Harnessing confinement and driving to tune active particle dynamics
Aniruddh Murali, Pritha Dolai, Ashwini Krishna, K. Vijay Kumar, Shashi Thutupalli
HHarnessing confinement and driving to tune active particledynamics
Aniruddh Murali, ∗ Pritha Dolai, ∗ Ashwini Krishna, K. Vijay Kumar, † and Shashi Thutupalli
1, 2, ‡ Simons Centre for the Study of Living Machines,National Centre for Biological Sciences,Tata Institute of Fundamental Research,GKVK Campus, Bellary Road, Bengaluru 560065, India International Centre for Theoretical Sciences,Tata Institute of Fundamental Research, Survey no 151,Shivakote, Hesaraghatta Hobli, Bengaluru 560089, India
A distinguishing feature of active particles is the nature of the non-equilibriumnoise driving their dynamics. Control of these noise properties is, therefore, of bothfundamental and applied interest. We demonstrate emergent tuning of the activenoise of a granular self-propelled particle by confining it to a quasi one-dimensionalchannel. We find that this particle, moving like an active Brownian particle (ABP) intwo-dimensions, displays run-and-tumble (RTP) characteristics in confinement. Weshow that the dynamics of the relative orientation co-ordinate of the particle mapsto that of a Brownian particle in a periodic potential subject to a constant force, inanalogy to the dynamics of a molecular motor. This mapping captures the essen-tial statistical characteristics of the one-dimensional RTP motion. Specifically, ourtheoretical analysis is in agreement with the empirical distributions of the relativeorientation co-ordinate and the run-times (tumble-rates) of the particle. Finally, weexplicitly control these emergent run-and-tumble like noise parameters by externaldriving. Altogether, our work illustrates geometry-induced tuning of the active dy-namics of self-propelled units thus suggesting an independent route to harness theirinternal dynamics. a r X i v : . [ c ond - m a t . s o f t ] J a n I. INTRODUCTION
Active matter is a generic class of nonequilibrium systems wherein the driving and dissi-pation occur at the level of the constituent individual units. Some of the earliest studies ofactive matter focused on the hydrodynamic description of collections of self-propelled unitsto explain coherent phenomenon seen in bird flocks, wildebeest herds, fish schools or insectswarms [1–5]. The orientational degrees of freedom of the critters are an integral part oftheir dynamics in these studies, with the interaction between motility and the orientationaldegrees of freedom leading to complex phases with rich behavior [5]. However, it is not somuch the orientational degrees of freedom nor the anisotropic interactions, but rather thebroken detailed balance in the dynamics of the individual units that is the crucial aspectof an active system [6]. Minimal models of self-propelled particles that incorporate thisessential aspect, in addition to isotropic interactions between individuals, have been of in-terest in recent years [7]. Synthetic systems, such as self-propelled droplets [8, 9], colloidalsystems [10, 11] and granular media [12, 13], provide experimental realisations of such scalaractive matter .Theoretical models of scalar active matter are typically of three broad types: Active Brow-nian particles (ABP), exemplified by the motion of self-phoretic colloids [10, 11] and drivengranular particles [12, 13], move with a constant speed and their motility direction undergoesrotational diffusion. Run-and-tumble-particles (RTP), inspired by the swimming motion ofbacteria [14], also move with a constant speed along a given direction for a certain “run-time”, after which they undergo an instantaneous “tumble” that randomises the directionof motion [15]. Active Ornstein-Uhlenbeck particles (AOUP) are driven by a self-propellednoise which has a finite-correlation time with a Gaussian strength [16]. The essential distin-guishing feature in these three models is the characteristics of the active noise driving theself-propelled motion. The changes in propulsion velocity are governed by rotational diffu-sion for ABPs, by a Poisson process for RTPs, and by an exponentially correlated Gaussiannoise for AOUPs. In spite of these contrasting active noise contributions, collections ofinteracting particles of all three kinds show motility-induced phase-separation (MIPS) [17]and scaling behavior in a single-file geometry [18], pointing to underlying universal features. ∗ Equal contribution † [email protected] ‡ [email protected] In this work, we seek to control the noise characteristics of a scalar active particle andthereby its self-propelled motion. Using an experimental system comprised of a drivengranular particle, we demonstrate that laterally confining an ABP-like particle in a narrowchannel leads to the emergence of an active noise reminiscent of an RTP-like motion, and thatthe characteristics of this noise can be controlled by our empirical parameters. To understandthis dynamics, the effects of the confining channel are modeled using a simple potential forthe orientation of the active particle and we then show that this maps the dynamics of therelative internal coordinate to that of a Brownian particle moving in a periodic potentialsubject to a constant force. Our results are a clear experimental demonstration of tuningthe noise characteristics of an active particle by employing lateral confinement, suggestingnew ways to control active particle dynamics. A FIG. 1. Experimental setup: A circular 3-D printed granular particle of diameter d is drivenusing an electromagnetic shaker (non-dimensional acceleration Γ). The particle displays activeself-propelled motion (blue trajectory) due to frictional asymmetry with the substrate. This two-dimensional motion is well captured by an ABP model describing a chiral particle with position r and orientation ϕ . Quasi one-dimensional confinement of the particle is achieved using concentriccircular channels of varying widths w (confinement δ = d/w ). This one-dimensional motion istracked using the co-ordinate θ and the relative internal co-ordinate ψ . II. EXPERIMENTAL SETUP
Our experimental system is comprised of circular 3-D printed disks (particles) of diameter d , with an asymmetric leg on one side (Fig. 1 and Supplementary Information); when placedon a flat surface (aluminum disk) and driven using an electromagnetic shaker (SupplementaryInformation), the particles propel in-plane along a fixed body axis. Owing to its design, aslight shape anisotropy is evident in the 2-dimensional projected diameter of the particlealong the propulsion direction (Fig. 1). In unconfined two-dimensional space, the particleis an active, motile object executing a self-propelled random walk (Supplementary Movie1) with continuously and noisily varying position r = ( x, y ) and orientation ϕ (Fig. 1,Supplementary Information). In addition to self-propulsion, the particles exhibit chiralmotion characterized by an angular speed ω . Altogether, the motion of the active particle inthe unconfined 2-dimensional space is well characterized by an ABP model [13] for a chiralactive particle (see MSD analysis in the Supplementary Information): the particle positionevolves according to ˙ r = v ˆ e ( ϕ ) + √ D t η ( t ) where ˆ e ( ϕ ) = cos ϕ ˆ x + sin ϕ ˆ y is the localdirection of motion, the over-dot indicates a time-derivative, v is an active speed, D t isthe translational diffusion coefficient, and the stochastic term η ( t ) is a Gaussian white-noiseprocess with zero-mean and unit-variance. The angular coordinate ϕ of the ABP performsa random walk governed by ˙ ϕ = ω + √ D r ζ ( t ) with ω the (chiral) angular velocity, D r the rotational diffusion coefficient of the ABP while ζ ( t ) is a zero-mean and unit-varianceGaussian white-noise process. In our experimental set-up, we are able to control thesevarious parameters of the ABP model, viz. ω , D r , v and D t , using the driving Γ of theelectro-magnetic shaker (Supplementary Information). III. EMERGENT STOCHASTIC SWITCHING DYNAMICS INCONFINEMENT
We now turn our attention to the behaviour of these active particles when they areconfined to quasi one-dimensional channels – grooves of width w that run along the peripheryof the aluminum disk as concentric circles (Fig. 1). The geometric configuration of thechannels is such that the particles are constrained radially to within ≤
1% of the channelradius (Supplementary Information) and can still explore all possible values of ϕ . When theconfinement δ = d/w ≈ .
