Non-Gaussian behaviour of a self-propelled particle on a substrate

The overdamped Brownian motion of a self-propelled particle which is driven by a projected internal force is studied by solving the Langevin equation analytically. The active particle under study is restricted to move along a linear channel. The direction of its internal force is orientationally...

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Datum:	2009
Hauptverfasser:	ten Hagen, B., van Teeffelen, S., Löwen, H.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2009
Schriftenreihe:	Condensed Matter Physics
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spelling	irk-123456789-1205562017-06-13T03:05:54Z Non-Gaussian behaviour of a self-propelled particle on a substrate ten Hagen, B. van Teeffelen, S. Löwen, H. The overdamped Brownian motion of a self-propelled particle which is driven by a projected internal force is studied by solving the Langevin equation analytically. The active particle under study is restricted to move along a linear channel. The direction of its internal force is orientationally diffusing on a unit circle in a plane perpendicular to the substrate. An additional time-dependent torque is acting on the internal force orientation. The model is relevant for active particles like catalytically driven Janus particles and bacteria moving on a substrate. Analytical results for the rst four time-dependent displacement moments are presented and analysed for several special situations. For a vanishing torque, there is a significant dynamical non-Gaussian behaviour at finite times t as signalled by a non-vanishing normalized kurtosis in the particle displacement which approaches zero for long time with a 1/t long-time tail. На основi знаходження аналiтичного розв’язку рiвняння Ланжевена дослiджується згасаючий броунiвський рух самохiдної частинки, що керується вiдпроектованою внутрiшньою силою. Рух такої “активної” частинки обмежується вздовж лiнiйного каналу, а напрямок внутрiшньої сили, що дiє на неї, орiєнтацiйно дифундує на одиничному колi в площинi, перпендикулярнiй до субстрату. Додатковий залежний вiд часу момент сили також впливає на орiєнтацiю внутрiшньої сили. Така модель є актуальною для активних частинок на кшталт каталiтично керованих частинок Януса або ж бактерiй, що рухаються на поверхнi субстрату. Для чотирьох перших моментiв змiщення частинки отримано аналiтичнi результати, якi аналiзуються для кiлькох спецiальних ситуацiй. Для моменту сил, що прямує до нуля, спостерiгається цiкава негаусова динамiчна поведiнка при скiнчених часах t, про що сигналiзує незникаюча величина нормалiзованого коефiцiєнта ексцесу , який спадає як 1/t при великих часах. 2009 Article Non-Gaussian behaviour of a self-propelled particle on a substrate / B. ten Hagen, S. van Teeffelen, H. Löwen // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 725-738. — Бібліогр.: 42 назв. — англ. 1607-324X PACS: 82.70.Dd, 05.40.Jc DOI:10.5488/CMP.12.4.725 http://dspace.nbuv.gov.ua/handle/123456789/120556 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The overdamped Brownian motion of a self-propelled particle which is driven by a projected internal force is studied by solving the Langevin equation analytically. The active particle under study is restricted to move along a linear channel. The direction of its internal force is orientationally diffusing on a unit circle in a plane perpendicular to the substrate. An additional time-dependent torque is acting on the internal force orientation. The model is relevant for active particles like catalytically driven Janus particles and bacteria moving on a substrate. Analytical results for the rst four time-dependent displacement moments are presented and analysed for several special situations. For a vanishing torque, there is a significant dynamical non-Gaussian behaviour at finite times t as signalled by a non-vanishing normalized kurtosis in the particle displacement which approaches zero for long time with a 1/t long-time tail.
format	Article
author	ten Hagen, B. van Teeffelen, S. Löwen, H.
spellingShingle	ten Hagen, B. van Teeffelen, S. Löwen, H. Non-Gaussian behaviour of a self-propelled particle on a substrate Condensed Matter Physics
author_facet	ten Hagen, B. van Teeffelen, S. Löwen, H.
author_sort	ten Hagen, B.
