Active Brownian Motion in two-dimensions under Stochastic Resetting
AActive Brownian Motion in two-dimensions under Stochastic Resetting
Vijay Kumar, Onkar Sadekar, and Urna Basu Department of Chemical Engineering, Indian Institute of Science, Bengaluru, India Indian Institute of Science Education and Research, Homi Bhabha Road, Pashan, Pune, India Raman Research Institute, C. V. Raman Avenue, Bengaluru, India
We study the position distribution of an active Brownian particle (ABP) in the presence ofstochastic resetting in two spatial dimensions. We consider three different resetting protocols : (I)where both position and orientation of the particle are reset, (II) where only the position is reset,and (III) where only the orientation is reset with a certain rate r. We show that in the first two casesthe ABP reaches a stationary state. Using a renewal approach, we calculate exactly the stationarymarginal position distributions in the limiting cases when the resetting rate r is much larger ormuch smaller than the rotational diffusion constant D R of the ABP. We find that, in some cases,for a large resetting rate, the position distribution diverges near the resetting point; the nature ofthe divergence depends on the specific protocol. For the orientation resetting, there is no stationarystate, but the motion changes from a ballistic one at short-times to a diffusive one at late times.We characterize the short-time non-Gaussian marginal position distributions using a perturbativeapproach. I. INTRODUCTION
Stochastic resetting refers to intermittent interruptionand restart of a dynamical process. Introduction of suchresetting mechanism to a stochastic process changes bothstatic and dynamical properties of the system drastically[1]. Study of resetting is relevant in a wide range of ar-eas including search problems [2–5], population dynam-ics [6, 7], computer science[8, 9], and biological processes[10–12]. The paradigmatic example of stochastic reset-ting is that of a Brownian diffusive particle which is resetto its initial position with some rate [13]. The presence ofthe resetting drives the system out of equilibrium, whichleads to a lot of interesting behaviour including nonequi-librium steady states, dynamical transition in the tem-poral relaxation and non-monotonic mean first passagetime [13–15]. Effect of resetting on various other diffu-sive processes have also been studied over the last decade[16–28]. A natural question that arises is what happenswhen resetting is introduced to a system where the un-derlying stochastic process is ‘active’ instead of passivediffusion.Active processes refer to a class of dynamics which areintrinsically out of equilibrium due to self-propulsion [29–34]. Since the seminal work of Vicsek [35], there hasbeen a huge surge of interest in active matter systemswhich show a set of novel collective behaviour like flock-ing [36, 37], clustering [38–40], motility induced phaseseparation [34, 41–43]. Theoretical attempts to under-stand the properties of active matter focuses on studiesof simple yet analytically tractable models, like Run andTumble particles (RTP), active Brownian particle (ABP)and their many variations [31]. In such models, the activenature of the dynamics emerges due to a coupling of thespatial motion with some internal ‘orientation’ degree offreedom which itself evolves stochastically. The presenceof an intrinsic time-scale associated with the internal ori-entation leads to a lot of interesting behaviour even at asingle particle level which includes spatial anisotropy and ballistic motion at short-times [44–46], non-Boltzmanstationary state and clustering near the boundaries ofthe confining region [47–53] and unusual relaxation andfirst-passage properties [44, 54, 55].The first step to study the effect of resetting on ac-tive processes is to investigate the behaviour of a singleactive particle under stochastic resetting. Since activeparticles are characterized by both position and orienta-tion degrees, the resetting can be defined in the phasespace instead of position space, which opens up variouspossibilities regarding resetting protocols. The presenceof stochastic resetting introduces an additional time-scalegiven by the inverse of the resetting rate. For active par-ticles, the interplay between the internal time-scale andthat of the resetting is expected to lead to a richer be-haviour compared to its passive counterpart. Indeed, ithas recently been shown that introduction of a stochasticresetting to the dynamics of an RTP leads to non-trivialstationary distribution and first passage properties [56].The first-passage properties of ABP and RTP under var-ious resetting mechanisms have also been investigated re-cently [57–59].In this article we study the effect of stochastic resettingon active Brownian motion in two spatial dimension. Anactive Brownian particle (ABP) is an overdamped parti-cle with an internal orientation which undergoes a rota-tional diffusion. Consequently, in two spatial dimension,an ABP is characterized by its position ( x, y ) as well asits orientation θ. We study three different resetting proto-cols: (I) The position and the orientation of the particleare reset to their initial values with rate r , (II) only theposition is reset, and (III) only the orientation is reset.In the first two cases, i.e. , where the resetting protocolinvolves the resetting of the position, the particle posi-tion reaches a stationary state. We show that dependingon whether the resetting rate r is larger or smaller com-pared to the rotational diffusion constant D R , the sta-tionary position distribution is very different. We com-pute exactly the marginal position distributions for the a r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug (a) (b) (c) FIG. 1: Typical trajectories of an ABP for the three different resetting protocols: position–orientation resetting (a), positionresetting (b) and orientation resetting (c). In all the cases the particle starts from x = 0 = y along θ = 0 . two liming scenarios, namely, r (cid:28) D R and r (cid:29) D R . Itturns out that, for protocol I, the position distribution isstrongly anisotropic for r (cid:29) D R ; while for the protocolII, the distribution remains isotropic. Moreover, we showthat, for large r (cid:29) D R , in some cases, the stationary dis-tribution diverges near the resetting position; the natureof the divergence depends on the resetting protocol.For purely orientational resetting, i.e. , for protocolIII, the particle does not reach a stationary state, butshows an anisotropic motion with a ballistic to diffusivecrossover as time progresses. We show that, at late times,the typical fluctuations of the position around its meanvalues are characterized by a Gaussian distribution. Inthe short-time regime, the position fluctuations are non-Gaussian; we adopt a perturbative method to computethe same for small values of the resetting rate.In the next section we define the resetting protocols indetails and present a brief summary of our results. Sec-tions III and IV are devoted to the study of the position-orientation resetting and position resetting cases, respec-tively. The behaviour of the ABP under orientation re-setting only is discussed in Sec. V. We conclude withsome general remarks in Sec. VI. II. MODEL AND RESULTS
Let us consider an active Brownian particle movingwith a constant speed v on a two-dimensional plane.Apart from the position coordinates ( x, y ) , the parti-cle also has an internal degree of freedom, characterizedby the orientation θ , which itself undergoes a rotationalBrownian motion. The Langevin equations describingthis active Brownian motion are,˙ x ( t ) = v cos θ ( t ) , ˙ y ( t ) = v sin θ ( t ) , ˙ θ ( t ) = (cid:112) D R η ( t ) , (1)where η ( t ) is a delta-correlated white noise and D R isthe rotational diffusion constant. The coupling betweenthe position and orientation degrees leads to the ‘active’nature of the motion. The activity, in turn, gives rises to various intriguing behaviour including non-trivial posi-tion distributions at short-times which crosses over to aneffective diffusive behaviour at late-times. For the sake ofcompleteness, a brief review of the behaviour of ordinaryABP is provided in Appendix A.In this article we study the effect of stochastic resettingon the dynamics of such an active Brownian particle.Since the ABP is characterized by both the position andorientation degrees, the resetting might affect both thesedegrees. In the following, we focus on three differentresetting protocols.I. Resetting of the position and orientation : Inthis case, the position of the particle, along with itsorientation is reset to the corresponding initial val-ues with rate r. We assume that the particle startsfrom the origin, oriented along the x -axis, so that,at any time t, the particle is reset to x = 0 = y = θ ,with rate r. The system reaches a stationary statein the long-time limit. We investigate the station-ary marginal position distributions as well as thetime evolution of the moments of the position.II.