88 is sufficiently small ( i.e. , a wide channel) the orientation co-ordinate ψ samples all directions uniformly ( P ( ψ ) for various confinements δ are not shown)and since this co-ordinate is coupled to the position co-ordinate θ , the particle executes a one-dimensional persistent random walk along the channel (Fig. 2 A , left panel; Supplementary AB
550 600 650 700 750 80010121416 - /2- /40/4
10 15 15 - /40 /2 /2/2- FIG. 2. Emergent run-and-tumble dynamics in quasi one-dimensional confinement. (A) Parametricplot of the particle trajectories in the translational co-ordinate θ and the relative orientation co-ordinate ψ for varying confinements δ , with vertical grid lines marking multiples of π/
2. Discretejumps in ψ are seen at sufficiently high δ . A constant drift of ψ , due to the particle chirality, tothe right (“downhill” direction) is evident. (B) The particle trajectory in θ is reminiscent of thatof a run-and-tumble particle motion with a run-duration τ . Red and blue arrows on the trajectorymark particle flips in the “uphill” and “downhill” directions respectively. Movie 2). The chirality of particle is evident in the steady drift of the orientation ψ .A striking difference is manifest when δ is increased to ≈ .
9: instead of evolving contin-uously, the orientation of the particle fluctuates noisily along one of the channel directions i.e. ψ = ± π/
2, only to stochastically and abruptly switch direction (Fig. 2 A middle panel).The discrete orientation jumps | ∆ ψ | ∼ π occur either “downhill” in a direction dictated bythe chirality of the particle or less frequently “uphill” (blue and red arrows, respectively inFig. 2 B ). As the confinement δ is further increased to ≈ .
92, the switches in the particleorientation become much less frequent – in the particular run shown (Fig. 2 A , right panel),they occur only “downhill”. When ψ ∼ ± π/
2, the particle executes a “run” along thechannel for a typical “run-time” duration τ until it stochastically switches to a “run” in theopposite direction (Fig. 2 B ; Supplementary Movie 3).Taken together, the characteristics discussed above are reminiscent of a run-and-tumbleparticle in one-dimension, with position co-ordinate θ . As such, the relevant equation de-scribing the dynamics of θ is that of an ABP confined to move on a circle of radius R (Supplementary Information), viz. ˙ θ = ν sin ψ + √ D ξ ( t ) ≡ ν σ ( t ) + √ D ξ ( t ) (1)where ν = v /R , D = D t /R , ξ ( t ) is a Gaussian white-noise process with zero-mean and unit-variance, and σ ( t ) = sin[ ψ ( t )] is the active noise. This model is validited by first computingthe empirical two point correlation C ( | t − t (cid:48) | ) = (cid:104) σ ( t ) σ ( t (cid:48) ) (cid:105) of the active noise, assuming σ ( t ) is a stationary stochastic process. The correlation function decays exponentially witha time-constant τ θ ∼ s (Fig. 3 A ). For ψ ∼ ± π/
2, the active noise σ ( t ) ∼ ±
1, and thus ν would correspond to the active speed of the RTP-like motion for θ . The bounded values σ ( t ) ∈ [ − ,
1] and the exponential decay of C ( | t − t (cid:48) | ) thus justify the effective dynamics of θ in equation (1) as the position coordinate of an RTP-like particle.We next compare the empirically measured mean-squared-displacement (MSD) (cid:104) [∆ θ ( t )] (cid:105) with that predicted from equation (1), using the values of v and D t inferred from the two-dimensional experiments for the ABP model: (cid:104) [∆ θ ( t )] (cid:105) = ν (cid:90) t dz (cid:90) t dz (cid:48) C ( | z − z (cid:48) | ) + 2 D t. (2)An excellent agreement is found between the empirical MSD and that given by equation(2) (Fig. 3 B ). Of note in the MSD are the crossovers from an initial diffusive regime(governed by passive translational diffusion), to a super-diffusive regime (governed by theself-propelled motion of the particle), and then, eventually, to a diffusive regime for time-scales t (cid:29) τ θ ( ∼ s ) corresponding to the decorrelation of the persistent motion driven bythe active noise σ ( t ). The effective diffusion coefficient in this asymptotic regime has anactive contribution ∼ ν τ θ in addition to the (angular) translational diffusion coefficient D [19, 20]. This additional contribution to the effective diffusivity could dramatically increasewith increasing confinement as the particle becomes more RTP-like. Indeed, the effectivediffusion coefficient does increase with confinement, and in fact for the highest confinements,a crossover to the eventual diffusive regime is not seen on the timescale of our experiments(Fig. 3 B , inset). These results conclusively suggest that equation (1) is indeed a gooddescriptor for the dynamics of θ , and also that the parameters describing the motion inconfinement remain reasonably similar to the particle motion in two-dimensions (more detailsin the Supplementary Information). BA C -1 -4 -2 -10 -5 0 5 100.30.40.50.60.70.8 -6 -4 -2 FIG. 3. Characterisitics of the run-and-tumble dynamics. (A) Two-point correction function C ( | t − t (cid:48) | ) = (cid:104) σ ( t ) σ ( t (cid:48) ) (cid:105) where σ ( t ) = sin[ ψ ( t )] is the active noise. (B) The empirical mean-squared-displacement (cid:104) [∆ θ ( t )] (cid:105) follows the RTP model in equation (1). MSDs for varying confinementsare shown in the inset. (C) The run-time distribution P ( τ ) of θ , equivalently the dwell-timedistribution of ψ , exhibits two-time scales fit by the convolution of the statistics of two Poissonprocesses (equation (3)). The empirically measured dwell time distribution of ψ (equivalently, the run time distri-bution of θ ) distribution of ψ is well fit by a bi-exponential function P ( τ ) = k f k b k f − k b (cid:0) e − k b τ − e − k f τ (cid:1) . (3)where k f / b are the forward (“downhill”) and backward (“uphill”) rates of the switchingbetween the states ψ ∼ ± π/ C ). It is clear from the above expression that asymp-totically, i.e., for k f τ (cid:29) k