Effect of reorientation statistics on torque response of self propelled particles
EEffect of reorientation statistics on torque response of self propelled particles
Benjamin Hancock ∗ and Aparna Baskaran † Martin Fisher School of Physics, Brandeis University, Waltham, MA 02453, USA (Dated: September 10, 2018)We consider the dynamics of self-propelled particles subject to external torques. Two models forthe reorientation of self-propulsion are considered, run-and-tumble particles, and active Brownianparticles. Using the standard tools of non-equilibrium statistical mechanics we show that the runand tumble particles have a more robust response to torques. This macroscopic signature of theunderlying reorientation statistics can be used to differentiate between the two types of self propelledparticles. Further this result might indicate that run and tumble motion is indeed the evolutionarilystable dynamics for bacteria.
I. INTRODUCTION
Self-propelled particles are inherently out of equilib-rium objects that consume energy at the scale of individ-ual entities to produce persistent self generated motion.Models of self propelled particles have become the proto-type theoretical system used to understand the physicsof active materials. These models are directly applica-ble to micron sized objects such as motile bacteria andself-phoretic colloids [1–9].There have been numerous theoretical investigationsof the collective behavior of active particles in the re-cent years. Phenomenology uncovered includes clusterformation, phase separation, motility induced segrega-tion [9–16], anomalous behavior of mechanical propertiessuch as pressure [17, 18] and rheology [19, 20]. Further, ithas been established that the presence of external forces[21–23] or confinement [24, 25] dramatically alters theobserved phenomenology in these systems. One strikingexample has been rectification [23, 26–30], in which asym-metric barriers induce directed transport of self-propelledparticles.Generally, self-propelled particles travel along a bodyaxis ˆ u that has an intrinsic reorientation process. Twomodel classes of self-propelled particles that have beenused extensively are Run-and-Tumble particles (RTPs)and Active Brownian particles (ABPs). The only dif-ference in the dynamics of the two classes is the natureof their reorientation process. The dynamics of an RTPconsists of periods of straight line motion followed bya sudden tumble event occurring at some mean tumblerate α . The tumble event completely decorrelates theorientation and a new orientation is selected at random.There are several examples of motile bacteria that followrun-and-tumble dynamics [31], the most famous exam-ple being Escherichia coli .[32]. The second class, ABPs,reorient their body axis through gradual rotational diffu-sion and this diffusion is Brownian with rotational diffu-sion coefficient D r . These dynamics have been observedin experiments of synthetic self-phoretic colloids [5–9]. ∗ [email protected] † [email protected] For times much longer than the angular reorientationtime, that is, t (cid:29) α for RTPs and t (cid:29) D r for ABPs, bothclasses of particles exhibit diffusive behavior. Equatingthe diffusivities at this stage would yield identical phe-nomenology for both models. This idea has been exploredextensively in [1, 13, 21, 23, 33, 34] where the authors findequivalence between the two models at the level of a driftdiffusion equation when α = ( d − D r , where d is thespatial dimension. FIG. 1. (color online) Orientation probability distributionfunctions for RTPs (red/solid) and ABPs (blue/dashed) inthe case of both a nematic aligning (left) field and a polaraligning field (right). RTP distributions are sharply peakedabout the optimal direction while the ABP distribution is thesmooth equilibrium result.