Resetting of the position : In the second sce-nario we reset the position of the particle to theorigin with rate r, but the orientation is not af-fected – it evolves as a free Brownian motion. Inthis case also the ABP reaches a stationary state.We characterize the moments and the stationarymarginal position distributions.III. Resetting of the orientation : In this scenario,only the orientation θ is reset with rate r, the po-sition degrees are not affected. In this case theposition distribution does not reach a stationarystate; we study the short-time and long-time lim-iting behaviour of the marginal distributions alongwith the position moments.Figure 1 shows typical trajectories of an ABP in thepresence of these three resetting protocols. In the firstcase (protocol I), the particle preferably visits the righthalf-plane x > -5 0 5 x -5 0 5 y -2 -1 0 1 2 3 x -2-1 0 1 2 y -0.25 0 0.25 0.5 x -0.25 0 0.25 y FIG. 2: Plot of the stationary position probability distribution P st ( x, y ) in the x − y plane for resetting protocols I (upperpanel) and II (lower panel). The darker region corresponds to higher value of the probability density. The left, middle andright columns correspond to r = 0 . , D R = 1 , r = 1 , D R = 1 and r = 10 , D R = 1 respectively, with v = 1 for all the cases. isotropic. For protocol III, the particle runs along the x -axis, away from the origin.In the absence of resetting, the active Brownian par-ticle shows an interesting dynamical crossover depend-ing on the value of the rotational diffusion constant D R . Starting from the origin, and with θ = 0 , at short-times t (cid:28) D − R , the motion is strongly non-diffusive and theposition distribution remains anisotropic with the vari-ance along the x and y directions showing very differenttemporal growths [44, 45]. At long times t (cid:29) D − R , how-ever, the motion becomes diffusive and the typical posi-tion fluctuations become Gaussian in nature, with onlythe tails retaining signatures of activity [52]. The pres-ence of stochastic resetting introduces another timescale r − , i.e. , the inverse of the resetting rate. We expect thatthe interplay of the two time scales D − R and r − wouldlead to a rich behaviour for ABP under resetting.Figure 2 illustrates the qualitative nature of the 2D sta-tionary position distribution P st ( x, y ) for resetting proto-cols I (upper panel) and II (lower panel). The left columnshows the distribution for r (cid:28) D R , where, in both cases,the distribution is isotropic. The middle column showsthe same for r ∼ D R , where for protocol I the distribu-tion becomes anisotropic. For protocol II, the distribu- tion remains isotropic, but the width decreases as r isincreased. The anisotropy becomes stronger for protocolI as r is increased, as can be seen from the right panel( r (cid:29) D R ). The anisotropy for the position-orientationresetting arises due to the fact that after each resetting,the orientation is brought back to θ = 0 , and the particlerestarts motion along the x -axis. On the other hand, forprotocol II, i.e. , when the resetting does not affect theorientation of the particle, the stationary distribution re-mains isotropic for all values of r and D R . As mentioned already, for protocol III, i.e. , for orien-tational resetting, the particle position does not reacha stationary state. In this case, the nature of the mo-tion changes from ballistic at short-times to diffusive atlate times. This is qualitatively illustrated in Fig. 3where P ( x, y, t ) is shown for three different values oftime. At short-times (left panel) the distribution remainsstrongly anisotropic, similar to the free ABP case. Theanisotropy decreases as time is increased (middle panel),ultimately reaching a Gaussian-like distribution at latetimes t (cid:29) ( r + D R ) − , as we will demonstrate later.Before going to the details of the computations we firstpresent a brief summary of our results. FIG. 3: Plot of position distribution P ( x, y, t ) in the x − y plane for orientation resetting for different values of time t = 0 . t = 1 (middle) and t = 200 (right). Darker regions correspond to higher values of the probability. Here r = 1 = D R and v = 1 . • We show that the position distribution reaches astationary state if the resetting protocol involveschanging the position directly, i.e. , for protocolsI and II. We study the corresponding stationarymarginal distributions and show that depending onwhether the time scale r − associated with reset-ting is larger or smaller than the inherent rotationaltime-scale D − R of the ABP, the stationary distri-bution has very different forms. • For protocol I, the stationary distribution isstrongly anisotropic for r (cid:29) D R . In this case, the x -marginal distribution falls off exponentially forlarge x > , while approaching a finite value nearthe origin [see Eq. (15)]. The x < y -distribution, however, turnsout to be symmetric and shows an algebraic diver-gence | y | − / near the origin, while decaying as acompressed exponential for large | y | [see Eqs. (25)and (26)].For r (cid:28) D R , on the other hand, the anisotropy dis-appears, both x and y -marginal distributions attainexponential forms; see Eqs. (12) and (20). • For protocol II, the stationary distribution remainsisotropic for all parameter values. In this case, for r (cid:28) D R the distribution is exponential in nature,similar to protocol I; see Eq. (35).For r (cid:29) D R , the distribution becomes independentof D R , and shows a log-divergence near the origin[see