title	Non-Gaussian behaviour of a self-propelled particle on a substrate
title_short	Non-Gaussian behaviour of a self-propelled particle on a substrate
title_full	Non-Gaussian behaviour of a self-propelled particle on a substrate
title_fullStr	Non-Gaussian behaviour of a self-propelled particle on a substrate
title_full_unstemmed	Non-Gaussian behaviour of a self-propelled particle on a substrate
title_sort	non-gaussian behaviour of a self-propelled particle on a substrate
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/120556
citation_txt	Non-Gaussian behaviour of a self-propelled particle on a substrate / B. ten Hagen, S. van Teeffelen, H. Löwen // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 725-738. — Бібліогр.: 42 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT tenhagenb nongaussianbehaviourofaselfpropelledparticleonasubstrate AT vanteeffelens nongaussianbehaviourofaselfpropelledparticleonasubstrate AT lowenh nongaussianbehaviourofaselfpropelledparticleonasubstrate
first_indexed	2025-07-08T18:07:41Z
last_indexed	2025-07-08T18:07:41Z
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fulltext	Condensed Matter Physics 2009, Vol. 12, No 4, pp. 725–738 Non-Gaussian behaviour of a self-propelled particle on a substrate B. ten Hagen, S. van Teeffelen, H. Löwen∗ Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany Received June 16, 2009 The overdamped Brownian motion of a self-propelled particle which is driven by a projected internal force is studied by solving the Langevin equation analytically. The “active” particle under study is restricted to move along a linear channel. The direction of its internal force is orientationally diffusing on a unit circle in a plane perpendicular to the substrate. An additional time-dependent torque is acting on the internal force orientation. The model is relevant for active particles like catalytically driven Janus particles and bacteria moving on a substrate. Analytical results for the first four time-dependent displacement moments are presented and analysed for several special situations. For a vanishing torque, there is a significant dynamical non-Gaussian behaviour at finite times t as signalled by a non-vanishing normalized kurtosis in the particle displacement which approaches zero for long time with a 1/t long-time tail. Key words: Brownian dynamics, self-propelled particle, substrate, swimmer, active particles, diffusion PACS: 82.70.Dd, 05.40.Jc 1. Introduction The Brownian motion of self-propelled (“active”) particles [1,2] bears much richer physics than the traditional diffusive dynamics of passive particles. Active particles can be modelled by moving under the action of an internal force sometimes combined with an internal or external torque. Realizations in nature are certain bacteria [3–7] and spermatozoa [8–10] which swim in circles when confined to a surface [11]. In the colloidal world, it is possible to prepare catalytically driven Janus particles [12–15] or biometric particles [16] which perform self-propelled Brownian motion. For a recent investigation including confinement see [17]. On the macroscopic scale, the vibrating polar granular rods [18] on a planar substrate and even the trajectories of completely blinded and ear-plugged pedestrians [19] can be considered as rough realizations of self-driven Brownian particles. If the particle is embedded in a liquid (a “swimmer”), as characteristic for colloids, the direction of its driving force fluctuates, in general, according to orientational Brownian motion [20– 22]. This gives rise to a non-ballistic translational motion of the particles which is coupled to the fluctuating orientational degree of freedom. In most cases the direction of the self-propelling force is within the plane of motion. For colloidal particles, however, it is possible to confine the particle on a substrate by using, e. g., strong gravity such that the particles are still freely rotating [15,23,24] though they are confined in a planar monolayer. In this