In this work we seek to identify signatures of the re-orientation statistics on the long time behavior of self-propelled particles. In particular we ask the followingquestion : If the swimming parameters are equivalent,how does the long time behavior of ABPs and RTPs dif-fer when subject to external torques? External torquesmodel gradient seeking behavior exhibited by microor-ganisms such as chemotaxis [31, 32], and aerotaxis [35].Such phenomena need not be restricted to biologicalsystems, as micron sized platinum-gold rods have beenshown to exhibit directed movement towards regions ofhigher hydrogen-peroxide concentrations [36]. By usingthe framework of non-equilibrium statistical mechanicswe show that run-and-tumble particles align with exter-nal torques (or gradients) much more robustly than ac-tive Brownian particles, indicating that the reorientationbehavior of bacteria is ideal for successful gradient seek-ing. a r X i v : . [ c ond - m a t . s o f t ] A ug II. THEORY
Let us consider non-interacting self propelled particlesmoving in unconstrained 2D space. The statistical me-chanics of this system is given in terms of ψ ( r , ˆ u , t ), theprobability of finding a particle at a point r at a time t oriented in a direction ˆu = cos θ ˆ x + sin θ ˆ y in space.When the particle reorients its direction through rota-tional diffusion, the dynamics of this probability distri-bution obeys a diffusion equation of the form ∂ t ψ ( r , ˆ u , t ) = −∇ · [ v ˆ u ψ ( r , ˆ u , t )] + ∇ · [ D t ∇ ψ ( r , ˆ u , t )]+ ∂ θ [ D r ∂ θ ψ ( r , ˆ u , t )] (1)where v is the self propulsion speed of the particle, D t isa translational diffusion constant and D r is a rotationaldiffusion constant. When the particle undergoes run andtumble dynamics, the probability distribution obeys amaster equation of the form ∂ t ψ ( r , ˆ u , t ) = −∇ · [ v ˆ u ψ ( r , ˆ u , t )] + ∇ · [ D t ∇ ψ ( r , ˆ u , t )] − αψ ( r , ˆ u , t ) + α π (cid:90) d ˆ u (cid:48) ψ ( r , ˆ u (cid:48) , t ) (2)where α is the tumble frequency. This description is validwhen the duration of a tumble is short compared to thetime scales set by the self propulsion and the tumble fre-quency of the particle. We are interested in the dynamicsof this system in the presence of uniform external torques.So, in the following, we consider the homogeneous limitof Eqs. (1-2) in the presence of a θ dependent potential.The dynamical equations of interest then are of the form ∂ t ψ ( θ, t ) = ∂ θ [ D r ∂ θ ψ ( θ, t )] + 1 ξ r ∂ θ [ ∂ θ V ( θ ) ψ ( θ, t )]] (3)for active Brownian particles and ∂ t ψ ( θ, t ) = − αψ ( θ, t ) + α π (cid:90) dθ (cid:48) ψ ( θ (cid:48) , t )+ 1 ξ r ∂ θ [ ∂ θ V ( θ ) ψ ( θ, t )]] (4)for run and tumble particles. In the above ξ r is a rota-tional friction constant characteristic of the medium inwhich the particles move.We will now consider different choices for the externalpotential and construct series solutions to Eqs. (3-4) ofthe form ψ ( θ ) = a + (cid:80) ∞ n =1 a n cos( nθ ) + b n sin( nθ ) fordifferent choices of external potential V ( θ ). Other use-ful measures to characterize the response of the systemare the first two moments of the orientation distribution,namely, the polarization P defined as P α = (cid:90) dθ ˆ u α ψ ( θ ) and the nematic order parameter tensor ←→ Q Q αβ = (cid:90) dθ (ˆ u α ˆ u β − δ αβ ) ψ ( θ )These will be calculated as well. III. NEMATIC TORQUE
First we consider an external potential of the form V ( θ ) = − γ cos(2 θ ). Such a field will induce a nematicaligning torque on the particles because of the two min-ima located at 0 and π . In this case a good characteriza-tion of the degree of order will be given by the componentof the nematic order parameter tensor describing align-ment in the ˆ x direction, namely Q xx . FIG. 2. (color online) RTPs exhibit higher nematic order thanABPs as shown by the nematic order parameter being higherfor RTPs (red) than for ABPs (blue/dashed)
In the following, the parameter κ is the ratio of thefield strength of the external potential to the friction co-efficient in units of the angular reorientation time andis given by κ = γξ r D r for active Brownian particles and κ = γξ r α for run and tumble particles. Computing Q xx from (3), one can show (see Appendix A for details) thatfor active Brownian particles, we obtain Q xx = φ I ( κ ) I ( κ ) (5)which is the equilibrium result for a thermal nematogenat temperature ξ r D r in the presence of an external field.Here the I i ’s are modified Bessel functions of the secondkind [37]. For run and tumble particles, the exact value ofthe order parameter turns out to be a continued fractionof the form Q xx = φ κ + κ + κ + κ + . . . (6)As seen in Fig.2, the nematic order parameter is largerfor run-and-tumble particles.We can also exactly compute the orientation distribu-tion functions in the presence of this external torque. Foractive Brownian particles the distribution is of the form(Appendix A) ψ ( θ ) = e κ cos(2 θ ) πI ( κ ) (7)which is precisely the equilibrium result and is usuallycalled a Von Mises distribution [38, 39], the analog ofthe Gaussian distribution on a unit sphere. For run andtumble particles we obtain the following formally exactseries solution ψ ( θ ) = a + ∞ (cid:88) n =1 a n ( κ ) cos(2 nθ ) (8)where the coefficients are given by a = 2 π Q xx and a n = a n − − a n − (2 n − κ : n ≥ IV. POLAR TORQUE