Eq. (38)] while decaying as an exponential forlarge | x | [see Eq. (39)]. • For the protocol III, the position distribution doesnot reach a stationary state. We show thatat late times, the particle shows a diffusive be-haviour; the typical position fluctuations are char-acterized by Gaussian distributions in this limit.We compute the corresponding effective diffusion constants, which turn out to be different for x and y components, signaling presence of an anisotropyeven at late times.At short times, for this protocol, the motion re-mains ballistic. We compute the position distribu-tion for small values of the resetting rate r , usinga perturbative approach. The perturbative correc-tions corresponding to x and y -distributions are ob-tained in Eqs. (54) and (57), respectively.In the following sections we study the three protocolsseparately and characterize the fluctuations of the posi-tion by computing the moments and marginal distribu-tions. III. ABP WITH POSITION ANDORIENTATION RESETTING
The simplest resetting protocol is when both the posi-tion and the orientation of the particle are reset to theirinitial values, with rate r. This is referred to as protocolI in Sec. II. For the sake of simplicity we assume thatthe ABP starts at the origin x = y = 0 , oriented alongthe x -axis, i.e. , with θ = 0 at time t = 0 . Then, at anytime t, the ABP is reset to x = y = 0 = θ with rate r ; between two consecutive resetting events, the parti-cle position evolves according to the Langevin equations(1). In the following we refer to this resetting protocolas ‘position-orientation reset’.We are interested in the position distribution P ( x, y, t ) = (cid:82) d θ P ( x, y, θ, t ) where P ( x, y, θ, t ) denotesthe probability that the particle is at the position ( x, y )with orientation θ, at time t. It is straightforward to writea renewal equation for P ( x, y, θ, t ) which reads, P ( x, y, θ, t ) = e − rt P ( x, y, θ, t ) + r (cid:90) t d s e − rs P ( x, y, θ, s ) , -1 t -4 -2 σ x r = 0.01 r = 0.1 r = 1.0 r = 5.0 -1 t -2 σ y r = 0.01 r = 0.1 r = 1.0 r = 5.0 (a) (b) FIG. 4: Position-orientation resetting: Mean squared dis-placements of x and y components as a function of time for D R = 1 and different values of the resetting rate r. Sym-bols represent the data from simulations while the solid blackcurves indicate the analytical predictions from Eqs. (B2) and(B3). Here v = 1 . where P ( x, y, θ, t ) denotes the probability that in theabsence of resetting, the ABP is at a position ( x, y ) withorientation θ at time t, starting from x = 0 = y = θ. Herethe first term corresponds to the situation when there areno resetting events up to time t and the second term cor-responds to the probability that the last resetting eventoccurred at a time t − s. A corresponding renewal equation for the position dis-tribution is obtained by integrating over the orientation θ,P ( x, y, t ) = e − rt P ( x, y, t ) + r (cid:90) t d s e − rs P ( x, y, s ) . (2)From this renewal equation, the position distribution can,in principle, be calculated for any time t , if the free ABPdistribution is known. Unfortunately, no closed form forthe full distribution P ( x, y, t ) of an ABP is known sofar. However, the short-time and long-time marginal po-sition distributions are known explicitly [44, 52], and inthis section we use these to investigate the effect of theposition-orientation resetting on an ABP using the re-newal equation (2). A. Moments
To get an idea about how the presence of the position-orientation resetting affects the dynamical behaviour ofthe ABP, let us first look at the moments of the posi-tion coordinates. It is straightforward to see that, in thepresence of the resetting, the moments would also satisfya renewal equation similar to Eq. (2). For example, bymultiplying both sides by x n and integrating over x and y, we get, (cid:104) x n ( t ) (cid:105) = e − rt (cid:104) x n ( t ) (cid:105) + r (cid:90) t d s e − rs (cid:104) x n ( s ) (cid:105) , (3)where (cid:104) x n ( t ) (cid:105) denotes the n th moment of the x -component of the position in the absence of the reset-ting which can be calculated explicitly for any n [44, 60]. The renewal equation for (cid:104) y n ( t ) (cid:105) also has a similar form.In the following we calculate explicitly the first two mo-ments of x and y -components using the known expres-sions for the same for free ABP (see Appendix A).Let us first look at the time-evolution of the averageposition. Using Eq. (3) for n = 1 along with Eq. (A6),we get, (cid:104) x ( t ) (cid:105) = v r + D R (1 − e − ( r + D R ) t ) , (4)while (cid:104) y ( t ) (cid:105) = 0 at all times. Here we see the first ev-idence of a new time-scale emerging as a result of thepresence of the resetting. Clearly, at short-times, i.e. ,for t (cid:28) ( r + D R ) − , the particle moves along x -axiswith an effective velocity v which is reminiscent of thefree ABP. On the other hand, at late-times the parti-cle reaches a stationary position which comes closer tothe origin as the resetting rate r is increased. Next,we calculate the second moments using Eq. (3) with n = 2 and Eq. (A10). The resulting exact (and long)expressions are provided in Appendix B. Here we explorethe behaviour of the mean squared displacement (MSD) σ x ( t ) = (cid:104) x ( t ) (cid:105) − (cid:104) x ( t ) (cid:105) and σ y ( t ) = (cid:104) y ( t ) (cid:105) in the short-time and the long-time regimes. At short-times, we have, σ x ( t ) = v rt + v