f (cid:29) k b , the dwell-time distribution P ( τ ) ∼ k b e − k b τ , decayingexponentially with the rate k b . For the same driving strength Γ, the time-scales that governthe decay of the two-point correlation C ( | t − t (cid:48) | ) and the crossover of the MSD (cid:104) [∆ θ ( t )] (cid:105) from the super-diffusive to the diffusive regime are comparable to the backward (“uphill”)switching rate i.e. /k b ∼ τ θ .Several remarks are in order. First, it should be noted that unlike the standard RTPmodel with an exponentially distributed run time between tumbles, our active particle dy-namics demonstrates a non-monotonic distribution P ( τ ) of run-times τ . Second, the discretejumps in the trajectory of ψ (Fig. 2 B ) are reminiscent of the positions of a molecular motorstepping along a polymeric track [21, 22]. Third, the fit of equation (3) (Fig. 3 C ) suggeststhe presence of (at least) two characteristic switching rates in the RTP-like dynamics of θ driven by the active noise σ ( t ). Incidentally, equation (3) is used to model the dwell timedistributions of a two-state molecular motor [22]. All the above points hint at a possible sim-ilarity between the dynamics of our active particle in confinement and those of a processivemolecular motor. We next show that this is indeed the case. IV. ANALOGY TO THE STEPPING DYNAMICS OF A MOLECULAR MOTOR
We conjecture that the effects of the confining channel on the particle can be captured bya periodic force of the form F wall = λ sin 2 ψ in the equation for the orientation co-ordinate,with λ being the strength of the wall-particle interaction. In doing this, we simplify theinteractions which are quite complicated indeed – they not only depend on the particle andchannel surface roughness and the resultant friction, but possibly also on the amplitude ofdriving Γ. However, the step-like trajectories, and the preponderance of the values of ψ closeto ± π/
2, motivate us to test this functional form of F wall to describe the particle motion.Following this intuition, the dynamics of ψ is modelled by the following equation (details inSupplementary Information):˙ ψ = ω + λ sin 2 ψ − ν sin ψ + √ D ξ ( t )= − V (cid:48) ( ψ ) + √ D ξ ( t ) (4)where D = D t /R + D r with R the (mean) radius of the confining channel, ξ is a zero-mean unit-variance Gaussian white noise process, and the prime denotes differentiationwith respect the argument of its function. The effective potential V ( ψ ) = − ω ψ + λ ψ − ν cos ψ (5)thus governs the dynamics of ψ . Notice that U ( ψ ) = V ( ψ ) + ω ψ is a periodic functionof ψ . Thus, the Langevin equation (4) describes the stochastic dynamics of a Brownianparticle with a position coordinate ψ moving in a periodic potential U , subject to a con-stant “force” ω (Fig. 4 A ). It is straightforward to solve the Fokker-Planck equation for P ( ψ ) in the steady-state, corresponding to this motion (Supplementary Information). Thistheoretical P ( ψ ) agrees well with the empirical distribution, relying on a single fitting pa-rameter λ (Fig. 4 B , Supplementary Information). However, λ depends on the driving Γ(Supplementary Information) which is altogether not surprising given that it encapsulatesthe effective strength of the complicated interactions between the active particle and theconfining channel. A C - - /2 0 /2 B ' ' -1 simulationsexperiments FIG. 4. Dynamics and control of the stochastic switching. ( A ) Effective potential for the internalco-ordinate ψ and the transition rates between the neighbouring minima of the potential. In ouranalysis, k (cid:48) f / b ≈ k f / b . ( B ) The empirical steady-state distribution P ( ψ ) (data points) comparedwith the analytical solution obtained from the Fokker-Planck equation for the stochastic dynamicsof ψ (red line). (C) Control of the forward and backward switching rates, k f and k b by modulatingthe driving Γ. Dashed lines are a guide to the eye. Simulations are performed with parametersextracted using a heuristic fitting procedure (SI Text). The inset shows the ratio k f /k b and itscorrespondence with the prediction from Kramers’ rate theory analysis (SI Text). Having concluded that the effective potential V ( ψ ) is sufficient to capture the steady-state distribution of the internal coordinate ψ , we next show that it can also capture thedynamical properties of ψ . The two point correlation function C ( | t − t (cid:48) | ) computed fromsimulations of the Langevin equation (4) compares well with empirical measurements (Sup-plementary Information). Further, the switching rates, k f and k b , as inferred from the fitsof equation (3) to the run-time distributions obtained from both the experimental and sim-ulation data compare well. These rates and other characteristics of the RTP-like motioncan be controlled by the driving amplitude Γ (Fig. 4C, Supplementary Information). The0transition rates governing the jumps of ψ across successive minima of the potential can beexplicitly computed in the Kramers’ approximation [23]. When ν (cid:54) = 0 there are in-principlefour transition rates, k f / b and k (cid:48) f / b , across the potential minima (Fig. 4 A ); however, in theapproximation ν ≈ k f / b remain. The variation of the ratio k f /k b from the empirical (experimen-tal and simulation) data compares well with the analytical results obtained in the Kramers’approximation (Fig. 4 C inset). Further, in analogy with two-state dynamics of molecularmotors, the chirality of our active particle makes the forward and reverse transition ratesdifferent, concomitant with the bi-exponential distribution of P ( τ ) (Supplementary Infor-mation). Taken together, the effective potential used to describe the confining effects of thechannel is a good predictor for the dynamical properties of ψ . V. DISCUSSION AND CONCLUSIONS