In this section we consider particles subject to an ex-ternal potential of the form V ( θ ) = − γ cos( θ ), i.e., atorque that tends to align their direction of motion alongone direction in space. In this case, the relevant measureof the response of the system is the polarization. Foractive Brownian particles the polarization is found to be(Appendix A) P x = φ I ( κ ) I ( κ ) (9) while for run and tumble particles we have P x = φ κ κ κ + κ + . . . (10)where κ is as defined earlier. As in the nematic case insection 3, the result for the active Brownian particle is thesame as that of equilibrium. The orientation distributionfunctions for the two classes are found to be ψ ( θ ) = e κ cos( θ ) πI ( κ ) (11)for ABPs, while for RTPs ψ ( θ ) = a + ∞ (cid:88) n =1 a n ( κ ) cos( nθ ) (12)where the coefficients are given by a = 1 π P x and a n = a n − − a n − ( n − κ : n ≥ κ in Fig 3a and Fig 1. Unlike in thenematic case, at the level of the moments of the distri-bution, the ABPs exhibit higher polar ordering than theRTPs even though the distribution itself is more sharplypeaked about the optimal direction. A better measureof the response in this case is the percentage of particlesthat find the optimal direction. We find that in terms ofthis measure the response of the run-and-tumble particleis indeed much more robust (see Fig(3b). FIG. 3. (color online) (a): The polarization as a functionof the parameter κ . The polarization for RTPs (red/solid)is less than for ABPs (blue/dashed) (b):The percentage ofparticles oriented within some range [ π/ , π/
18] as a func-tion of the parameter κ . It is clear that the percentage ofRTPs (red/square) oriented in that range is higher than ABPs(blue/circle). V. DISCUSSION
We have shown that the response of self-propelled par-ticles to external torques is sensitive to the nature of thereorientation statistics of the propulsion direction. Wefind that run and tumble particles exhibit a more robustresponse to applied torques in the following sense. Whenthere is one optimal direction for the particle’s orienta-tion, such as in the polar case, the fraction of particlesthat find the optimal direction is greater for those parti-cles using a run-and-tumble search strategy, even thoughthe average polarization is higher for particles undergo-ing Brownian rotational diffusion. For cases when there ismore than one optimal direction for the particles to movein (such as the nematic case, where there exists 2 optimaldirections) the run-and-tumble strategy always leads tobetter alignment as the particles are able to sample thedifferent optimal directions better than through Brow-nian diffusion. These external torques model gradientseeking behavior exhibited by microorganisms and themore robust response of run-and-tumble particles mightindicate a biological preference toward this search strat-egy.Even though the torque response is dramatically differ-ent based on the reorientation statistics, a number of bulkphenomena such as phase separation and clustering areidentical in the two classes of particles. This dichotomycan be understood by noting that the two models of selfpropulsion considered here have identical correlations buttheir response is very different. While the response ofthe Brownian particle has the conventional fluctuation-dissipation relationship to the correlation, the run-and-tumble particles do not have this property (see AppendixC). Therefore the nature of the reorientation statisticsbecomes important when considering external perturba-tions such as confining fores or aligning torques.Finally we note that the exact expressions obtained inthis work for run-and-tumble particles are given in termsof formal infinite series. These series solutions convergevery slowly (see Appendix B) and any truncation at loworders drastically fails to capture the true nature of thesolution. Therefore, low moment closure estimates, con-ventionally used in the literature of active particles, is un-reliable for the case of run-and-tumble particles thoughthey work very well for the case of active Brownian par-ticles.