12 (4 D R − D R r − r ) t + O ( t ) ,σ y ( t ) = 2 v D R t − v D R (5 D R + 3 r ) t + O ( t ) . (5)It is interesting to compare this short-time behaviourwith that of ABP in the absence of resetting. Startingfrom the origin, oriented along the x -axis, for the or-dinary ABP, the MSD along the x -direction grows ∼ t while along y , it shows a t temporal growth. In the pres-ence of the position-orientation resetting, however, we seethat both σ x,y grow as t ; while the resetting changes theleading order behaviour of the MSD along x -direction, itdoes not affect the same for the MSD along y. At long-times, the particle is expected to reach a sta-tionary state, and the MSD does not depend on the timeanymore, lim t →∞ σ x = v (4 D R + 2 rD R + r ) r ( r + D R ) ( r + 4 D R ) , lim t →∞ σ y = 4 v D R r ( r + D R )( r + 4 D R ) . (6)It is to be noted that, the stationary values of the MSDare different for x and y -components, indicating that theanisotropy survives. This is not surprising as the reset-ting to θ = 0 introduces strong anisotropy at each epochs.Figure 4 show plots of σ x and σ y as functions of time t for different values of r ; as expected, the MSD saturatesfaster to its stationary value with increasing r . B. Marginal x -distribution Let us consider the marginal x -distribution in the pres-ence of position-orientation resetting. It satisfies a re- -10 -5 0 5 10 x -4 -2 P st ( x ) r = 0.01 r = 0.05 r = 0.1 r = 0.2 x -4 -2 P st ( x ) r = 1 r = 2 r = 5 r = 10 (a) (b) -9 -6 -3 0 3 6 9 x -3 -2 -1 P st ( x ) D R = 0.001 D R = 0.1 D R = 1 D R = 10 (c) FIG. 5: Stationary x -marginal distribution for position-orientation resetting: (a) Plot of P st ( x ) versus x for different values of r in the regime r (cid:28) D R and a fixed D R = 10 . (b) Similar plot in the regime r (cid:29) D R with D R = 0 . . (c) The crossover betweenthe two regimes for a fixed value of r = 0 . D R . The solid black lines indicate the analytical predictions[see Eq. (12) for (a) and Eq. (14) for (b)], the symbols show the data from numerical simulations and the red dashed line in(b) shows the exponential trend at large values of x . For all the plots v = 1 . newal equation obtained by integrating Eq. (2) over y,P ( x, t ) = e − rt P ( x, t ) + r (cid:90) t d s e − rs P ( x, s ) , (7)where P ( x, s ) denotes the x -marginal distribution in theabsence of the resetting. Note that for the sake of sim-plicity we use the same letter P for both the 1 − d and2 − d position distributions.At late-times t → ∞ , the particle position is expectedto reach a stationary state. We concentrate on the sta-tionary position distribution, which is given by, P st ( x ) = r (cid:90) ∞ d s e − rs P ( x, s ) . (8)As mentioned already, no closed form expressions areavailable for P ( x, t ) . However, the short-time ( t (cid:28) D − R )and long-time ( t (cid:29) D − R ) behaviour of P ( x, t ) are knownseparately [44, 52]. In the following we show that, theseshort-time and long-time behaviour can be used to calcu-late the distribution in the presence of resetting in somecases. To this end, let us first recast Eq. (8) as, P st ( x ) = r (cid:90) ∞ d u e − u P ( x, u/r ) . (9)Because of the presence of the e − u factor, the dominatingcontribution to the integral comes from u ∼ O (1) . Then,depending on whether u/r is large or small compared to D − R , the dominant contribution comes from the large orshort-time regime of the free ABP distribution. In thefollowing, we discuss the two limiting cases separately. Small resetting rate ( r (cid:28) D R ): In this case the typi-cal interval between two consecutive resetting events r − is longer than the rotational time-scale D − R , and the par-ticle evolves as a free ABP for a long-time before beingreset to the origin. Consequently, the dominant contri-bution to the integral in Eq. (9) comes from the regime, ur (cid:29) D − R . In other words, we can use the long-timedistribution of free ABP in Eq. (8) to compute the dis-tribution in the presence of resetting. It has been shown that for t (cid:29) D − R , the free ABP distribution admits alarge-deviation form, P ( x, t ) ∼ exp (cid:20) − D R s Φ (cid:18) xv t (cid:19)(cid:21) , (10)where the large deviation function Φ( z ) = z + O ( z )[52]. We are particularly interested in the typical fluctu-ations around x = 0 , and it suffices to take the leadingterm, which, when normalized, leads to a Gaussian dis-tribution, P ( x, t ) = (cid:115) D R πv t exp (cid:20) − D R x v t (cid:21) . (11)Substituting the above equation in Eq. (8) and perform-ing the integral over s, we get an exponential stationarydistribution in the presence of resetting, P st ( x ) = 1 v (cid:114) rD R (cid:20) − (cid:112) rD R | x | v (cid:21) . (12)This distribution is symmetric around x = 0 , and forlarge x, falls faster as either r or D R is increased. Figure5 shows a plot of the predicted P st ( x ) for different(small) values of r along with the same obtained fromnumerical simulations; an excellent match confirms thatthe prediction (12) is valid for a substantial range of r. Large resetting rate ( r (cid:29) D R ): In this case, the typi-cal interval between two resetting events is much smallercompared to the rotational diffusion time-scale of the freeABP dynamics. Consequently, most trajectories evolvefor a short-time before being reset to the origin. Inother words, the dominant contribution to the integral(8) comes from the short-time regime of free ABP. It hasbeen shown that, at short-times t (cid:28) D − R , the x -marginaldistribution is given by a scaling form, P ( x, t ) = 1 v D R t f x (cid:18) v t − xv D R t (cid:19) , for x ≤ v t. (13)Here the scaling function f x ( u ) is given by the sum of aninfinite series. The explicit form of f x ( u ) is known andquoted in Appendix A; we use that to calculate P st ( x )using Eq. (8). Note that as P ( x, s ) defined only in theregime x ≤ v s, the lower limit of the integral becomes s = x/v . This integral can be computed explicitly andyields a sum of exponentials, P st ( x ) = r √ D R v ∞ (cid:88) k =0 ( − k (4 k + 1)2 k (cid:18) kk (cid:19) √ a k + √ a k + r (cid:112) a k ( a k + r )exp (cid:34) − (cid:18) √ a k + √ a k + r (cid:19) xv (cid:35) , (14)with a k = (4 k + 1) D R / . Note that, this expression isvalid for x > . In fact, x < r case. Figure 5(b) compares the analytical pre-diction (14) with P st ( x ) obtained from numerical simula-tions for large values of r which show perfect agreement.To understand the asymptotic behaviour for large x, we note that for large x, the exponential term with thesmallest coefficient, i.e. , with k = 0 , would contribute.Hence, we expect that the tail of the distribution willhave the form, P st ( x ) (cid:39) rv √ (cid:0) √ D R + √ D R + 8 (cid:1) √ D R + 8 × exp (cid:20) − x v (cid:18) r + D R + (cid:112) D R ( D R + 8 r ) (cid:19)(cid:21) . (15)The exponential tails predicted in Eq. (15) are indicatedby red dashed lines in Fig. 5(b).To explore how the stationary distribution looks forintermediate values of r, we take recourse to numericalsimulations. Figure 5(c) shows a plot of the same fordifferent values of r which shows the crossover from theasymmetric (one sided exponential for x >
0) to the sym-metric (exponential decay on both sides) distribution.We see that as D R is increased, the x < x = 0 . The asymmetry disappears only for verylarge D R (cid:29) r. To understand this crossover from a strongly asym-metric to symmetric behaviour of P st ( x ), we computethe skewness of P ( x, t ), γ ( t ) = (cid:104) x ( t ) (cid:105) − (cid:104) x ( t ) (cid:105) σ x ( t ) − (cid:104) x ( t ) (cid:105) σ x ( t ) . (16)The third moment of x ( t ) can be calculated using Eq. (3)and Eq. (A11). The explicit expression for (cid:104) x ( t ) (cid:105) is pro-vided in Appendix B. However, since we are interestedin the stationary distribution, it suffices to look at thestationary limit of the skewness γ st = lim t →∞ γ ( t ) . Sub-stituting the expressions for moments and then takingthe long-time limit we get, γ st = 2 r (cid:112) r (4 D R + r ) (30 D R + 7 D R r + r )(9 D R + r ) (4 D R + 2 D R r + r ) / . (17) -4 -2 D R γ st r = 0.1 r = 1 r = 10 -2 D R -4 -2 γ st D R -3/2 FIG. 6: Position orientation resetting: Plot of the steady stateskewness γ st as a function of D R for different values of r [seeEq. (17)]. For small values of D R (cid:28) r γ st → . The algebraicdecay for large D R (cid:29) r is shown in the inset [see Eq. (18)]. Figure 6 shows a plot of γ st as a function of D R for a setof values of r . From Eq. (17) it is clear that for smallvalues of D R (cid:28) r, γ st → , which indicates a stronglyasymmetric distribution, as seen in Fig. 5(b). On theother hand, for large values of D R , we have, γ st (cid:39) (cid:18) rD R (cid:19) / . (18)Hence, in the limit r (cid:28) D R , the symmetric distribution( γ st = 0) is approached with an algebraic decay. C. Marginal y -distribution The anisotropic nature of the position distribution, asseen in Fig. 2, indicates that the marginal distributionalong y -direction is very different than the same along x -direction, at least for r ≥ D R . In this section we in-vestigate the behaviour of the marginal y -distribution inthe presence of position-orientation resetting.The marginal distribution P ( y, t ) satisfies a renewalequation similar to the x -component, P ( y, t ) = e − rt P ( y, t ) + r (cid:90) t d s e − rs P ( y, s ) , (19)where P ( y, t ) denotes the marginal distribution of ABPin the absence of resetting. Once again, we focus on thestationary distribution, and use the known short-timeand long-time behaviours of the P ( y, t ) to computethe position distribution in the presence of resetting.As before, we consider the two limiting cases wherethe resetting rate is much larger and smaller than therotational diffusion constant. Small resetting rate ( r (cid:28) D R ): In this case, as be-fore, we can use the late time expression for the ordi-nary active Brownian particle. In fact, at late times -10 -5 0 5 10 y -4 -2 P st ( y ) r = 0.01 r = 0.05 r = 0.1 r = 0.5 -2 -1 0 1 2 y -2 P st ( y ) r = 1 r = 2 r = 5 r = 10 -6 -5 y P st ( y ) (a) (b) r y -1/3 -6 -3 0 3 6 y -4 -2 P st ( y ) D R = 0.01 D R = 0.1 D R = 1 D R = 10 (c) FIG. 7: Marginal stationary distribution P st ( y ) for position-orientation resetting: (a) Plot of P st ( y ) versus y for different valuesof r in the regime r (cid:28) D R and a fixed D R = 10 . (b) Similar plot in the regime r (cid:29) D R with D R = 0 . . The inset shows thealgebraic divergence near the origin [see Eq. (25)]. Panel (c) shows the crossover between the two regimes for a fixed value of r = 1. In all the plots the solid black lines indicate the analytical predictions [see Eq. (20) for (a) and Eq. (23) for (b)]. Thesymbols indicate the data obtained from numerical simulations. For all the plots v = 1. t (cid:29) D − R , free ABP loses the anisotropy, and themarginal y -distribution becomes same as the marginal x -distribution. Thus, the typical y -fluctuations are alsoGaussian, and we can use Eq. (11) for P ( y, t ) . Obviously,this leads to the same stationary exponential distribu-tion, P st ( y ) = 1 v (cid:114) rD