui−μ
j=i∇riV(ri−rj), (A.1)
˙
θi=√2DRηi.(A.2)
Here riis position vector and ui≡(cos θi,sinθi) the unit orientation vector of particle i,DRdenotes its
rotational diffusion constant, and ηiis Gaussian white noise of zero mean and unit variance: ηi=0,
ηi(t)ηj(t)=δijδ(t−t). We perform simulations in the regime of large v, which is physically most
interesting. This allows us to neglect the effect of translational thermal diffusivity in (A.1). Active particles
interact with each other through pairwise forces, which are given by the negative gradient of the
Weeks–Chandler–Andersen (WCA) potential
V(r)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
4εσ
|r|12
−σ
|r|6+ε,|r|⩽21/6σ,
0, |r|>21/6σ.
Here εis the strength of the potential and σis the characteristic length where the potential takes the value ε.
We carry out simulations in a rectangular box of size L×Land use periodic boundary conditions [37].
We use σas the unit of length, persistence time τR=D−1
R=σ2/(3μkBT) of an active particle as the unit
of time, and we measure energies in units of kBT,whereTis the temperature of the solvent surrounding
active particles (not to be confused with the torque Tused in the main text). We introduce the persistence
number Per =vτR/σ, which measures the distance an active particle travels in approximately the same
direction. The equations (A.1)and(A.2) can be transformed into a dimensionless form with two
independent dimensionless parameters: the persistence number Per and the potential strength ε/kBT.The
persistence number Per, together with the area packing fraction of active particles, φ=Nσ2π/(4L2),
determinethepropertiesoftheactivebath.
Here we consider a polar tracer immersed in the bath of interacting active particles (see figure 1). We
imagine our tracer as a semicircle of radius Rcomposed of particles having effective diameter σ.Then,an
active particle interacts with a particle of the semicircle through a repulsive contact force, derived from the
WCA potential, provided that the distance between them is smaller than 21/6σ. The position of the polar
tracer is described by the coordinates of its center of mass, r=(x,y), and the angle its symmetry axis makes
with the x-axis of the lab frame (figure 1(a)). Now the equations of motion of the tracer can be written in
the form
˙
x=μFcos θ−μ⊥F⊥sin θ,(A.3)
˙
y=μFsin θ+μ⊥F⊥cos θ,(A.4)
˙
θ=κT.(A.5)
Here, Fand F⊥are the projections on eand e⊥of the resulting force exerted by active particles on the
tracer, and similarly Tis the projection on the unit vector ez=e×e⊥of the resulting torque on the tracer.
The translational mobilities of the tracer are denoted by μand μ⊥, while its rotational mobility is denoted
by κ.
ThenumberofactiveparticlesisfixedtoN=104,andtheareaL2of the simulation box is adjusted to
obtain the required packing fraction φ.Wesetε/kBT=100, R/σ =5, μ/μ =0.2, μ⊥/μ =0.1and
κσ2/(3 μ)=10−3(unless otherwise stated). Equations (A.1)–(A.5) are integrated using a simple Euler
scheme with a time step of δt/τR=10−5. The simulation time goes up to t/τR=5000, and all results are
averaged over 15 independent simulation runs.
A typical snapshot from our Brownian dynamics simulation is presented in figure 1(b). The probability
distributions of F,F⊥and Tand their time auto-correlation functions are shown in figure 2,whilethe
cross-correlation function F⊥(t0)T(t0+t)obtained in this approach is presented in figure 3.Infigure11
we give the probability distributions Pxof tracer displacement Δxfor several characteristic times. The
MSDs Δr2
(t)and Δr2
⊥(t), introduced in the section 3.3 of the main text, are displayed in figure A1.The
auto-correlation function Co(t)=e(t+t0)e(t0)t0, quantifying the correlation of tracer’s orientation
vector e(t)=cos θ(t)ex+sin θ(t)ey,isshowninfigureA2 anddiscussedinthemaintext.
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Figure A1. The MSDs Δr2
(t)and Δr2
⊥(t),definedinthesection3.3 of the main text, measured in units of σ2,asafunction
of time t/τR. The inset shows the same quantities for small times t/τR. The black solid lines are guides to the eye. The active bath
is characterized by the parameters Per =80/3andφ≈0.08.
Figure A2. The orientational auto-correlation function Co(t)=e(t+t0)e(t0)t0versus time t/τR. The black solid line is the
best fit of the data to the form Co(t)=e−t/τo,whereτois the orientational correlation time. The active bath is characterized by
the parameters Per =80/3andφ≈0.08.
Figure A3. The average self-propulsion force Facting on the tracer, measured in units of kBT/σ, as a function of packing
fraction φof active particles in the bath, for two selected values of the persistence number Per =40, 80/3.
The influence of bath parameters φand Per on the average force Fexperienced by the tracer is
presented in figure A3.TheforceFincreases with active particles’ persistence Per, while it saturates as the
packing fraction φofactiveparticlesisincreasedforafixedPer. Note that we considered only homogeneous
active baths with particle packing fractions φsmaller than the threshold value above which the
motility-induced phase separation of the bath takes place [1].
Finally, we consider tracers of three different sizes R/σ =5, 27, 80 immersed in a bath characterized by
Per =80/3andφ≈0.08. Their MSD Δr2(t)is presented in figure A4.Wesimulatedthetracer
trajectories up to times t/τR=5000. The tracer of size R/σ =5 reaches a diffusive regime, characterized by
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Figure A4. The MSD Δr2(t)of the tracer, measured in units of σ2, as a function of time t/τR, for three selected tracer sizes
R/σ =5, 27, 80. The black solid lines are guides to the eye. The active bath is characterized by the parameters Per =80/3and
φ≈0.08.
Δr2(t)∼t,alreadyattimest/τR100. On the other side, even for much larger times t/τR1000 the
MSD of tracers of size R/σ =27 and R/σ =80 still displays super-diffusive motion, characterized by
Δr2(t)∼tαwith α>1. This effect we attribute to a significant decrease of tracer rotational mobility
when increasing R. It is expected, however, that the tracers of large size should reach the diffusive regime for
sufficiently large times; due to high computational costs these times were not accessible in our simulations.
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