Engineering sensorial delay to control phototaxis and emergent collective behaviors

Mite Mijalkov,1, 2 Austin McDaniel,3 Jan Wehr,3 and Giovanni Volpe1, 2

1Soft Matter Lab, Department of Physics, Bilkent University, Cankaya, 06800 Ankara, Turkey2UNAM – National Nanotechnology Research Center, Bilkent University, Ankara 06800, Turkey

3Department of Mathematics and Program in Applied Mathematics,University of Arizona, Tucson, Arizona 85721 USA

Collective motions emerging from the interaction of autonomous mobile individuals play a key rolein many phenomena, from the growth of bacterial colonies to the coordination of robotic swarms.For these collective behaviors to take hold, the individuals must be able to emit, sense and reactto signals. When dealing with simple organisms and robots, these signals are necessarily veryelementary, e.g. a cell might signal its presence by releasing chemicals and a robot by shininglight. An additional challenge arises because the motion of the individuals is often noisy, e.g. theorientation of cells can be altered by Brownian motion and that of robots by an uneven terrain.Therefore, the emphasis is on achieving complex and tunable behaviors from simple autonomousagents communicating with each other in robust ways. Here, we show that the delay betweensensing and reacting to a signal can determine the individual and collective long-term behavior ofautonomous agents whose motion is intrinsically noisy. We experimentally demonstrate that thecollective behavior of a group of phototactic robots capable of emitting a radially decaying light fieldcan be tuned from segregation to aggregation and clustering by controlling the delay with whichthey change their propulsion speed in response to the light intensity they measure. We track thistransition to the underlying dynamics of this system, in particular, to the ratio between the robots’sensorial delay time and the characteristic time of the robots’ random reorientation. Supportedby numerics, we discuss how the same mechanism can be applied to control active agents, e.g.airborne drones, moving in a three-dimensional space. Given the simplicity of this mechanism, theengineering of sensorial delay provides a potentially powerful tool to engineer and dynamically tunethe behavior of large ensembles of autonomous mobile agents; furthermore, this mechanism mightbe already at work within living organisms such as chemotactic cells.

I. INTRODUCTION

The interaction between several simple autonomousagents can give rise to complex collective behaviors. Thisis observed at all scales, from the organization of bacte-rial colonies [1, 2] and the foraging of ants and bees [3]to the assembly of schools of fish [4] and the collectivemotion of human crowds [5]. Inspired by these naturalsystems, the same principles have been applied to en-gineer autonomous robots capable of performing taskssuch as search-and-rescue in disaster zones, surveillanceof hazardous areas and targeted object delivery in com-plex environments [6–11].

Complex behaviors can emerge even if each agent fol-lows very simple rules, senses only its immediate sur-roundings and directly interacts only with nearby agents,without having any knowledge of an overall plan [12, 13].For example, while performing their swim-and-tumblemotion, chemotactic bacteria are able to climb a chemo-tactic gradient, e.g. in order to move towards re-gions rich in nutrients, by simply adjusting their tum-bling rate depending on the chemical concentration theysense [2, 14]. Furthermore, by releasing chemoattrac-tant molecules into their surroundings, they are capableof generating a chemical gradient around themselves towhich other cells can respond, e.g. in order to createbacterial colonies [1]. Similarly, simple mechanisms areat work in the organization of flocks of birds, schools offish, and herds of mammals, whereby complex collective

behaviors result from each animal reacting to signals sentby its neighbors. A similar approach has also been fruit-fully explored in order to build artificial systems withrobust behaviors arising from interactions between verysimple constituent agents [6, 10, 11, 15–18]. Complex be-haviors emerging from agents obeying simple rules havethe advantage of being extremely robust: for example,even if one or more agents are destroyed, the others cancontinue to work together to complete the task at hand;agents can also be removed or added mid-task withoutsignificantly affecting the final result.