situation the component of the self-propelling force which is normal to the surface is compensated by the substrate, i.e., only the projection of the self-propelling force onto the plane is driving the particle. Therefore, the translational motion is coupled to the (Brownian) orientational motion [25]. In this paper we consider a one-dimensional model [26] for the Brownian dynamics of a self- propelled particle on a substrate. The particle is self-propelled along its orientational axis, which itself is subjected to Brownian orientational diffusion. The particle is confined to a channel, however, such that only the projected force in channel direction is acting to drive the particle. The present ∗E-mail: hlowen@thphy.uni-duesseldorf.de c© B. ten Hagen, S. van Teeffelen, H. Löwen 725 B. ten Hagen, S. van Teeffelen, H. Löwen study is more general than the earlier work in reference [25]: first of all, the present calculation resolves the Cartesian components of the isotropic model on an unconfined plane. Second, an arbitrary time-dependence of the external torque is included here while this torque was constant in [25]. Finally, we calculate time-dependent moments of the particle displacement up to the fourth order as compared to the results up to the second order in reference [25]. The results are discussed for several special cases. In general, long-time self-diffusion is found. Non-Gaussian behaviour is found for intermediate times as signalled in the corresponding fourth cumulant. The normalized kurtosis is positive for small times, then it changes the sign and approaches zero from below at long times with a 1/t long-time tail. This can be compared to recent investigations for an undriven Brownian ellipsoid [27]. In the latter case, the kurtosis was found to be positive approaching zero from above for long times with the same 1/t long-time tail. This work represents a first step towards a many-body situation of interacting self-propelled particles. These are also realizable in experiments (see, e. g., [12,15,18]). The suitable theoretical framework is the many-body Smoluchowski equation [28], from which one can derive a coupled hierarchy of equations for the set of many-body distribution functions similar in spirit to the traditional BBGKY (Bogolyubov-Born-Green-Kirkwood-Yvon) hierarchy [29–31] for Liouville dy- namics, see also Felderhof [32] for a discussion in the context of Brownian motion. Therefore, we think that this paper is particularly appropriate for this issue dedicated to the 100th anniversary of Prof. N.N. Bogolyubov. This paper is organized as follows: In section 2, we propose and motivate the model. The first four displacement moments are calculated analytically for the torque-free case in section 3, while section 4 contains the results for a general time-dependent torque. Finally, in section 5, we conclude and give an outlook on possible future activities. 2. The model The model system under study consists of a self-propelled colloidal sphere of radius R, which is confined to an infinite linear channel in the x-direction, where it undergoes completely overdamped Brownian motion (for a sketch see figure 1). Whereas the motion of the center-of-mass position x φ linear channel Me x z y u 0F cosφ F = F u00 Figure 1. Sketch of the model system: A spherical colloidal particle (dark grey) is confined to a linear channel (light grey) along the x-direction. The self-propulsion is modelled by a constant effective force ~F0 along the particle orientation ~̂u. The latter is constrained to rotate in the xy-plane. Only the projected force F0 cos φ drives the particle along the channel. A systematic, time-dependent torque ~M(t) = M(t)~̂ez is also indicated. is constrained to one dimension, the orientation vector ~̂u = (cos φ, sin φ, 0) is constrained to rotate in the xy-plane. The self-propulsion of the particle is modelled by a constant effective force along the particle orientation ~F = F0~̂u and a generally time-dependent effective torque in the z-direction ~M = M~̂ez. Because the particle is confined, only the projected force ~F · ~̂ex = F0 cosφ~̂ex drives 726 Dynamics of a self-propelled particle the particle systematically along the channel. Based on these considerations, the translational and orientational motion is modelled by a Langevin equation for the center-of-mass position x and the orientation vector ~̂u: dx dt = βD [F0 cosφ + f(t)] , (2.1) d~̂u dt = βDr [M(t) + g(t)] ~̂ez × ~̂u , (2.2) where f(t) is a zero-mean, Gaussian white noise random force, which is characterized by 〈f(t)〉 = 0 and 〈f(t)f(t′)〉 = 2δ(t−t′)/(β2D), where angular brackets denote a noise average. Correspondingly, g(t) is a Gaussian white noise random torque with 〈g(t)〉 = 0 and 〈g(t)g(t′)〉 = 2δ(t − t′)/(β2Dr). Here, β−1 = kBT denotes the thermal energy. D and Dr are the translational and rotational short- time diffusion constants, respectively. For a sphere of radius R in the three-dimensional bulk the two quantities fulfill the relationship D Dr = 4R2 3 . (2.3) Due to the constraint on the orientational motion, the vector equation (2.2) reduces to a Langevin equation for the orientational angle φ, which is given by dφ dt = βDr [M(t) + g(t)] . (2.4) If the initial time t0 is set to be zero, the solutions of the Langevin equations (2.1) and (2.4) are given by φ(t) = βDr ∫ t 0 [M(t′) + g(t′)] dt′ + φ0 (2.5) and x(t) = βD [ F0 ∫ t 0 cosφ(t′)dt′ + ∫ t 0 f(t′)dt′ ] + x0 (2.6) with φ0 ≡ φ(t0) and x0 ≡ x(t0). The translation-rotation-coupling between these two equations, which is due to the cosine in equations (2.1) and (2.6), leads to nontrivial results for the mean position 〈x − x0〉 and the mean square displacement 〈(x − x0) 2〉 of the particle position, as is shown in the following sections. Furthermore, the presence of the coupling term leads to non-Gaussian behaviour, which is reflected in a non-zero kurtosis. The latter is obtained by calculating the fourth moment of the particle displacement distribution further down. We start our analysis in section 3 by studying the special case of a vanishing systematic torque M = 0. The more complex situations of a constant torque M(t) = M and a generally time- dependent torque M(t) are considered in section 4. 3. Results for a vanishing torque In this section, the simplest case with a vanishing torque is covered. Solving equation (2.4) for M(t) ≡ 0 and averaging gives 〈φ(t)〉 = φ0 (3.1) and for the second moment 〈 (φ(t) − φ0) 2 〉 = 2Drt. (3.2) As φ(t) is a linear combination of Gaussian variables g(t′), according to Wick’s theorem [20], φ(t) is Gaussian as well. Thus the probability distribution of φ proves to be P (φ, t) = 1√ 4πDrt exp ( − (φ − φ0) 2 4Drt ) . (3.3) 727 B. ten Hagen, S. van Teeffelen, H. Löwen Now the mean position of the particle can be calculated. From 〈cosφ(t)〉 = ∫ ∞ −∞ cos(φ)P (φ, t)dφ = e−Drt cosφ0 (3.4) follows 〈x(t) − x0〉 = 4 3 βF0R 2 cos(φ0) [ 1 − e−Drt ] , (3.5) where we made use of equation (2.3). Thus for short times one obtains 〈x(t) − x0〉 = 4 3 βF0R 2 cos(φ0)Drt + O ( t2 ) (3.6) and for t � D−1 r the φ0-dependent mean position converges towards lim t→∞ 〈x(t) − x0〉 = 4 3 βF0R 2 cos(φ0). (3.7) The trajectory of the mean position 〈x(t)〉 is shown in figure 2 where the time t is given in units Figure 2. Mean position of a spherical particle without external torque for βRF0 = 10 and different values of φ0. of D−1 r , while the length x is scaled by the particle radius R. To calculate the mean square displacement, the following integrals have to be solved: 〈 (x(t) − x0) 2 〉 = β2D2 [ F 2 0 ∫ t 0 dt1 ∫ t 0 dt2〈cosφ(t1) cosφ(t2)〉 +2F0 ∫ t 0 dt1 ∫ t 0 dt2〈cos φ(t1)f(t2)〉 + ∫ t 0 dt1 ∫ t 0 dt2〈f(t1)f(t2)〉 ] . (3.8) The third summand can be calculated easily and equals 2t/(β2D). As 〈cosφ(t)〉 only depends on the random torque g(t), 〈cosφ(t)〉 and f(t) are statistically independent. Therefore the second summand vanishes. To calculate the first summand in equation (3.8), the time correlation function is used. With φ1 ≡ φ(t1) and φ2 ≡ φ(t2) the required average can be written as 〈cosφ1 