R (cid:20) − (cid:112) rD R | y | v (cid:21) . (20)This analytical prediction is verified in Fig. 7(a) whichshows a plot of the predicted P st ( y ) versus y for differentvalues of r in the regime r (cid:28) D R along with the sameobtained from numerical simulations. Large resetting rate ( r (cid:29) D R ): Following the sameargument as in the previous section, we expect that inthis case, the stationary distribution can be determinedfrom the short-time behaviour of P ( y, s ) . Note that, be-cause of the strong anisotropic nature of the free ABP atshort-times, P ( y, s ) is very different than P ( x, s ) usedin the previous section. In fact, it has been shown [44]that, at short-times s (cid:28) D − R , the y -dynamics of theABP can be mapped to a Random Acceleration Processand P ( y, s ) has a Gaussian form with variance v D R s [see Appendix A 2 for more details]. Then the stationary y -distribution in the presence of resetting is given by, P st ( y ) = √ r v √ πD R (cid:90) ∞ d s e − rs s / exp (cid:20) − y v D R s (cid:21) . (21)It is useful to use a change of variable u = rs, which leadsto, P st ( y ) = √ r / v √ πD R (cid:90) ∞ d u e − u u / exp (cid:20) − r y v D R u (cid:21) . (22)Clearly, the stationary distribution is a function of thescaled variable z = r / yv √ D R . In fact, this integral can becomputed exactly using Mathematica and the stationarydistribution can be expressed in a scaling form, P st ( y ) = 2 πr / v √ D R F (cid:18) r / | y | v √ D R (cid:19) , (23) where the scaling function, F ( z ) = 3 π (cid:20) ker / (cid:18) (cid:114) z (cid:19) + kei / (cid:18) (cid:114) z (cid:19) (cid:21) . (24)Here kei ν ( w ) and ker ν ( w ) are Kelvin functions (see Eq.10.61.2 in Ref. [61]). It can be shown that the stationarydistribution given by Eqs. (23) and (24) is identical toEq. (19) of Ref. [27] obtained in the context of resettingof Random Acceleration Process.Figure 7(b) shows a plot of the predicted stationarydistribution for different (large) values of r along withthe same measured from numerical simulations.It is interesting to look at the asymptotic behaviourof this stationary distribution. The behaviour near theorigin can be obtained using the series expansion of theKelvin functions. The details are provided in the Ap-pendix C; here we just quote the final result. As | y | → ,P st ( y ) shows an algebraic divergence, P st ( y ) = 2 πr ( v D R ) / / Γ( ) | y | − / + O (1) . (25)The inset in Fig. 7(b) shows a log-log plot of P st ( y ) nearthe origin where this divergence is illustrated.To understand the decay of the distribution for large | y | , we use the asymptotic expansion of the Kelvin func-tions for large argument; see Appendix C for the details.This exercise leads to a compressed exponential form forlarge z,P st (cid:18) z = r / yv √ D R (cid:19) (cid:39) √ π √ z exp − (cid:114) z . (26)Figure 7(b) shows a plot of P st ( y ) versus y for differentvalues of r (cid:29) D R obtained from numerical simulationsalong with the analytical predictions.To investigate the crossover between the limiting cases( r (cid:28) D R and r (cid:29) D R ), we use numerical simulations.Figure 7(c) shows a plot of P st ( y ) versus y for differ-ent values of D R and fixed r = 1 . As expected, the di-vergence near the origin disappears as D R is increased.Moreover, we see that, with increasing D R , the width ofthe distribution first increases, and then decreases again,consistent with Eq. (6). IV. ABP WITH POSITION RESETTING
In this Section we focus on the behaviour of the ABPunder resetting protocol II, i.e. , the position-resetting.In this case, the particle position is reset to the origin x = y = 0 with rate r , but the orientation is not affectedby the resetting events. As before, we consider that theparticle starts from the origin with θ = 0 at time t = 0 . Hence, at any time t, the θ distribution remains Gaussianwith zero-mean and variance 2 D R t. Our objective is to find the position distribution P ( x, y, t ). We can derive a renewal equation for the samein the following way. Let us consider the evolution of theparticle trajectory during the interval [0 , t ] . If there are noresetting events during this interval, the position evolvesunder ordinary active Brownian motion. For the trajec-tories with at least one resetting, let us consider thatthe time elapsed since the last resetting event is givenby s. Then, the position at time t is dictated by the freeABP evolution during this interval s, but starting fromsome arbitrary orientation θ t − s , which itself is dictatedby the Brownian motion of θ. Then the position distri-bution is obtained by integrating over all possible valuesof 0 ≤ s ≤ t, and θ t − s ∈ [ −∞ , ∞ ] . Combining all thesecontributions, we get the renewal equation, P ( x, y, t ) = e − rt P ( x, y, t ) + r (cid:90) t d s e − rs × (cid:90) ∞−∞ d θ P θ ( x, y, s ) e − θ DR ( t − s ) (cid:112) πD R ( t − s ) , (27)where we have used the notation P θ ( x, y, s ) to denotethe probability that the free ABP is at ( x, y ) at time s, starting from an initial orientation θ at s = 0 . Thestructure of the above renewal equation is different thanthe same obtained for the position-orientation resetting[see Eq. (2)], and the behaviour is also expected to bedifferent.The renewal equations for marginal distribution canbe obtained by integrating over either x or y. We willinvestigate the stationary marginal position distributionslater in Sec. IV B. In the following, we first look at themoments to get an idea about the nature of the motion.