Here, we experimentally and theoretically demonstratethat it is possible to engineer the individual and collectivebehavior of autonomous agents whose motion is intrinsi-cally noisy by making use of the delay in their sensorialfeedback cycle. That is, we show how the delay betweenthe time when an agent senses a signal and the time whenit reacts to it can be used as a new parameter for the engi-neering of large-scale organization of autonomous agents.This proposal is inspired by the motion of chemotacticcells, which are able to climb a chemical gradient by ad-justing a different parameter, i.e. their tumbling rate, inresponse to the concentration of molecules in their sur-roundings. We demonstrate that the collective behaviorof a group of phototactic robots, capable of emitting aradially decaying light field, can be tuned from segre-gation to aggregation and clustering by controlling thedelay with which they adjust their propulsion speed tothe light intensity. More precisely, we show that thistransition occurs as the ratio between the robots’ senso-

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rial delay time and characteristic time of their randomreorientation crosses a certain critical value.

II. SINGLE AGENT

We start by considering a single autonomous agentthat moves in a plane and whose orientation is sub-ject to noise. This happens naturally in the case ofmicroswimmers — microscopic particles capable of self-propulsion such as motile bacteria and cells [19, 20] —as the direction of their motion changes randomly overtime because of the presence of rotational Brownian mo-tion [2]. Similarly, autonomous robots, animals, and evenhumans can undergo a random reorientation when mov-ing in the absence of external reference points (a strik-ing example of this is an experiment where blindfoldedpeople who were asked to walk in a straight line sponta-neously moved along bent trajectories [21]). Such motionis known as active Brownian motion and can be modelledby the following system of stochastic differential equa-tions [12, 20, 22, 23]:

dxtdt

= v cosφt

dytdt

= v sinφt

dφtdt

=

√2

τηt

(1)

where (xt, yt) is the position of the agent in the plane attime t, φt is its orientation, v is its speed, τ is the re-orientation characteristic time (i.e., the time after whichthe standard deviation of the agent’s rotation is 1 rad),and ηt is a white noise driving the agent’s reorientation,as shown in Fig. 1a. The reorientation time τ can beassociated to an effective reorientation diffusion constantDR = τ−1, which, in the case of microswimmers, oftencoincides with the rotational diffusion constant of theparticle.

Furthermore, we will assume that this agent moves inthe presence of an external intensity field to which itreacts by adjusting its speed as a function of the instan-taneous intensity it senses. We have realized this ex-perimentally by using a phototactic robot (Elisa-3 [24])moving within the light gradient generated by a 100 Winfrared lamp, which emitted a radially symmetric lightintensity radially decaying with a characteristic lengthR = 35 cm, as shown in Fig. 1b. This robot measuresthe local light intensity It = I(xt, yt) corresponding toits position (xt, yt) at time t using 8 infrared sensorsevenly distributed around its circumference, and adjustsits propulsion speed v (I) accordingly, while randomlychanging its orientation with a characteristic reorienta-tion time τ = 1 s. Its motion can be described by modi-

fying Eqs. (1) as

dxtdt

= v (It) cosφt

dytdt

= v (It) sinφt

dφtdt

=

√2

τηt

(2)

Fig. 1b shows also a sample trajectory (line) superim-posed onto the picture of the robot. The function v(I) isplotted in Fig. 1c; its functional form is

v(I) = (v0 − v∞)e−I/Ic + v∞ , (3)

where v0 = 60 cm s−1 is the maximum speed (correspond-ing to a null intensity), Ic = 90 mV is the characteristicintensity scale (measured in volts) over which the velocitydecays, and v∞ = 3 cm s−1 is the residual velocity (in thelimit of infinite light intensity). It can be seen in Fig. 1bthat the runs between consecutive turns are longer in thelow-intensity (high-speed) regions while they are shorterin the high-intensity (low-speed) regions. The result isthat over a long period of time, the robot spends moretime in the high-intensity regions. As we will see, this be-havior is in agreement with our theoretical results givenin Eq. (7). This is also in agreement with the behaviorof chemotactic cells whose explorative behavior decreaseswhen they reach regions with ideal conditions and reducetheir locomotion activity in favor of other metabolic ac-tivities [2].