cosφ2〉t1>t2 = ∫ dφ1 ∫ dφ2 cosφ1 cosφ2G(φ1, φ2, t1 − t2)P (φ2, t2)\|t1>t2 . (3.9) Here, G(φ1, φ2, t1 − t2) is the Green function, which is given by G(φ1, φ2, t1 − t2) = 1 √ 4πDr(t1 − t2) exp ( − (φ1 − φ2) 2 4Dr(t1 − t2) ) . (3.10) 728 Dynamics of a self-propelled particle This yields 〈cosφ1 cosφ2〉t1>t2 = 1 2 e−Dr(t1−t2) [ 1 + cos(2φ0)e −4Drt2 ] . (3.11) The expression for 〈cos φ1 cosφ2〉t2>t1 is obtained in exactly the same way by replacing t1 and t2 with each other. Now, the first summand in formula (3.8) is calculated by simple integration and the mean square displacement can be written in the final form 〈 (x(t) − x0) 2 〉 = 2Dt + ( 4 3 βF0R 2 )2 [ e−Drt + Drt − 1 + 1 12 cos(2φ0) ( e−4Drt − 4e−Drt + 3 ) ] . (3.12) The long-time diffusion coefficient Dl is given by Dl = lim t→∞ 1 2t 〈 (x(t) − x0) 2 〉 = D + 8 9 ( βF0R 2 )2 Dr . (3.13) Figure 3 displays the results for the same cases that were examined in figure 2. The graph for φ0 = π coincides with the graph for φ0 = 0. As can be seen in the logarithmic plots and from the expression (3.13), the initial angle φ0 is not relevant for times much longer than D−1 r . Figure 3. Mean square displacement of a spherical particle for βRF0 = 10 and different values of φ0. In what follows, the non-Gaussian behaviour of the particle is investigated. For this purpose skewness S and kurtosis γ are calculated. The non-Gaussian behaviour is clearly signalled in the nonzero value of these quantities. In general, the skewness is given by S = 〈 (x − 〈x〉)3 〉 〈(x − 〈x〉)2〉3/2 = 〈x3〉 − 3〈x〉〈x2〉 + 2〈x〉3 (〈x2〉 − 〈x〉2)3/2 , (3.14) and the kurtosis is calculated as γ = 〈 (x − 〈x〉)4 〉 〈(x − 〈x〉)2〉2 − 3 = 〈x4〉 − 4〈x〉〈x3〉 + 6〈x〉2〈x2〉 − 3〈x〉4 (〈x2〉 − 〈x〉2)2 − 3. (3.15) For the third and fourth moments of x – by analogy with equation (3.8) – one has to solve the integrals 〈 (x(t) − x0) 3 〉 = β3D3 ∫ t 0 dt1 ∫ t 0 dt2 ∫ t 0 dt3 [ F 3 0 〈cos φ(t1) cosφ(t2) cosφ(t3)〉 +3F0〈cosφ(t1)〉〈f(t2)f(t3)〉 ] (3.16) 729 B. ten Hagen, S. van Teeffelen, H. Löwen and 〈 (x(t) − x0) 4 〉 = β4D4 ∫ t 0 dt1 ∫ t 0 dt2 ∫ t 0 dt3 ∫ t 0 dt4 [ F 4 0 〈cosφ(t1) cosφ(t2) cosφ(t3) cosφ(t4)〉 +6F 2 0 〈cosφ(t1) cosφ(t2)〉〈f(t3)f(t4)〉 + 〈f(t1)f(t2)f(t3)f(t4)〉 ] , (3.17) respectively. Before solving the time-integrals over the first summands, the time correlation func- tions 〈cosφ1 cosφ2 cosφ3〉t1>t2>t3 = 1 2 cos(φ0)e −Dr(t1−t2+t3) + 1 4 e−Dr(t1+3t2−4t3) [ cos(φ0)e −Drt3 + cos(3φ0)e −9Drt3 ] (3.18) and 〈cosφ1 cosφ2 cosφ3 cosφ4〉t1>t2>t3>t4 = 1 4 e−Dr(t1−t2+t3−t4) [ 1 + cos(2φ0)e −4Drt4 ] + 1 4 e−Dr(t1+3t2−4t3) {1 2 e−Dr(t3−t4) [ 1 + cos(2φ0)e −4Drt4 ] + 1 2 e−9Dr(t3−t4) [ cos(4φ0)e −16Drt4 + cos(2φ0)e −4Drt4 ] } (3.19) have to be evaluated. Here the notation φi ≡ φ(ti) with i ∈ {1, 2, 3, 4} is used again. Both in equation (3.16) and in equation (3.17), the remaining terms can be easily calculated using the expressions already obtained for the first and second moments. The complete analytical results for the third and fourth moments (and for the skewness and kurtosis) are presented in the appendix. Figure 4. Skewness S(t) for βRF0 = 10 and different values of φ0. Figure 5. Skewness S(t) for φ0 = 0 and differ- ent values of βRF0. Figures 4 and 5 display the skewness S of the probability distribution of the particle position for different values of the initial angle φ0 and the dimensionless quantity βRF0 which determines whether the self-propulsion or the motion due to the interaction with the solvent molecules is dominant. Figure 4 shows that the sign of the skewness depends on φ0. If the x-component of the initial orientation is positive (−0.5π < φ0 < 0.5π), the skewness is negative, while initial angles between 0.5π and 1.5π lead to positive S. For symmetry reasons the skewness is zero for φ0 = 0.5π. Further analysis of formula (0.3) (see the appendix) gives the leading long-time behaviour of the skewness S(t) as S(t) = 8 3 a3 cos (φ0) ( (cos (φ0)) 2 − 3 ) (3 + 2 a2) √ 4 a2 + 6 (Drt) −3/2 + o ( 1 t3/2 ) , (3.20) 730 Dynamics of a self-propelled particle where the abbreviation a ≡ βRF0 