A. Moments
The time-evolution of the moments of the position canbe obtained from Eq. (27) in a straightforward manner.Let us first look at the moments of the x -position. Mul-tiplying Eq. (27) by x n and integrating over both x and y, we get a renewal-like equation for the n − th moment t -2 σ x2 r = 0.01 r = 0.1 r = 1 r = 5 t -2 σ y2 r = 0.01 r = 0.1 r = 1 r = 5(a) (b) FIG. 8: Position resetting: Variance of x -coordinate (a) and y -coordinate (b) as functions of time t for different values of r and a fixed D R = 1 . The symbols correspond to the data fromnumerical simulations whereas the solid black lines indicatethe analytical results [see Eqs. (D2) and (D4)]. v = 1 here. of the x -component of the position, (cid:104) x n ( t ) (cid:105) = e − rt (cid:104) x n ( t ) (cid:105) + r (cid:90) t d s e − rs (cid:90) d θ (cid:104) x n ( s ) (cid:105) θ e − θ DR ( t − s ) (cid:112) πD R ( t − s ) . (28)Here (cid:104) x n ( s ) (cid:105) θ denotes the corresponding n th moment forthe free ABP, starting from the origin, but oriented alongsome arbitrary direction θ and (cid:104) x n ( t ) (cid:105) , as before, de-notes the moment starting from θ = 0 . Similarly, wecan also write an equivalent renewal equation for the y -moments. The free ABP moments appearing in Eq. (28)can be calculated exactly, and Appendix A provides ex-plicit form for n = 1 and 2 . We use these expressionsto calculate the first two moments of x and y for thisposition-resetting protocol.Using Eqs. (A5) and (A6) in Eq. (28), we get the time-evolution of the average position, (cid:104) x ( t ) (cid:105) = v D R − r ( e − rt − e − D R t ) , (29)and (cid:104) y ( t ) (cid:105) = 0 . Note that, for r = D R the above equationremains well defined, with (cid:104) x ( t ) (cid:105) = v te − D R t . Clearly, theaverage position approaches the origin in the stationarystate t → ∞ . At short times, i.e. , for t (cid:28) min( r − , D − R ) , (cid:104) x ( t ) (cid:105) = v t − v ( r + D R ) t + O ( t ) , (30)indicating that the resetting does not change the effectivevelocity to the leading order.The second moment of the x and y -components canalso be calculated exactly using Eq. (28). The explicitexpressions are provided in Eqs. (D1) and (D4) in theAppendix D, here we quote the short-time and long-timebehaviour of Mean squared displacements of x and y com-ponents. At short-times t (cid:28) min( r − , D − R ) , we have, σ x ( t ) = v rt + v
12 (4 D R − D R r − r ) t + O ( t ) ,σ y ( t ) = 2 v D R t − v D R (5 D R + 2 r ) t + O ( t ) , (31)indicating a superdiffusive behaviour. Moreover, eventhough both the variances show t growth in this regime,0 -10 -5 0 5 10 x -3 -2 -1 P st ( x ) r = 0.01 r = 0.05 r = 0.1 r = 1 -10 -5 0 5 10 x -6 -3 P st ( x ) r = 1 r = 2 r = 5 r = 10 -4 -3 -2 -1 x P st ( x ) π r -log(| x |) (a) (b) -6 -3 0 3 6 x -4 -2 P st ( x ) D R = 0.01 D R = 0.05 D R = 0.1 D R = 1 D R = 10 (c) FIG. 9: Position resetting: Stationary x -marginal probability distribution (a) Plot of P st ( x ) versus x for different values of r in the regime r (cid:28) D R with D R = 10 . (b) P st ( x ) versus x in the regime r (cid:29) D R with D R = 0 . . Inset shows the logarithmicdivergence near the origin [see Eq. (38)]. (c) Plot of P st ( x ) covering both the limiting cases with r = 1 and for different valuesof D R . In all the plots the numerical simulation results are indicated by symbols and the solid black lines indicate the analyticalpredictions; see Eq. (35) for (a) and Eq. (37) for (b). Panel (c) shows that, for r (cid:29) D R the distribution becomes independentof D R . v = 1 here. the coefficients are different, which is a signature of theanisotropy present in the short-time regime. On the otherhand, at late times t (cid:29) max( r − , D − R ) , , both σ x ( t ) and σ y ( t ) reach the same stationary value, σ x = σ y = v r ( D R + r ) , (32)indicating that the anisotropy disappears in the steadystate. Figure 8(a) and (b) show plots of σ x ( t ) and σ y ( t )for different values of r along with the same obtainedfrom numerical simulations.In the next section we discuss the stationary probabil-ity distribution for this position-resetting protocol. B. Marginal position distribution
In the presence of position resetting only, the positiondistribution satisfies the renewal equation (27). As be-fore, we focus on the stationary distribution, which isobtained by taking t → ∞ limit. Clearly, the first termdrops off in this limit. In the second term, the presenceof the e − rs implies that the dominant contribution of theintegrand comes from the regime s < r − . For any fi-nite r, then, in the limit of large t, t − s (cid:39) t, and theGaussian factor becomes flat. Now, since P θ ( x, y, s ) is aperiodic function of θ, we can reduce the θ -integral overone period, say, to the interval [ − π, π ] where θ is dis-tributed uniformly. The stationary distribution can thenbe expressed as, P st ( x, y ) = r π (cid:90) ∞ d s e − rs (cid:90) π − π d θ P θ ( x, y, s ) . (33)The θ -integration makes the stationary distributionisotropic and it suffices to look at the marginal distri-bution along x -axis only. Integrating over y, we get fromEq. (33), P st ( x ) = r π (cid:90) ∞ d s e − rs (cid:90) π − π d θ P θ ( x, s ) . (34) We proceed as in the previous section, looking at thetwo limiting cases, namely, r (cid:28) D R and r (cid:29) D R . Wealso follow the same reasoning outlined in the previoussection, and identify the region which contributesdominantly to the integral in (34) in the two limitingcases.