We now proceed to add a delay δ in the agent’s re-sponse to the measured intensity, which is the main nov-elty of our work. With this addition, the equations de-scribing the motion of the robot become:

dxtdt

= v (It−δ) cosφt

dytdt

= v (It−δ) sinφt

dφtdt

=

√2

τηt

(4)

The idea of introducing a sensorial delay is inspired bythe way in which bacteria react to a chemotactic gradient;in fact, chemotactic bacteria make a comparison of thenumber of molecules they detect around themselves atconsecutive times in order to decide how to adapt theirmotion [2, 14, 25]. The presence of sensorial delays istypically ignored, or treated as a nuisance to be con-trolled [26], while only few theoretical works have con-sidered its possible constructive effects but in situationsdifferent from the one studied in this work [27, 28]. Byintroducing a delay long enough so that the robot hasenough time to randomize its direction of motion beforeresponding to the sensorial input by changing its speed,we can observe that the motion becomes more directed

3

0 10 20 30

10

20

30

x (cm)

y (cm)

0 100 200 300 400

10

20

30

40

50

60

I (mV)

v (cm/s)(a) (b) (c)

φt

(xt, yt)

(xt, yt)

x

y v0

v∞

FIG. 1. (a) An autonomous agent, whose position at time t is (xt, yt), moves with speed v in the direction described byφt, corresponding to its instantaneous orientation (arrow), which varies randomly with a characteristic time τ . (b) Pictureof a phototactic robot in a light gradient generated by an infrared lamp. The propulsion speed of the robot depends on theinstantaneously measured light intensity, while its orientation changes randomly. A sample trajectory is shown by the graysolid line. (c) Relation between the measured light intensity I and the robot’s speed v [Eq. (3)].

towards the high-intensity (low-speed) region, as can beobserved by comparing the trajectories in Fig. 2a (withdelay δ = +5τ) to that in Fig. 1b (without delay).

Things become even more interesting if a “negative”delay is introduced, i.e. if a prediction of the futuremeasured intensity is employed to determine the currentrobot speed. While it is straightforward to see how apositive delay is introduced (e.g. by a delay in the trans-mission of the signal or by a lapse time before reacting tothe signal), the introduction of a negative delay is less in-tuitive. In fact, a negative delay can be rationalized as aprediction of the future state of the system, which can bedone based on the signal received up to the present time.For example, in the case of our robots, a negative delay isintroduced by linearizing the light intensity measurementas a function of time and extrapolating it into the future,i.e. I(t − δ) ≈ I(t) − δI ′(t), where both I(t) and I ′(t)are known at time t; higher order predictor algorithmsare also possible making use of more information aboutthe evolution of the intensity measured up to the present.We show the corresponding trajectory in Fig. 2b, whereδ = −5τ . In this case, the robot escapes from the high-intensity region and moves towards the edge, where theinfrared lamp intensity is lower (and the speed higher).

In order to quantify these observations, we have mea-sured the effective radial drift of the robots, which iscalculated [29] as

D(r) =1

∆t〈rn+1 − rn | rn ∼= r〉 , (5)

where r is the radial coordinate, rn are samples of therobot’s radial position and ∆t is the time step betweensamples. The results are shown in Figs. 2c and 2d. Forpositive delay (red circles), the negative drift for largeradial distance shows that the robot tends to move to-

wards the central high-intensity region. For negative de-lay (blue diamonds), the positive drift shows that therobot escapes from the central high-intensity region. Wehave also theoretically calculated the radial drift for anautonomous agent whose motion is governed by Eqs. (4)(see Appendix A), obtaining