is used, i.e., the skewness decreases proportionally to t−3/2. Similar analysis of formula (0.4) for the kurtosis γ(t) reveals a long time behaviour as γ(t) = −21a4 9 + 12 a2 + 4 a4 (Drt) −1 + o ( 1 t ) . (3.21) First of all, as can be seen from this formula and in figures 6–8, the kurtosis does not depend on φ0 for long times. The long-time tail, being proportional to 1/t, is more pronounced than that for the skewness. Moreover, as displayed in figures 6 and 7, for initial angles φ0 6= 0.5π the distribution is leptokurtic (positive kurtosis) for relatively short times and platykurtic (negative kurtosis) for relatively long times. Thus for intermediate times a change of sign is induced such that the kurtosis approaches its asymptotic value zero from below. This is in contrast to passive ellipsoidal particles in two dimensions [27] where non-Gaussian behaviour is due to dissipatively coupled translational and rotational motion. In the latter case, the same scaling of the long-time tail proportional to 1/t is found for the kurtosis but it approaches zero from above. Figure 6. Kurtosis γ(t) for βRF0 = 10 and different values of φ0. Figure 7. Kurtosis γ(t) for φ0 = 0 and differ- ent values of βRF0. Figure 8. Kurtosis γ(t) for φ0 = 0.5π and different values of βRF0. We expect that the different sign is linked to the one-dimensionality of our model rather than to the qualitatively different translation-rotation coupling, which is due to the driving force in our 731 B. ten Hagen, S. van Teeffelen, H. Löwen model as opposed to the different transverse and parallel short-time translational diffusivities in the passive ellipsoidal particle model. In particular, we expect the negative kurtosis at long times t � D−1 r to reflect a broad translational van Hove function [33] with shorter tails as compared to a Gaussian distribution, which is attributed to the non-linear cos-term in equation (2.1). 4. Results for a time-dependent torque Let us now assume an additional internal or external torque. Before considering the case of an arbitrarily time-dependent torque M(t), we first consider a constant torque M . Solving the Langevin equations (2.1) and (2.4) under this assumption, one obtains 〈φ(t)〉 = φ0 + βDrMt = φ0 + ωt (4.1) with the frequency ω = βDrM and 〈 (φ(t) − 〈φ(t)〉)2 〉 = 2Drt. (4.2) By replacing φ0 in formula (3.3) with φ0 + ωt, the updated probability distribution of φ is gained. The mean position is obtained as 〈x(t) − x0〉 = βD (D2 r + ω2) F0 [ Dr cos(φ0) − ω sin(φ0) +e−Drt(ω sin(φ0 + ωt) − Dr cos(φ0 + ωt)) ] . (4.3) In figure 9 this result is plotted for different values of the dimensionless quantity βM , which is the ratio of the external torque over the thermal energy. The long-time mean position is given by lim t→∞ 〈x(t) − x0〉 = βD (D2 r + ω2) F0 [Dr cos(φ0) − ω sin(φ0)] , (4.4) while the behaviour for short times is the same as in formula (3.6) for a vanishing torque. Figure 9. Mean position of a spherical particle with additional constant external torque for βRF0 = 10, φ0 = 0 and different values of βM . Following the notation introduced in formula (3.9) the Green function is now given by G(φ1, φ2, t1 − t2) = 1 √ 4πDr(t1 − t2) exp ( − (φ1 − φ2 − ω(t1 − t2)) 2 4Dr(t1 − t2) ) . (4.5) This leads to 〈cosφ1 cosφ2〉t1>t2 = 1 2 e−Dr(t1−t2) [ cos(ω(t1 − t2)) + cos (2φ0 + ω(t1 + t2)) e−4Drt2 ] (4.6) 732 Dynamics of a self-propelled particle and by integration one obtains 〈 (x(t) − x0) 2 〉 = 2Dt + β2F 2 0 D2 { Drt D2 r + ω2 − D2 r − ω2 (D2 r + ω2)2 − e−Drt (D2 r + ω2)2 [ (ω2 − D2 r ) cos(ωt) + 2ωDr sin(ωt) ] + 1 (9D2 r + ω2)(D2 r + ω2) [ e−Drt ( (−3D2 r + ω2) cos(2φ0 + ωt) +4Drω sin(2φ0 + ω) ) − (−3D2 r + ω2) cos(2φ0) − 4Drω sin(2φ0) ] + 1 (9D2 r + ω2)(16D2 r + 4ω2) [ e−4Drt ( (12D2 r − 2ω2) cos(2φ0 + 2ωt) −10Drω sin(2φ0 + 2ωt) ) − (12D2 r − 2ω2) cos(2φ0) + 10Drω sin(2φ0) ] } . (4.7) The result is displayed in figure 10. In this case, the long-time diffusion coefficient is given by Dl = D + 8 9 (βF0R 2)2Dr (1 + (βM)2) . (4.8) Figure 10. Mean square displacement of a