By confining a self-propelled granular particle to a quasi one-dimensional channel, wehave demonstrated emergent active noise properties, qualitatively distinct from those for thetwo-dimensional unconfined motion of the particle. While the two-dimensional dynamics iswell described by an ABP model, RTP-like behavior emerges due to lateral confinement.This dynamics of the translational coordinate θ is driven by the internal coordinate ψ ofthe particle, which in turn has similarities to the stepping dynamics of a molecular motor;most notably, this manifests in the non-monotonic nature of the dwell-time distribution of ψ which is identical to that found in two-state processive motors. To complete the pictureand capture the dynamics of ψ , we reduced the effects of confinement to a simple periodicforce, thus mapping it to the dynamics of a Brownian particle moving in a periodic potentialsubject to a constant external force. Rather surprisingly, the introduction of this periodicforce is sufficient to capture all features of the active particle dynamics.We emphasize that there is no fundamental reason to expect that the approach taken hereto describe the active particle dynamics in confinement – particularly, one that works formicroscopic systems, in or close to equilibrium [24, 25] – should have successfully describedthe emergent noisy dynamics of our active particle. Our experimental system is athermal,far from equilibrium, and the various interactions between the granular surfaces are com-plicated, thus possibly requiring a priori , a dynamical description more complex than that1given by equations (4) and (5) for the emergent statistical features of the particle motion.Therefore, the agreement we find between empirical measurements and theoretical analysis,in particular the Kramers’ transition rates is remarkable.We conclude with a couple of comments. First, the fundamental character of an activeparticle is the inseperable coupling between its internal (chemical) co-ordinates and its po-sitional degrees of freedom, while being constantly driven out of equilibrium by an energythroughput [6]. The resulting mobility in the parametric space of the internal and positionalco-ordinates is non-diagonal implying that a net current along one of these co-ordinates nec-essarily drives a corresponding current along the other co-ordinate – our experimental tra-jectories (Fig. 2) are an explicit manisfestation of this picture. Second, the RTP-like discretedynamics that we describe here is not directly imposed, for example, using explicit externalmeans [26–28] but is rather an emergent behavior resulting from confinement. This opensup an avenue to exploit the effects of geometric constraints and non-equilibrium driving tocontrol the noise characteristics of active particles. VI. ACKNOWLEDGEMENTS
We acknowledge support from the Department of Atomic Energy, Government of India,under project no. 12-R&D-TFR-5.04-0800 and 12-R&D-TFR-5.10-1100, the Simons Foun-dation (Grant No. 287975 to S.T.) and the Max Planck Society through a Max-Planck-Partner-Group at NCBS-TIFR (S.T.), the Department of Biotechnology, India, through aRamalingaswami re-entry fellowship (K.V.K.) and by the Max Planck Society through aMax-Planck-Partner-Group at ICTS-TIFR (K.V.K.). We thank Pawan Nandakishore forhelp with the initial set-up and experiments. S.T. and K.V.K acknowledge discussions dur-ing the KITP 2020 online program on Symmetry, Thermodynamics and Topology in ActiveMatter. [1] J. Toner and Y. Tu, Long-range order in a two-dimensional dynamical XY model: How birdsfly together, Phys. Rev. Lett. , 4326 (1995).[2] J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev.E , 4828 (1998). [3] J. Toner, Y. Tu, and S. Ramaswamy, Hydrodynamics and phases of flocks, Annals of Physics , 170 (2005).[4] S. Ramaswamy, The Mechanics and Statistics of Active Matter, Annual Review of CondensedMatter Physics , 323 (2010).[5] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A.Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. , 1143 (2013).[6] S. Ramaswamy, Active matter, Journal of Statistical Mechanics: Theory and Experiment , 054002 (2017).[7] C. Bechinger, R. Di Leonardo, H. L¨owen, C. Reichhardt, G. Volpe, and G. Volpe, Activeparticles in complex and crowded environments, Rev. Mod. Phys. , 045006 (2016).[8] S. Thutupalli, R. Seemann, and S. Herminghaus, Swarming behavior of simple model squirm-ers, New Journal of Physics , 073021 (2011).[9] Z. Izri, M. N. van der Linden, S. Michelin, and O. Dauchot, Self-propulsion of pure waterdroplets by spontaneous marangoni-stress-driven motion, Phys. Rev. Lett. , 248302 (2014).[10] J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian,Self-motile colloidal particles: From directed propulsion to random walk, Phys. Rev. Lett. ,048102 (2007).[11] J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and P. M. Chaikin, Living crystals oflight-activated colloidal surfers, Science , 936 (2013).[12] N. Kumar, S. Ramaswamy, and A. K. Sood, Symmetry properties of the large-deviation func-tion of the velocity of a self-propelled polar particle, Phys. Rev. Lett. , 118001 (2011).[13] L. Walsh, C. G. Wagner, S. Schlossberg, C. Olson, A. Baskaran, and N. Menon, Noise anddiffusion of a vibrated self-propelled granular particle, Soft Matter , 8964 (2017).[14] H. C. Berg and D. A. Brown, Chemotaxis in Escherichia coli analysed by Three-dimensionalTracking, Nature , 500 (1972).[15] M. E. Cates and J. Tailleur, When are active brownian particles and run-and-tumble particlesequivalent? consequences for motility-induced phase separation, EPL (Europhysics Letters) , 20010 (2013).[16] ´E. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, and F. van Wijland, HowFar from Equilibrium Is Active Matter?, Physical Review Letters , 10.1103/Phys-RevLett.117.038103 (2016). [17] M. E. Cates and J. Tailleur, Motility-induced phase separation, Annual Review of CondensedMatter Physics , 219 (2015).[18] P. Dolai, A. Das, A. Kundu, C. Dasgupta, A. Dhar, and K. V. Kumar, Universal scaling inactive single-file dynamics, Soft Matter , 7077 (2020).