D(r) =τ

2

(1− δ

τ

)v(r)

dv

dr(r) +

τv(r)2

r, (6)

where v(r) = v(I(r)) and we have assumed a radiallysymmetric intensity distribution. The solid lines plot-ted in Figs. 2c and 2d show that there is a good agree-ment between these theoretical predictions and the ex-perimentally measured data. We have further corrob-orated these results with numerical simulations, whoseresults are shown in Fig. S1 in the Supplementary Infor-mation and are in good agreement with the experimentalresults shown in Fig. 2. The numerical simulations wereperformed by solving the finite difference approximationof Eqs. (4) [20, 30]. The delayed sensorial measurementwas evaluated by Taylor-expanding the measured inten-sity about the agent’s location and extrapolating the cor-responding past/future value.

We can also theoretically derive the approximatesteady-state probability distribution of the agent’s po-sition (see Appendix A), which exists and equals

ρ0(x, y) =1

N v(x, y)1+δτ

, (7)

provided that the normalization constant

N =

∫v(x, y)−(1+ δ

τ ) dx dy <∞.

Eq. (6) confirms our initial observations that the largerthe positive delay is (solid lines in Fig. 3a), the more

4

5 10 15 20

-5

0

5

10

15

r (cm)

D (r ) (cm s−1)

-5 -3 -1 1 3 5

-4

0

4

8

12

δ /τ

D (30 cm) (cm s−1)

δ = +5τ

δ = -5τ

t = 10s t = 20s t = 30s

t = 10s t = 20s t = 26s

(a)

(b)

(c)

(d)

FIG. 2. The long-term behavior of a robot in the light gradi-ent generated by an infrared lamp changes depending on thedelay with which it adjusts its speed in response to the sen-sorial input, i.e. the measured light intensity. The sensorialdelay was introduced by linearizing the measured light inten-sity as a function of time and by extrapolating its past/futurevalue. (a) For positive delays (δ = +5τ), the tendency of therobot to move towards the high-intensity (low-speed) regionsis enhanced, when compared to the case without delay pre-sented in Fig. 1b. (b) For negative delays (δ = −5τ) therobot tends to move towards the low-intensity (high-speed)regions. In both cases, the trajectories are shown for a periodof 10 s preceding the time indicated on the plot and the robotis shown at the final position. (c) Radial drift D(r) calculatedaccording to Eq. (5) from a 40-minute trajectory for the casesof positive (circles) and negative (diamonds) delays. (d) Ra-dial drift calculated according to Eq. (5) when the robots areat 30 cm from the center of the illuminated area as a func-tion of δ/τ . The solid lines in (c) and (d) correspond to thetheoretically predicted radial drifts given by Eq. (6).

time the agent spends in the low-speed (high-intensity)regions. On the other hand, the more negative the delayis (solid lines in Fig. 3b), the more time the agent spendsin the high-speed (low-intensity) regions. Interestingly,

20 40 60 80 1000

0.25

0.5

0.75

1

r (cm)

ρ0(r)/r (a.u.)

δ = −5τ

δ = −3τ

δ = −1τ

20 40 60 80 1000

0.25

0.5

0.75

1

r (cm)

ρ0(r)/r (a.u.)

δ = +5τ

δ = +3τ

δ = 0τ

(a)

(b)

FIG. 3. Theoretically predicted radial probability distributionof the position of an agent [Eq. (7)] moving in a radial inten-sity field (inset in (a), the agent is confined in a circular wellwith radius 100 cm indicated by the gray border) as a functionof the sensorial delay time: (a) for δ > −τ the agent tendsto spend more time in the low-speed (high-intensity) centralregion; (b) for δ < −τ the agent spends more time in the high-speed (low-intensity) peripheral region; for δ = −τ the prob-ability distribution is uniform (black line). These results arecorroborated by numerical simulations of autonomous agentsshown by the symbols. For each case, we have simulated avery long trajectory (108 s) to obtain an accurate and smoothdistribution.

we note that there is a cutoff value at δ = −τ for whichthe probability distribution of the agent is uniform (blacksolid line in Fig. 3b). We have further corroborated theseresults with numerical simulations shown by the symbolsin Figs. 3a and 3b.