spherical particle with additional constant external torque for βRF0 = 10, φ0 = 0 and different values of βM . To generalize the preceding considerations, the torque M(t) is assumed to be arbitrarily time- dependent now. Similarly to the two special cases investigated so far, it can be seen that the mean position of the particle is given by 〈x(t)〉 = βF0D ∫ t 0 cos [ φ0 + βDr ∫ t1 0 M(t2)dt2 ] e−Drt1dt1 + x0 . (4.9) The calculation of the mean square displacement starts with formula (3.8) again. The first summand is the most interesting one because the other ones can be treated as before. Based on the formula 〈cosφ1 cosφ2〉t1>t2 = 1 2 e−Dr(t1−t2) [ cos ( βDr ∫ t1 t2 M(t)dt ) + cos ( 2φ0 + 2βDr ∫ t2 0 M(t)dt + βDr ∫ t1 t2 M(t)dt ) e−4Drt2 ] (4.10) 733 B. ten Hagen, S. van Teeffelen, H. Löwen we introduce ωt1 := βDr ∫ t1 0 M(t)dt, ωt2 := βDr ∫ t2 0 M(t)dt. (4.11) Using this notation the problem can be solved in a similar way as for a constant M . The mean square displacement is now given by 〈 (x(t) − x0) 2 〉 = 2Dt + β2F 2 0 D2 ∫ t 0 dt1 ∫ t1 0 dt2e −Dr(t1−t2) × [ cos(ωt1 − ωt2) + cos (2φ0 + ωt1 + ωt2) e−4Drt2 ] . (4.12) 5. Conclusions In conclusion, motivated by recent experiments on catalytic colloidal particles [15,23,24], we have proposed and solved a model for a self-propelled colloidal particle on a substrate. An internal or external time-dependent torque is also included in the most general version of the model which can arise, e. g., from an external magnetic field. The first four moments of the particle displace- ment distribution were calculated analytically. Significant non-Gaussian behaviour was found for intermediate time. The normalized kurtosis changes sign and approaches zero from below with a massive long-time tail inversely proportional to time. Future work should address several generalizations of the model. First of all, the one-dimensio- nality of our model can be generalized towards higher dimensions both for the translational and orientational degrees of freedom. In particular, the translational degrees of freedom can be consi- dered to be two-dimensional (in a plane), and the orientational ones on a sphere. For the latter case, first analytical results have been obtained [34]. Also, e. g., for weak gravity, the third trans- lational dimension perpendicular to the substrate is getting important, which results in unusual sedimentation effects [35]. Furthermore, the self-propelled particle can be confined in the lateral direction [24] which leads to a finite mean square displacement. This effect should be incorporated into a model study as well. First results have been obtained for a circle-swimmer in planar circular geometry [36] and for swimmers in cuspy environments leading to self-rotating objects [37]. Last not least, the collective behaviour of many interacting self-propelled particles is expected to lead to novel characteristic nonequilibrium effects both without [38–41] and with confinement [37, 42]. As stated in the introduction, the Smoluchowski equation, suitably generalized to self-propelled particles [42], is an appropriate starting point here and the general hierarchy of Bogolyubov-Born- Green-Kirkwood-Yvon [29–31] is expected to be a valuable tool in deriving approximations in a systematic way. This fact, after all, clearly links the present paper to the 100th anniversary of N.N. Bogolyubov. Acknowledgements We thank L. Baraban, A. Erbe and P. Leiderer for helpful discussions which have stimulated the study of our model. We further thank H. H. Wensink and U. Zimmermann for helpful suggestions. This work has been supported by the DFG through the SFB TR6. We dedicate this work to the 100th anniversary of N.N. Bogolyubov. 