[19] S. Ebbens, R. A. L. Jones, A. J. Ryan, R. Golestanian, and J. R. Howse, Self-assembledautonomous runners and tumblers, Phys. Rev. E , 015304 (2010).[20] K. Malakar, V. Jemseena, A. Kundu, K. V. Kumar, S. Sabhapandit, S. N. Majumdar, SRedner, and A. Dhar, Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension, Journal of Statistical Mechanics: Theory and Experiment , 043215 (2018).[21] F. J¨ulicher, A. Ajdari, and J. Prost, Modeling molecular motors, Reviews of Modern Physics , 1269 (1997).[22] T. J. Purcell, H. L. Sweeney, and J. A. Spudich, A force-dependent state controls the coordi-nation of processive myosin v, Proceedings of the National Academy of Sciences , 13873(2005).[23] P. H¨anggi, P. Talkner, and M. Borkovec, Reaction-rate theory: fifty years after kramers,Reviews of Modern Physics , 251 (1990).[24] S. Ciliberto, Experiments in stochastic thermodynamics: Short history and perspectives, Phys.Rev. X , 021051 (2017).[25] L. I. McCann, M. Dykman, and B. Golding, Thermally activated transitions in a bistablethree-dimensional optical trap, Nature , 785 (1999).[26] H. Karani, G. E. Pradillo, and P. M. Vlahovska, Tuning the random walk of active colloids:From individual run-and-tumble to dynamic clustering, Phys. Rev. Lett. , 208002 (2019).[27] M. A. Fernandez-Rodriguez, F. Grillo, L. Alvarez, M. Rathlef, I. Buttinoni, G. Volpe, andL. Isa, Feedback-controlled active brownian colloids with space-dependent rotational dynam-ics, Nature Communications , 4223 (2020).[28] G. Vizsnyiczai, G. Frangipane, C. Maggi, F. Saglimbeni, S. Bianchi, and R. Di Leonardo, Lightcontrolled 3D micromotors powered by bacteria, Nature Communications , 15974 (2017).[29] D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, emcee: The MCMC hammer,Publications of the Astronomical Society of the Pacific , 306 (2013). [30] H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications , 2nd ed.(Springer-Verlag Berlin Heidelberg, 1996).[31] R. D. T. H. Tanuj Aggarwal, Donatello Materassi and M. Salapaka, Detection of steps insingle molecule data, Cell Mol. Bioeng. , 14 (2012). SUPPLEMENTARY INFORMATIONI. EXPERIMENTAL DETAILSA. Particle, Shaker and Imaging Setup
Particles
Particles used in the experiment were designed in-house and fabricated using a 3D printer(Form2 SLA 3D printer (minimum resolution of 0.025 mm) from FormLabs which usesFormLab proprietary clear resin for printing). Particles as shown in FIG. 5 A have a shapeasymmetry because of a protruding “leg” under the front side of the particle. Particles are4.5 mm in diameter ( d ) and 2.5 mm in height ( h ) which includes the leg of height 0.5 mm(FIG. 5 A ). The front half of the particle is marked using yellow paint for identification ofthe particle orientation. Ø4.5mm2mm2.5mm R1.95mm0.25mm1.35mm A Ø125mm5mmØ88mm4.9mm5.1mm5.3mm
C B
FIG. 5. A shows the design of our granular self-propelled particle. The overall experimental setupincluding the electromagnetic shaker and the camera is shown in B . The design of the shaker-headis shown in C . Shaker setup
A Br¨uel & Kjær electromagnetic shaker (LDS V406) is used to excite a patterned alu-minum disk on which the particles move. The shaker is suspended on a trunnion mount,which is connected to two sheets of stainless steel plates (FIG. 5 B). The two stainlesssteel plates are separated by a rubber pad (hardness 65, shore A) for isolation of mechan-ical vibration. Each stainless steel plate weighs 30 kg and the bottom plate is grouted tothe ground below. The steel plates along with the mount allow for leveling of the system.The aluminum disk measures 25 cm in diameter and 19 mm in thickness. Circular chan-nels of varying widths of 5.3 mm, 5.1 mm and to 4.9 mm are precision milled into thedisk (FIG. 5 C ). The channels are 5 mm in depth with a mean circular radius of 115.7 mm,105.5 mm and 95.5 mm respectively (from the outer perimeter of the disk inwards, FIG. 5 C),leaving a two-dimensional arena of radius 88 mm and depth of 5 mm in the center of thedisk. A separate head with a single channel of width 5 mm and mean circular radius 116.5mm was used for some of the experiments reported here. The aluminum disk is polishedfollowed by soft anodization to provide a uniform black background for imaging purposes.The disk is electrically grounded to discharge any static charges present in the system andmounted on the shaker. Imaging
The particle dynamics is recorded using a Nikon DSLR camera aligned above the centerof the aluminum disk. The setup is illuminated from above using an array of LED strips(FIG. 5 B) at the top. The videos are recorded at 30 fps for a duration of 10 – 20 mins. Anopensource program, Digicam Controller, is used to capture videos which are saved directlyto a computer.
B. Calibration, Data recording and Image processing
Leveling of the setup
Granular experiments are sensitive to tilt in any direction and the horizontal level of thealuminum disk needs to be maintained precisely. We use a trunnion mount along one of the7axis and screws placed at the corners of the steel plates at the bottom of the electromagneticshaker to further fine-tune the leveling. To measure the accuracy of the leveling, fine semolinaparticles are placed in the two-dimensional arena (FIG. 6) and excited by the shaker, causingthem to disperse in the two-dimensional plane. From a 10 mins video of the dynamics, theimage intensity due to the semolina particles is computed in four quadrants of the plate andthe pixel intensity difference is used as a measure of the uniformity of the distribution ofthe particles in the two-dimensional plane which is indicative of the horizontal leveling ofthe system.
FIG. 6. To ensure that the head of the shaker is leveled with respect to gravity, we monitor theintensity profiles of fine granular particles (semolina) in four regions of the domain. The intensityversus time plots show that the head is leveled to a good extent.