We emphasize that the qualitative change of the parti-cle’s behavior occurs at a negative delay, i.e. δ = −τ .Introduction of negative delays is thus crucial for thedescribed transition. On the other hand, positive de-lays also influence the system’s behavior strongly. Whilewithout delay the particle spends more time in slow re-gions, a positive delay makes this tendency more pro-nounced, as seen clearly at the quantitative level fromEq. (7). This tendency persists, albeit in a weaker form,for small negative delays −τ < δ < 0 and gets reversedat the critical value δ = −τ .

5

III. MULTIPLE AGENTS

We can now build on these observations to engineerthe large-scale organization of groups of robots. In orderto do this, each robot must be able not only to sensethe local intensity, but also to create a luminosity field.Thus, we have equipped each robot with 6 LEDs evenlyplaced around its circumference (EDEI-1LS3), as shownin Fig. 4a, which emit infrared light (wavelength 850 nm)so that each robot generates a decaying light intensityaround itself. The LEDs are arranged so that the robotmeasures only the light intensity emitted by the otherrobots. A phototactic robot capable of measuring thislight intensity will be able to move in the resulting fieldsimilarly to the case discussed above, i.e. that of the lightintensity generates by a static infrared lamp. We stressthat each robot only measures the local intensity withoutbeing aware of the positions of the other robots.

We have experimentally studied how three autonomousrobots organize by reacting to the cumulative light fieldcreated by all of them as a function of their sensorial de-lay. For a positive sensorial delay (δ = +3τ , Fig. 4b),the three robots gradually move towards each other andform a dynamic cluster, which remains stable over time.A single robot’s tendency to spend more time in the high-intensity regions when there is positive delay leads tomultiple robots forming clusters because of their pref-erence for high-intensity regions. For a negative delay(δ = −3τ , Fig. 4c), the three robots tend to move awayfrom each other, dispersing and exploring a much largerarea. In order to understand this behavior in a morequantitative way, we have also simulated a larger num-ber of trajectories for a group of three agents and plot-ted the average distance between the agents as a func-tion of time for various sensorial delays. The results arereported in Fig. 4d: for positive delays, as the agentstend to come together and form a cluster, their averagedistance decreases over time; for negative delays, as theagents move apart and explore a larger area, their av-erage distance increases. The qualitative change of theagents’ behavior occurs at a strictly negative value of thedimensionless parameter δ/τ = −1 (see Eq. (7)). Whileintroduction of negative delays is thus crucial for the de-scribed transition from aggregation to segregation, posi-tive delays also influence the system’s behavior stronglyby enhancing the tendency of the agents to aggregate.Importantly, not only a light field, but any radially de-caying scalar (e.g. chemical, acoustic) field created bythe autonomous agents can be used in order to achievethis kind of control over their behavior.

In order to explore the scalability of this mechanism,we have simulated the behavior of an ensemble of 100robots. Each robot emits around itself a Gaussian in-tensity field that decays radially, and responds to thelocally measured cumulative intensity by adjusting itsspeed. The long-term behavior and the large-scale orga-nization of these ensembles of agents significantly dependon the sensorial delay, as shown in Fig. 5. For positive de-

2500 5000 75000

40

80

120

160

t (s)

d (cm)

δ = −3τ

δ = +3τ

t = 25s t = 50s t = 90s

t = 25s t = 50s t = 75s

(a) (d)

(b)

(c)