ACKNOWLEDGMENTS
BH and AB acknowledge support from NSF-DMR-1149266, the Brandeis-MRSEC through NSF DMR-0820492 and NSF DMR-1420382. AB acknowledgesKITP for hospitality through NSF-PHY11-25915 andfruitful conversations with Mike Cates and Julien Tail-luer. BH also acknowledges support through IGERTDGE-1068620.
Appendix A: Derivation of order parameters anddistribution functions
In this appendix, the details of the derivation of steadystate solutions to equations (3-4) for different externaltorques are given. Let us begin by considering the ne-matic torque. The statistical mechanics in this case isgiven by a Fokker-Planck equation for active Brownianparticles ∂ t ψ ( θ, t ) = ∂ θ [ D r ∂ θ ψ ( θ, t )] + 2 γξ r ∂ θ [sin(2 θ ) ψ ( θ, t )]] (A1)and the following master equation for run and tumbleparticles ∂ t ψ ( θ, t ) = − αψ ( θ, t ) + α π (cid:90) dθ (cid:48) ψ ( θ (cid:48) , t )+ 2 γξ r ∂ θ [sin(2 θ ) ψ ( θ, t )]] (A2)As described earlier we search for series solutions ψ ( θ ) = a + (cid:80) ∞ n =1 a n cos( nθ ) + b n sin( nθ ). This Fourier decom-position is equivalent to the following moment expansion ψ ( θ ) = 12 π ( φ + 2 P a ˆ u a + 4 Q αβ (ˆ u α ˆ u β − δ αβ ) + . . . )In this form it is apparent that Q xx = π a . Using thiswith Eq.(A1) we arrive at the following hierarchy of equa-tions for active Brownian particles ∂ t a = 0 (A3) ∂ t a n δ n,m = γξ r ma n ( δ ,n + m + sgn ( m − n ) δ , | m − n | ) − n D r a n δ n,m + γξ r ma δ ,m (A4) ∂ t b n δ n,m = − γξ r mb n ( δ ,n + m + sgn ( n − m ) δ , | m − n | ) − n D r b n δ n,m (A5)And similarly we find the corresponding hierarchy for runand tumble particles, ∂ t a = 0 (A6) ∂ t a n δ n,m = γξ r ma n ( δ ,n + m + sgn ( m − n ) δ , | m − n | ) − αa n δ n,m + γξ r ma δ ,m (A7) ∂ t b n δ n,m = − γξ r mb n ( δ ,n + m + sgn ( n − m ) δ , | m − n | ) − αb n δ n,m (A8)First note that all odd coefficients are zero by the require-ment the ψ ( θ ) = ψ ( θ + π ). By truncating at arbitrary a n +2 or b n +2 and taking the steady state solution, onefinds that all a n or b n couple to the zeroth coefficient a or b . Since b does not exist, the b n vanish and byusing an iterative procedure, we solve for a and find foractive Brownian particles a = 2 a κ κ κ κ . . . (A9)while for run and tumble particles we have a = 2 a κ + κ + κ + κ + . . . (A10)Beyond Eq.(8) no further analytic treatment for run andtumble particles is available. For active Brownian par-ticles the continued fraction Eq.(A9) is of the form of aGauss continued fraction. In general we have the follow-ing identity for the ratio of modified Bessel functions I ν ( z ) I ν − ( z ) = z ν + z ν + 1) + z ν + 2) + z ν + 3) + . . . (A11)One can find continued fraction representations for all thecoefficients in our series solution and by using Eq.(A11)one finds that a n = I n ( κ ) I ( κ ) (A12)thus the orientation distribution function is given by theexact form ψ ( θ ) = 12 π + 1 πI ( κ ) ∞ (cid:88) n =1 I n ( κ ) cos(2 nθ ) (A13)With the help of the following identity, an example of aJacobi-Anger expansion e z cos(2 θ ) = I ( z ) + 2 ∞ (cid:88) n =1 I n ( κ ) cos(2 nθ )Eq.(A13) is able to be summed exactly to the form ofEq.(7). The same process holds for the polar aligningtorque and repeating the calculation in that case yieldsEq.(9)-(12) Appendix B: Truncation error and perturbationtheory