Measuring relative acceleration Γ To calculate the relative acceleration of the aluminum disk resulting from the actuation,we measure the vertical displacement of the disk using a high speed video recorded at 300fps (FastCam Mini AX-200, Photron). The high speed camera is positioned perpendicularto the direction of displacements of aluminum disk parallel to the ground and points aremarked along the sides of the aluminum disk. The displacement of the points is trackedusing image processing. The relative acceleration Γ is then calculated using Γ = A Ω /g ,where A is the maximum vertical displacement of the disk, Ω is the shaking frequency and g = 9 . m/s is the acceleration due to gravity.8 Image Processing
Videos obtained from the experiments are processed using MATLAB. The video is slicedinto frames and a mask is applied to specify the region of interest either to the 1D channel orthe 2D arena. Then the image is passed through the imfindcircles function to detect thecenter of the particle (FIG. 7 A ). Once the center of particle is detected, tracking is achievedusing a least displacement method using custom written MATLAB code. The position ofthe particle represented by the polar co-ordinate θ is calculated using the center of diskand the center of the particle and is measured with respect to x − axis of the image. Todetect the orientation ϕ of the particle, the image is passed through a yellow channel filterto identify the front of the particle with respect to its center (FIG. 7 B ). The image is thenbinarized and the regionprops function is used to find the orientation of the semicircularregion (yellow mark present at top of the particle, FIG. 7 C ). Output of the orientation from regionprops is within [ − π/ , π/
2] which are then changed to span [ − π, π ]. To calculateerror in detection, particle positions are recorded without being actuated and their centersand orientations are processed as previously described. From this method, we estimate anerror in the detection of θ to ± . ϕ to ± . A B C
FIG. 7. The image analysis procedures allow us to detect both the 2D position and the orientationof our active granular particle. II. ACTIVE BROWNIAN MOTION IN 2D
We start with the equations for the position r and the orientation ϕ of an active Brownianparticle (ABP) in two-dimensions˙ r = v ˆ e ( ϕ ) + (cid:112) D t η ( t ) , (6)˙ ϕ = ω + (cid:112) D r ζ ( t ) , (7)where the overdot indicates a time-derivative, v is the active speed, ˆ e ( ϕ ) = cos ϕ ˆ x + sin ϕ ˆ y is the instantaneous direction of the active force, D t the translational diffusion coefficient, ω the (chiral) angular velocity, and D r the rotational diffusion coefficient. The stochastic terms η ( t ) and ζ ( t ) are uncorrelated Gaussian white noises with zero-mean and unit-variance.From (7), it is straightforward to compute the angular mean-squared-displacement (cid:104) [∆ ϕ ( t )] (cid:105) = ω t + 2 D r t, (8)where (cid:104)· · · (cid:105) indicates an average over the realizations of η ( t ) and ζ ( t ). To compute themean-squared displacement (cid:104) [∆ r ( t )] (cid:105) of the position r , we proceed as follows. The formalsolution of (6) is ∆ r ( t ) ≡ r ( t ) − r = v (cid:90) t ds ˆ e [ ϕ ( s )] + (cid:112) D t (cid:90) t ds η ( s ) (9)which leads to (cid:104) [∆ r ( t )] (cid:105) = v (cid:90) t ds (cid:90) t ds (cid:48) (cid:104) ˆ e [ ϕ ( s )] · ˆ e [ ϕ ( s (cid:48) )] (cid:105) + 2 D t (cid:90) t ds (cid:90) t ds (cid:48) (cid:104) η ( s ) · η ( s (cid:48) ) (cid:105) = v (cid:90) t ds (cid:90) t ds (cid:48) (cid:104) ˆ e [ ϕ ( s )] · ˆ e [ ϕ ( s (cid:48) )] (cid:105) + 4 D t t, = 2 v D (cid:2) D t cos α − cos 2 α + cos( ωt + 2 α ) e − D r t (cid:3) + 4 D t t, (10)where D = (cid:112) ω + D r and tan α = ω/D r . The intermediate steps for evaluating the double-integral C ϕϕ ( t ) = (cid:90) t ds (cid:90) t ds (cid:48) (cid:104) ˆ e [ ϕ ( s )] · ˆ e [ ϕ ( s (cid:48) )] (cid:105) (11)are detailed in the next subsection.0 A. Evaluating C ϕϕ ( t ) We simplify C ϕϕ ( t ) = (cid:90) t ds (cid:90) t ds (cid:48) (cid:104) cos[ ϕ ( s ) − ϕ ( s (cid:48) )] (cid:105) (12)To proceed further, we use the formal solution ϕ ( t ) = ωt + (cid:112) D r (cid:90) t ds ζ ( s ) (13)of equation (7) and write C ϕϕ ( t ) = (cid:90) t ds (cid:90) t ds (cid:48) (cid:104) cos [ ω ( s − s (cid:48) ) + F ( s, s (cid:48) )] (cid:105) = 12 (cid:90) t ds (cid:90) t ds (cid:48) (cid:104) e iω ( s − s (cid:48) ) (cid:104) e iF ( s,s (cid:48) ) (cid:105) + e − iω ( s − s (cid:48) ) (cid:104) e − iF ( s,s (cid:48) ) (cid:105) (cid:105) (14)where F ( s, s (cid:48) ) = (cid:112) D r (cid:90) ss (cid:48) dz ζ ( z ) . (15)We thus need to evaluate (cid:104) exp [ ± iF ( s, s (cid:48) )] (cid:105) . To do so, we note that ζ ( t ) is a zero-meanunit-variance Gaussian white-noise process. This means that (cid:104) [ F ( s, s (cid:48) )] n (cid:105) vanishes for allodd n . And for even n = 2 m , we write (cid:104) [ F ( s, s (cid:48) )] m (cid:105) = (2 D r ) m (cid:90) ss (cid:48) dz (cid:90) ss (cid:48) dz . . . (cid:90) ss (cid:48) dz m (cid:104) ζ ( z ) ζ ( z ) . . . ζ ( z m ) (cid:105) . (16)Furthermore, using the properties of Gaussian stochastic variables to (cid:104) ζ ( z ) ζ ( z ) . . . ζ ( z m ) (cid:105) can be decomposed into N = (2 m − m − . . . · z i into all possible pairs. For instance, for m = 2, we have (cid:104) ζ ( z ) ζ ( z ) ζ ( z ) ζ ( z ) (cid:105) = δ ( z − z ) δ ( z − z ) + δ ( z − z ) δ ( z − z ) + δ ( z − z ) δ ( z − z )and (cid:104) [ F ( s, s (cid:48) )] (cid:105) = (2 D r ) (cid:90) ss (cid:48) dz (cid:90) ss (cid:48) dz (cid:90) ss (cid:48) dz (cid:90) ss (cid:48) dz (cid:20) δ ( z − z ) δ ( z − z )+ δ ( z − z ) δ ( z − z ) + δ ( z − z ) δ ( z − z ) (cid:21) = 3 (2 D r ) ( s − s (cid:48) ) (17)1In general, we get (cid:104) [ F ( s, s (cid:48) )] m (cid:105) = (2 D r ) m (2 m − m − . . . · s − s (cid:48) ) m (18)and thus (cid:104) exp [ ± iF ( s, s (cid:48) )] (cid:105) = ∞ (cid:88) m =0 ( ± i ) m (2 m )! (cid:104) [ F ( s, s (cid:48) )] m (cid:105) = ∞ (cid:88) m =0 ( − m (2 m )! (2 D r ) m (2 m − m − . . . · s − s (cid:48) ) m = ∞ (cid:88) m =0 ( − D r ) m m ! ( s − s (cid:48) ) m = e − D r ( s − s (cid:48) ) . (19)Now, we use the above result in equation (14) to get C ϕϕ ( t ) = 12 (cid:90) t ds (cid:90) t ds (cid:48) (cid:104) e ( iω − D r )( s − s (cid:48) ) + e − ( iω + D r )( s − s (cid:48) ) (cid:105) = 2 t D cos α − D cos 2 α + 2 e − D r t D cos( ωt + 2 α ) (20)where D = (cid:112) ω + D r and tan α = ω/D r which finally leads to (10). B. Bayesian parameter inference in 2D