FIG. 4. (a) Picture of a phototactic robot equipped with sixinfrared LEDs so that it can emit a radially decaying light in-tensity around itself. (b) A group of three such robots, whichadjust their speed as a function of the sensed light intensity,aggregate and form a dynamic cluster if their sensorial de-lay is positive (δ = +3τ) and (c) segregate if it is negative(δ = −3τ). In each panel in (b) and (c) the trajectories areshown for a period of 10 s preceding the time indicated on theplot and the dot indicates the final position of the robot. (d)Average distance d between agents in a group of three sim-ulated autonomous agents as a function of time: for positivedelays, as the agents tend to come together and form a clus-ter, their average distance decreases over time; for negativedelays, as the agents move apart and explore a larger area,their average distance increases.

lay, they move collectively by forming clusters (Figs. 5aand 5b). On the other hand, for negative delays, theymove away from each other in order to reduce the inten-sity each of them measures and are thus able to explorethe space more effectively (Figs. 5c and 5d). The possi-bility of tuning the sensorial delay can be exploited, forexample, in a search-and-rescue task by setting initially anegative delay so that the robots can thoroughly explorethe environment and, at a later stage, a positive delayso that the robots can be collected into clusters to sharethe gathered information. Collecting all robots can alsobe easily achieved by sending a strong signal capable ofeclipsing the signals emitted by the robots themselves.

It is also possible to adjust the behavior of the agentsby altering the intensity-speed relation to something dif-ferent than Eq. (3). For example, instead of a monoton-ically decreasing relation, it is possible to use a relation

6

(a)

(b)

(c)

(d)

δ=

+10τ

δ=

+3τ

δ=

−3τ

δ=

−10τ

t = 10s t = 100s t = 1000s t = 10000s

FIG. 5. Simulation of the long-term behavior of an ensembleof 100 autonomous agents that emit a radially decaying inten-sity field and adjust their speed depending on the measuredlocal intensity. Depending on the sensorial delay, the long-term behavior and large-scale organization are significantlydifferent. (a)-(b) In the case of positive delays, the agentscome together and form metastable clusters. (c)-(d) In thecase of negative delays, they explore the space, staying awayfrom each other.

with a minimum at some specific value. As can be seen inFig. S2 in the Supplementary Information, this alters theagent’s behavior so that it spends more time where theintensity corresponds to the minimum speed. In this way,it is possible to control where the agent will spend most ofits time, which may be useful, e.g., for targeted delivery.Furthermore, in the presence of multiple agents capableof emitting a radially decaying intensity field, changingthe intensity-speed relation permits one to control var-ious features of the clusters such as their characteristicsize, as shown in Fig. S3 in the Supplementary Informa-tion.

Our results can also be extended to the three-dimensional case, where they still hold with only minoradjustments. This could be important when consider-ing airborne objects (e.g. drones, flying insects, birds)or underwater objects (e.g. fish, submarine robots). Inthree dimensions, the autonomous agent motion can be

modelled by the set of equations

dxtdt

= v (It−δ) sin θt cosφt

dytdt

= v (It−δ) sin θt sinφt

dztdt

= v (It−δ) cos θt

dθtdt

=1

τcot θt +

√2

τη(1)t

dφtdt

=1

sin θt

√2

τη(2)t

(8)

where (xt, yt, zt) is the position of the agent at time t, θtand φt are its azimuthal and polar orientations respec-

tively, and η(1)t and η

(2)t are independent white noises.

Similar equations but without delay have already beenconsidered, e.g. in Ref. 31 to describe active Brownianmotion in three dimensions. The last two equations de-scribe (accelerated) Brownian motion on the surface ofthe unit sphere (see Supplementary Information). Fromthis model we obtain the approximate steady-state prob-ability distribution (see Appendix A and SupplementaryInformation), which exists and equals

ρ0(x, y, z) =1

M v(x, y, z)1+2 δτ

, (9)

provided that the normalization constant

M =

∫v(x, y, z)−(1+2 δ

τ ) dx dy dz <∞.