734 Dynamics of a self-propelled particle Appendix Using the notation a = βRF0 and a scaled time τ = Drt, we summarize here the analytical results for the third and fourth moments as well as for the skewness S and kurtosis γ: 〈 (x(t) − x0) 3 R3 〉 = 32 3 aτ cos (φ0) ( 1 − e−τ ) + 64 27 a3 ( −45 8 cos (φ0) + 1 24 cos (3 φ0) + 17 3 cos (φ0) e−τ − 1 16 cos (3 φ0) e−τ + 3 cos (φ0) τ − 1 24 cos (φ0) e−4 τ + 1 40 cos (3 φ0) e−4 τ + 5 2 cos (φ0) τe−τ − 1 240 e−9 τ cos (3 φ0) ) , (0.1) 〈 (x(t) − x0) 4 R4 〉 = 64 3 τ2 + 256 9 a2τ ( e−τ + τ − 1 + 1 12 cos (2 b) ( e−4 τ − 4 e−τ + 3 ) ) + 256 81 a4 ( 3 τ2 + 1 6720 e−16 τ cos (4 φ0) − 5 τe−τ − 45 4 τ + 261 16 + 1 600 cos (2 φ0) e−9 τ − 19 6 cos (2 φ0) + 1 240 cos (4 φ0) e−4 τ + 1 192 cos (4 φ0) + 1 48 e−4 τ − 49 3 e−τ − 7 450 cos (2 φ0) e−4 τ − 1 120 cos (4 φ0) e−τ + 3 2 τ cos (2 φ0) − 1 30 τ cos (2 φ0) e−4 τ + 229 72 cos (2 φ0) e−τ − 1 840 cos (4 φ0) e−9 τ + 5 3 τ cos (2 φ0) e−τ ) , (0.2) S = [ 8 3 τ + 16 9 a2 ( e−τ + τ − 1 + 1 12 cos (2 φ0) ( e−4 τ − 4 e−τ + 3 ) ) −16 9 a2 (cos (φ0)) 2 ( 1− e−τ )2 ] −3/2 × { −32 9 a3 cos (φ0) − 760 81 a3 cos (φ0) e−τ − 44 27 a3 cos (3 φ0) e−τ − 32 81 a3 cos (φ0) e−4 τ − 32 135 a3 cos (3 φ0) e−4 τ − 4 405 a3e−9 τ cos (3 φ0) + 32 81 a3 cos (3 φ0) + 352 27 a3 cos (φ0) τe−τ + 448 27 a3 cos (φ0) e−2 τ + 64 27 a3e−2 τ cos (3 φ0) − 32 27 a3e−3 τ cos (3 φ0) − 32 9 a3e−3 τ cos (φ0) + 8 27 a3e−5 τ cos (φ0) + 8 27 a3e−5 τ cos (3 φ0) } (0.3) and γ = [ 8 3 τ + 16 9 a2 ( e−τ + τ − 1 + 1 12 cos (2 φ0) ( e−4 τ − 4 e−τ + 3 ) ) −16 9 a2 (cos (φ0)) 2 ( 1 − e−τ )2 ] −2 × { −128 9 τa2e−2 τ cos(2 φ0) + 64 81 a4e−6 τ cos(2 φ0) + 64 3 τ2 − 15424 729 a4e−τ cos(2 φ0) + 32 81 a4e−6 τ cos(4 φ0) + 64 45 a4e−τ cos(4 φ0) + 2032 27 a4 − 128 3 a2τ + 256 9 a2τ2 − 1216 27 a4τ + 256 27 a4τ2 + 64 27 τa2e−4 τ cos(2 φ0) − 70976 18225 a4e−4 τ cos(2 φ0) 735 B. ten Hagen, S. van Teeffelen, H. Löwen −20480 243 a4e−τ − 688 243 a4e−4 τ − 2560 81 a4τe−τ + 512 9 τa2e−τ − 128 9 τa2e−2 τ + 2048 81 a4τe−2 τ − 1136 1215 a4e−4 τ cos(4 φ0) − 256 81 a4e−2 τ cos(4 φ0) + 32 81 a4e−6 τ −2560 243 a4τe−τ cos(2 φ0) + 4 8505 a4e−16 τ cos(4 φ0) − 256 243 a4e−5 τ − 2048 1215 a4e−5 τ cos(2 φ0) + 2048 81 a4τe−2 τ cos(2 φ0) + 1792 81 a4e−3 τ + 1088 81 a4 cos(2 φ0) − 20 81 a4 cos(4 φ0) + 2048 81 a4e−3 τ cos(2 φ0) + 256 81 a4e−3 τ cos(4 φ0) − 128 27 a4τ cos(2 φ0) − 64 9 a2τ cos(2 φ0) −2336 243 a4e−2 τ − 32 1215 a4e−10 τ cos(2 φ0) − 32 1215 a4e−10 τ cos(4 φ0) − 3104 243 a4e−2 τ cos(2 φ0) + 64 2025 a4e−9 τ cos(2 φ0) − 256 405 a4e−5 τ cos(4 φ0) + 512 27 a2τe−τ cos(2 φ0) − 128 1215 a4τe−4 τ cos(2 φ0) + 64 2835 a4e−9 τ cos(4 φ0) } − 3. (0.4) 736 Dynamics of a self-propelled particle References 1. Toner J., Tu Y., Ramaswamy S., Annals of Physics, 2005, 318, 170. 2. Hänggi P., Marchesoni F., Rev. Mod. Phys., 2009, 81, 387. 3. Berg H.C., Turner L., Biophys. J., 1990, 58, 919. 4. 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Löwen Негаусова поведiнка самохiдної частинки на поверхнi субстрату Б. тен Хаген, С. ван Тееффелен, Г. Льовен Iнститут теоретичної фiзики II: Унiверситет iм. Гайнрiха Гайне, вул. Унiверситетська,1, 40225 Дюссельдорф, Нiмеччина Отримано 16 червня 2009 р. На основi знаходження аналiтичного розв’язку рiвняння Ланжевена дослiджується згасаючий броу- нiвський рух самохiдної частинки, що керується вiдпроектованою внутрiшньою силою. Рух такої “активної” частинки обмежується вздовж лiнiйного каналу, а напрямок внутрiшньої сили, що дiє на неї, орiєнтацiйно дифундує на одиничному колi в площинi, перпендикулярнiй до субстрату. Додатко- вий залежний вiд часу момент сили також впливає на орiєнтацiю внутрiшньої сили. Така модель є актуальною для активних частинок на кшталт каталiтично керованих частинок Януса або ж бактерiй, що рухаються на поверхнi субстрату. Для чотирьох перших моментiв змiщення частинки отрима- но аналiтичнi результати, якi аналiзуються для кiлькох спецiальних ситуацiй. Для моменту сил, що прямує до нуля, спостерiгається цiкава негаусова динамiчна поведiнка при скiнчених часах t, про що сигналiзує незникаюча величина нормалiзованого коефiцiєнта ексцесу , який спадає як 1/t при великих часах. Ключовi слова: Броунiвська динамiка, самохiдна частинка, субстрат, плавець, активнi частинки, дифузiя PACS: 82.70.Dd, 05.40.Jc 738

Non-Gaussian behaviour of a self-propelled particle on a substrate

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