As stated in the main text, the series solution for runand tumble particles is slowly converging and a low mo-ment approximation is not valid in this case. One way to see this is to look at the % difference in the valueof the function when you truncate the series solution atsuccessive terms (see Fig.(4)). Even in the weak field( κ = . . n = 300. For both classes, the number of terms neededincreases as κ gets larger as should be expected. FIG. 4. (color online) For RTPs a large number of terms in theseries must be included. Above shows the percent differencein the value of ψ ( θ ) whether truncating at the nth term or n th + 1 term. Here we examine the value at θ = 0. The firstpoint on the horizontal axis represents the percent differencein ψ ( θ ) by truncating the series at n = 2 and n = 3 , the nextpoint represents the percent difference when truncating theseries at n = 3 and n = 4 this is continued up to the the lastpoint which represents the percent difference by truncatingthe series at n = 14 and n = 15. Appendix C: Relationship between Correlation andResponse
As another measure of the difference between the twoclasses of active particles one can compute the correla-tion and response functions. To illustrate, consider RTPsin the presence of a polar aligning torque. To computethe correlation and response functions we will need theaverage of cos( θ ) , i.e., the polarization given by the firstmoment of the angular orientation distribution function. (cid:68) cos( θ ) (cid:69) = (cid:90) dθ cos( θ )[ a + ∞ (cid:88) n =1 a n ( κ ) cos( nθ )] (C1)= πa ( κ ) = 2 πa κ κ κ + . . . = κ κ κ + . . . Where we used a = 1 / π . The response function isgiving by R = ∂∂κ (cid:68) cos( θ ) (cid:69) = 12 − κ + O ( κ ) (C2)The response function measures how the order parameterchanges in response to a changing field. The correlationfunction is given by C = (cid:68) cos ( θ ) (cid:69) − (cid:68) cos( θ ) (cid:69) (C3)= 12 + 12 a ( κ ) − a ( κ ) ; a = a κ κ + κ + . . . = ⇒ C = 12 − κ O ( κ )So up to quadratic order in κ we have CR = 1 + 32 κ + . . . (C4) FIG. 5. (color online) The ratio of correlation to response forRTPs in the presence of a polar aligning torque (Red/Dashed)and a nematic aligning torque (Blue/Dot-Dashed) as a func-tion of κ . The constant line at 1 represents the value of thisratio for ABPs Similary, for a nematic aligning field the relevant quan-tity is (cid:68) cos(2 θ ) (cid:69) = (cid:90) dθ cos(2 θ )[ a + ∞ (cid:88) n =1 a n ( κ ) cos(2 nθ )](C5)= πa ( κ ) = 2 πa κ + κ + κ + . . . = κ + κ + κ + . . . Where we used a = 1 / π . The response function isgiving by R = ∂∂κ (cid:68) cos(2 θ ) (cid:69) = 2 − κ + O ( κ ) (C6)The correlation function is given by C = (cid:68) cos (2 θ ) (cid:69) − (cid:68) cos(2 θ ) (cid:69) (C7)= 12 + 12 a ( κ ) − a ( κ ) ; a = a κ + κ + κ + . . . = ⇒ C = 12 − κ + O ( κ )So up to quadratic order in κ we have CR = 14 + 6 κ + O ( κ ) (C8)The exact expression for both aligning torques is ex-plored numerically and shown in Fig.5. 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