We now compute the mean-squared displacements of the position (cid:104) [∆ r ( t )] (cid:105) expt and ori-entation (cid:104) [∆ ϕ ( t )] (cid:105) expt from experimental data obtained as described in section I B by con-sidering trajectories that were at least 150 s long as shown in Fig. 8. This gave us 25 to 30statistically independent segments per amplitude Γ.Next, we used equations (8) and (10) to infer the parameters of the ABP model thatcorrespond to our self-propelled granular particle. To this end, we performed a Markov-Chain-Monte-Carlo (MCMC) Bayesian inference procedure in the four-dimensional param-eter space of ω , D r , v and D t . Specifically, we choose flat priors for all the parameterswithin suitable limits and assumed a Gaussian likelihood function with errors estimatedfrom the independent trajectories mentioned above. We used the Python package emcee [29] to implement the MCMC sampling. The posterior distributions obtained in this processare shown in Fig. 9A while the projection onto the data space of (cid:104) [∆ r ( t )] (cid:105) and (cid:104) [∆ ϕ ( t )] (cid:105) isshown in Fig. 9B. The agreement of the MSD obtained from the parameters inferred using2 FIG. 8. We considered particle trajectories in 2D which were at least 150 s and removed thesegments which are either shorter than this time or were close to the wall. The gray curvesindicates the entire trajectory of a particle while the overlaid color trajectories denote independentsegments used in our analysis. the Bayesian analysis with experimental data clearly shows that the ABP model capturesthe dynamics of our self-propelled granular particle quite adequately.In Fig. 9C, we plot the dependence of these parameters on the driving amplitude Γ. Asmentioned earlier, our granular particles have a net chirality , i.e., ω (cid:54) = 0. From Fig. 9C,we observe that ω is nearly constant with the amplitude Γ. However, the positive andnegatively chiral particles in our setup have different values of chirality. The rotationaldiffusion coefficient D r , the active speed v and the translational diffusion coefficient D t increase with Γ.We have thus mapped the stochastic dynamics of our granular particle moving freely intwo-dimensions to an ABP model and have also inferred the model parameters and theirdependence on the driving amplitude Γ. We next turn to characterizing the dynamics ofour granular active particle when it is confined to move in a narrow quasi one-dimensionalchannel. III. ACTIVE BROWNIAN MOTION IN A CONFINED CHANNEL
To develop a description for the quasi 1D motion in confinement, we first measured theradial motion of the particle in the channel. From FIG. 10, we see that the motion in theradial direction is constrained to within <
1% of the channel radius for all the channel3
AC B
FIG. 9. A Posterior distributions of the parameters ω , D r , v and D t inferred using a Bayesiananalysis of the 2D MSD. The lines indicate confidence intervals of the parameters. In B , we comparethe empirical MSD with the theoretical result. The variation of the ABP model parameters withthe driving amplitude is shown in C . widths that we have considered. As such, we can neglect the motion in the radial directioncompared to that along the channel.We transform equations (6) and (7) above to plane-polar coordinates ( r, θ ) and approx-imate the motion of our ABP in the confining channel of mean-radius R by assuming thatthe radial coordinate is a constant at r = R . With this approximation, we get an equation4 -0.01 -0.005 0 0.005 0.0100.020.040.060.080.1 FIG. 10. Probability distribution of particle displacement in the radial direction for differentconfinements. We note that the radial motion of the particle is constrained to within <
1% of thechannel radius. for θ as follows R ˙ θ = v sin( ϕ − θ ) + (cid:112) D t [ − sin θ η x ( t ) + cos θ η y ( t )] . (21)Since η ( t ) is an uncorrelated Gaussian white noise, we rewrite˙ θ = ν sin ψ + √ D ξ ( t ) = ν σ ( t ) + √ D ξ ( t ) , (22)where ν = v /R , D = D t /R , the relative orientation coordinate ψ ≡ ϕ − θ , ξ ( t ) is a Gaussianwhite noise of zero-mean and unit-variance, and σ ( t ) = sin[ ψ ( t )] is the active noise.As shown in the main text, laterally confining our ABP in a quasi 1D channel leadsto the emergence of a bi-modal peak in the probability distribution of P ( ψ ) of the relativeorientation coordinate. As argued in the main text, a simple way to capture the effect of thislateral confinement is to introduce an additional force λ sin 2 ψ for the angular coordinate ϕ .This confinement-induced force has a strength λ and a periodicity of π . Thus the equationfor ϕ changes to ˙ ϕ = ω + λ sin 2 ψ + (cid:112) D r ζ ( t ) (23)Using (22) and (23), we get the following Langevin equation for ψ :˙ ψ = − V (cid:48) ( ψ ) + √ D ξ ( t ) (24)5where D = D t /R + D r , ξ is a Gaussian white noise, and the effective potential is V ( ψ ) = − ω ψ + λ ψ − ν cos ψ. (25) A. Brownian motion in a periodic potential
It is clear from equations (24) and (25) that the dynamics of ψ maps onto the dynamicsof a Brownian particle with position coordinate ψ moving in a one-dimensional periodicpotential U ( ψ ) = λ/ ψ − ν cos ψ with a constant “force” ω applied to it [21]. Note,however, that V ( ψ ) is not a periodic function of ψ for ω (cid:54) = 0. However, the probabilitydensity P ( ψ, t ) should be a periodic function of ψ with a period L = 2 π . The Fokker-Planckequation governing the evolution of P ( ψ, t ) is ∂ t P = − ∂ ψ J, (26)where the current J = − [ ∂ ψ V − D∂ ψ ] P . (27)Following [30], the stationary solution to equation (26) is P ( ψ ) = N e − V ( ψ ) /D (cid:20) − I ( ψ ) I ( L ) (cid:0) − e − ωL/D (cid:1)(cid:21) , (28)with a constant current J = D NI ( L ) (cid:0) − e − ωL/D (cid:1) , (29)and 1 N = (cid:90) L dβ e − V ( β ) /D (cid:20) − I ( β ) I ( π ) (cid:0) − e − ωL/D (cid:1)(cid:21) , I ( z ) = (cid:90) z − L dx e V ( x ) /D . (30) B. Comparing simulations and experiments