Comparing Eq. (9) and Eq. (7), we note that the maindifference is that in the three-dimensional case the uni-form distribution occurs for δ = −0.5τ instead of forδ = −τ . Otherwise, the agents still exhibit a qualita-tively different behavior for positive and negative sen-sorial delay, corresponding, respectively, to an effectivedrift towards high-intensity and low-intensity regions, asillustrated in Figs. S4 and S5 in the Supplementary In-formation. As in the two-dimensional case, also in thethree-dimensional case it is possible to engineer this driftby changing the time delay in order to tune the collectivebehavior of a swarm from aggregation and clustering tosegregation.

IV. CONCLUSION

We have demonstrated the use of delayed sensorialfeedback to control the organization of an ensemble ofautonomous agents. We realized this model experimen-tally by using autonomous robots, further backed it upwith simulations, and finally provided a mathematicalanalysis which agrees with the results obtained in theexperiments and simulations. Our findings show that a

7

single robot, measuring the intensity locally, spends moretime in either a high or a low-intensity region depend-ing on its sensorial delay. Tuning the value of the delaypermits one to engineer the behavior of an ensemble ofrobots so that they come together or separate from eachother. The robustness and flexibility of these behaviorsare very promising for applications in the field of swarmrobotics [6, 10, 11, 16, 18] as well as in the assemblyof nanorobots, e.g., for targeted delivery within tissues.Furthermore, since some living entities, such as bacteria,are known to respond to temporal evolution of stimuli[2, 25], the presence of a sensorial delay could also explainthe swarming behavior of groups of living organisms.

ACKNOWLEDGMENTS

The authors thank Gilles Caprari (GCtronic) for hishelp with the robots. GV thanks Holger Stark foruseful discussions that led to the original idea for thiswork. JW and AM were partially supported by theNSF grants DMS 1009508 and DMS 0623941. GV waspartially supported by Marie Curie Career IntegrationGrant (MC-CIG) PCIG11 GA-2012-321726 and a Distin-guished Young Scientist award of the Turkish Academyof Sciences (TUBA). MM was partially supported by

TUBITAK grant 113Z556.

Appendix A: Mathematical derivation

We studied the limit of the system (4) as δ, τ → 0 atthe same rate so that δ = cε and τ = kε where c and kremain constant in the limit δ, τ, ε → 0. We expanded vabout t to first order in δ and solved the resulting equa-tions for x and y. We expanded the resulting system to

first order in the small parameter δ√τ

. We then consid-

ered the corresponding backward Kolmogorov equationfor the probability density ρ. We expanded ρ in powersof the parameter

√ε, i.e. ρ = ρ0 +

√ερ1 + ερ2 + ..., and

used the standard multiscale expansion method [32] toderive the backward Kolmogorov equation for the limit-ing density ρ0:

∂ρ0∂t

=τ

2

(1− δ

τ

)v

(∂v

∂x

∂ρ0∂x

+∂v

∂y

∂ρ0∂y

)+τv2

2∆ρ0 .

(A1)From this equation, we got the limiting SDE:{dxt = τ

2

(1− δ

τ

)v(xt, yt)

∂v∂x (xt, yt)dt+

√τv(xt, yt)dW

1t

dyt = τ2

(1− δ

τ

)v(xt, yt)

∂v∂y (xt, yt)dt+

√τv(xt, yt)dW

2t

(A2)where W 1 and W 2 are independent Wiener processes.Assuming that v is rotation-invariant, we got fromEq. (A2) the formula for the radial drift [Eq. (6)]:

D(r) =τ

2

(1− δ

τ

)v(r)

dv

dr(r) +

τv(r)2

r. (A3)

Setting the right-hand side of the forward (Fokker-Planck) equation corresponding to Eq. (A1) equal tozero, we got the formula for the stationary probabilitydensity ρ0 (if it exists) [Eq. (7)]

ρ0(x, y) =1

N v(x, y)1+δτ

, (A4)

where N is the normalization constant. A similar analy-sis follows for the three-dimensional case, leading to thethree-dimensional stationary probability density given byEq. (9). A more detailed derivation is provided in theSupplementary Information.

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