We performed explicit Langevin simulations of equations (22) and (24) with ∆ t = 10 − s and averaged over N = 10 realizations to get statistical measures of the dynamics. We thenasked for the optimal value of λ (the potential strength parameter) that would best capturethe dynamics of (cid:104) [∆ θ ( t )] (cid:105) , the steady-state probability distribution P ( ψ )and the correlation6 FIG. 11. Comparing experimental data (filled-circles) for the mean-squared-displacement (cid:104) [∆ θ ( t )] (cid:105) , the steady-state probability distribution P ( ψ ) and the correlation function C ( | t − t (cid:48) | ) = (cid:104) σ ( t ) σ ( t (cid:48) ) (cid:105) with Langevin simulation results (lines) of (22) and (24) at various driving amplitudes. function C ( | t − t (cid:48) | ) = (cid:104) σ ( t ) σ ( t (cid:48) ) (cid:105) of the active noise. A heuristic fitting procedure revealedno significant change in the values of ω and D r compared to the values inferred from our 2Dexperiments as reported in section II B, while v and D t changed by a factor close to 2-3.This is reasonable given our simplistic modeling of the particle-wall interactions in the quasione-dimensional channel. FIG. 11 shows the results of the Langevin simulations comparedwith empirical results at various driving amplitudes and FIG. 12 shows the variation of λ with the driving parameter Γ.7 FIG. 12. Variation of the potential strength parameter λ with the driving amplitude Γ. IV. FLIPPING DYNAMICS
With the agreement between experiments and simulations discussed above, we now com-pute the transition rates k f and k b from the time-series of ψ . A. Experimental data analysis
Using a custom written MATLAB code to implement a previously described algo-rithm [31], we detect steps in the ψ ( t ) time series data as shown in FIG. 13. From thefits obtained to the data, we identify flips (switches) in the “forward” (direction of theparticle chirality) and “backward” directions (FIG. 13). From the detections, the dwelltimes τ in each of the ψ ∼ ± π/ P ( τ ) are constructed. B. Waiting time distribution for a composite Poisson processes
The probability density function for the waiting times τ of a Poisson stochastic processwith a rate k is p ( τ ) = k e − kτ . (31)8 A CB
FIG. 13. Detection of flips in the particle relative orientation ψ . A . From the image processing,a time series of the orientation ψ is obtained. This time series is fit using custom MATLABcode to identify discrete jumps/switches in the orientation. The same data is plotted in B where ψ ∈ [ − π, π ]. Discrete steps and the preponderance of ψ ∼ ± π/ C . The distributionof the switch/flip sizes shows clear peaks at ± π . Flips corresponding to step sizes close to 0 aretreated as spurious and the detection of the run times τ and further analysis is performed aftersuch filtering. Consider two independent Poisson processes, each with distinct rates k f and k b . The prob-ability density functions of the waiting times corresponding to each process are p f ( τ ) = k f e − k f τ , (32)and p b ( τ ) = k b e − k b τ . (33)The waiting time distribution of a stochastic process comprising these independent subpro-cesses is then obtained via a convolution of (32) and (33) as P ( τ ) = (cid:90) τ dt k f k b e − k f t e − k b ( τ − t ) = e − k b τ − e − k f τ k f − k b . (34) C. Transition rates in a tilted periodic potential
We observe from FIG. 9, that the active speed v ∼ [4 − mm/s . Given that the mean-radius of the confining channel R = 116 . mm , the parameter ν = v /R is quite small, say9 FIG. 14. The dwell time distribution P ( τ ) between the flips seen in the relative orientation coor-dinate ψ . The solid lines are the fit of equation (34) to the data. BC A
FIG. 15. The tilted potential used for calculating the transition rates k f / b in the Kramers’ approx-imation. compared to ω . Further, we find that ν is small compared to λ as well FIG. 12. As such,we approximate the effective potential to V ( ψ ) = − ωψ + λ ψ. (35)We calculate the forward/backward (along/opposite-to ω ) transition rates in the Kramers’approximation for a particle moving in the potential V ( ψ ) [23]. We calculate the potentialheight difference ∆ V BA between points A and B, and ∆ V BC between points B and C as0shown in FIG. 15 . ∆ V BA = |V B − V A | = √ λ − ω − ω sin − ωλ + ω π , (36)∆ V BC = |V B − V C | = √ λ − ω − ω sin − ωλ − ω π . (37)The second derivative of the potential at the required points are V (cid:48)(cid:48) ( A ) = 2 √ λ − ω = V (cid:48)(cid:48) ( C ) = −V (cid:48)(cid:48) ( B ) . (38)Using Kramers’ escape rate theory, we then obtain the forward and backward flipping rates k f and k b respectively k f = (cid:112) V (cid:48)(cid:48) ( C ) |V (cid:48)(cid:48) ( B ) | π e − ∆ V BC /D , = √ λ − ω π exp (cid:18) − √ λ − ω − ω sin − ( ω/λ ) − πω/ D (cid:19) , (39)and k b = (cid:112) V (cid:48)(cid:48) ( A ) |V (cid:48)(cid:48) ( B ) | π e − ∆ V BA /D , = √ λ − ω π exp (cid:18) − √ λ − ω − ω sin − ( ω/λ ) + πω/ D (cid:19) . (40)Thus, the ratio of the two rates k f k b = e πω/D (41)is controlled by the constant “force” ω (in our case the chirality of the active particle).1 V. MOVIES
Supplementary Movie 1 (00Unconfined.avi):
Active self-propelled motion of disks in a 2-D arena. The particles have a diameter of4.5 mm. Here the driving Γ = 6 .
71. Previous positions are continuosly overlaid to indicatethe overall trajectory of the particles.2
Supplementary Movie 2 (01ConfinedABP.avi):
ABP-like motion in a confined channel. Here, the particle diameter is 4.5 mm, the channelconfinement δ = 0 .
88 and the driving Γ = 6 .
71. A red circle is used to highlight the particle.Insets show the time evolution of the position θ and the relative internal orientation ψ .3 Supplementary Movie 3 (02ConfinedRTP.avi):
RTP-like motion in a confined channel. Here, the particle diameter is 4.5 mm, the channelconfinement δ = 0 . .
71. A red circle is used to highlight the particle.Insets show the time evolution of the position θ and the relative internal orientation ψψ