Small resetting rate ( r (cid:28) D R ): In this case, the dom-inant contribution to the integral (34) comes from thelong-time behaviour of free ABP distribution P θ ( x, s ) . Atlong-times s (cid:29) D − R , the anisotropy disappears, and thedistribution does not depend on the initial orientation θ. In fact, as mentioned in the previous section, to the lead-ing order the long-time distribution is a Gaussian (seeAppendix A 2). Using this Gaussian form for P θ ( x, s ) inEq. (34), we get an exponential stationary distribution, P st ( x ) = 1 v (cid:114) rD R (cid:20) − (cid:112) rD R | x | v (cid:21) , (35)which is same as in the r (cid:28) D R regime for the position-orientation resetting case.Figure 9(a) compares the above prediction with thedata from numerical simulations for a set of (small)values of r and a fixed D R . An excellent match over alarge range of r illustrates the validity of Eq. (35), alongwith the underlying assumptions. Large resetting rate ( r (cid:29) D R ): In this case, the sta-tionary distribution is dominated by the contributionsfrom the short-time trajectories of the free ABP, butstarting from an arbitrary angle θ. As the behaviour offree ABP is ballistic at short-times s (cid:28) D − R , as a firstapproximation we can use (see Appendix A 2), P θ ( x, s ) (cid:39) δ ( x − v s cos θ ) . (36)Using the above equation in (34), and performing theintegrals (see Appendix E for details), we get, P st ( x ) = rπv K (cid:18) r | x | v (cid:19) , (37)1where K ( w ) is the modified Bessel function of secondkind [61]. Interestingly, within this approximation, thestationary distribution does not depend on the rotationaldiffusion constant D R at all in this large r limit. Thisis in contrast to the position-orientation resetting, wherethe limiting distribution depends on both r and D R . Fig-ure 9(b) shows a plot of P st ( x ) predicted in Eq. (37) fora set of (large) values of r and a fixed D R along with thesame obtained from numerical simulations; the excellentagreement confirms our analytical prediction.It is interesting to look at the asymptotic behaviour ofthe stationary distribution given in Eq. (37). Expanding K ( w ) near w = 0, we find that, the distribution showsa logarithmic divergence near the origin, P st ( x ) = − rπv log | x | + O (1) . (38)The inset in Fig. 9(b) illustrates this logarithmic diver-gence. On the other hand, for large x (cid:29) v /r the distri-bution falls off exponentially, P st ( x ) (cid:39) (cid:114) r v | x | exp (cid:20) − r | x | v (cid:21) . (39)It should be mentioned that we have restricted to theleading order approximate forms for the free ABP to cal-culate the position distribution in both the limiting sce-narios. We can improve the range of validity (in r ) ofthe analytical predictions by using next order corrections.However, in that case the integrals cannot be evaluatedanalytically and the qualitative behaviour remains thesame. Hence we skip this exercise here.We use numerical simulations to investigate thecrossover of the stationary distribution between the twolimiting cases discussed above. Figure 9(c) shows a plotof P st ( x ) for a fixed r and a range of values of D R ; as D R is increased from the regime D R (cid:28) r, the divergencenear the origin disappears, and the distribution crossesover to the exponential behaviour. The width of the dis-tribution also decreases continuously as D R is increased,as expected from Eq. (32). V. ABP WITH ORIENTATION RESETTING
In this Section we consider the third resetting proto-col where the orientation θ resets to θ = 0 with rate r while the position does not. In this case the position dis-tribution does not satisfy any renewal equation directlybut the θ -distribution does. Let P ( θ, t | θ (cid:48) , t (cid:48) ) denote theprobability that the orientation takes the value θ at time t given that it was θ (cid:48) at an earlier time t (cid:48) . P ( θ, t | θ (cid:48) , t (cid:48) )satisfies a renewal equation [13], P ( θ, t | θ (cid:48) , t (cid:48) ) = e − r ( t − t (cid:48) ) P ( θ, t | θ (cid:48) , t (cid:48) )+ r (cid:90) ( t − t (cid:48) )0 d s e − rs P ( θ, s | , , (40) -1 t -4 -2 σ x r =0.1 r =1.0 r =2.0 -1 t -2 σ y r =0.1 r =1.0 r =2.0 (a) (b) t t tt FIG. 10: Orientation resetting: Plot of σ x (a) and σ y (b) ver-sus time t for different values of r with D R = 1 and v = 1 Thesolid lines correspond to the analytical predictions Eqs. (F6)and (F7). The red dashed lines indicate the predicted be-haviour in the short-time and long-time regimes. where P ( θ, t | θ (cid:48) , t (cid:48) ) denotes the propagator for the stan-dard Brownian motion, given by Eq. (A1). At long-times,the orientation reaches a stationary state with an expo-nential distribution although the position does not. Asbefore, we look at the moments of x and y components,and the corresponding marginal position distributions. A. Moments
The Langevin equations (1) can be formally integratedto write, x ( t ) = v (cid:90) t d s cos θ ( s ) ,y ( t ) = v (cid:90) t d s sin θ ( s ) , (41)where we used the initial condition x (0) = y (0) = 0 . To calculate the position moments we need to know themean and the auto-correlations of cos θ and sin θ underresetting which can be calculated using the propagator(40). The details of this calculation is provided in theAppendix F, here we just quote the results. As in all theprevious cases, (cid:104) y ( t ) (cid:105) vanishes at all times due to sym-metry. Along x -axis, however, the average displacementis given by, (cid:104) x ( t ) (cid:105) = v r + D R (cid:20) rt + D R r + D R (cid:18) − e − ( r + D R ) t (cid:19)(cid:21) , (42)= v t for t (cid:28) ( r + D R ) − ,v rtD R + r for t (cid:29) ( r + D R ) − . (43)Clearly, the x -motion is ballistic at short-times withthe velocity v , which is reminiscent of the free ABP.Unlike the previous cases considered here, the effectivevelocity remains non-zero at late times, however, its valuechanges to v eff = v rD R + r due to the presence of the reset-ting.To understand the fluctuations around the mean po-sition, we also look at the mean-squared displacement.The exact and long expressions for (cid:104) x ( t ) (cid:105) and (cid:104) y ( t ) (cid:105) D R D eff x r = 0.5 r = 1 r = 2 D R D eff y r = 0.5 r = 1 r = 2 (a) (b) FIG. 11: Orientation resetting: Plots of D x eff (a) and D y eff (b) versus D R for different values of r . See Eqs. (46) for theanalytical expressions. We have taken v = 1 . are provided in Eqs. (F5) and (F7) respectively in Ap-pendix F. These analytical predictions are compared withnumerical simulation results in Fig. 10 for different val-ues of r and a fixed D R . As in the previous cases, wesee that both the x and y -variances show a crossoverfrom a superdiffusive to a diffusive behaviour as time t increases. To understand the nature of these crossovers,we look at the short-time and long-time behaviours of themean-square displacements. At very short-times, i.e. , for t (cid:28) ( r + D R ) − , we have, σ x ( t ) = v D R t − v D R (14 D R − r ) t + O ( t ) ,σ y ( t ) = 2 v D R t − v D R (5 D R + 2 r ) t + O ( t ) . (44)To the leading order, this behaviour is same as that offree ABP with strong anisotropy between x and y mo-tions [44]. The effect of resetting appears at higher or-ders, and it introduces an additional anisotropy. Thisis expected, as the resetting configuration θ = 0 is alsostrongly anisotropic. The effect of this anisotropy sus-tains at late-times also – even though both x and y mo-tions become diffusive, i.e. ,lim t →∞ σ x (cid:39) D x eff t, lim t →∞ σ y (cid:39) D y eff t, (45)the effective diffusion constants remain very different, D x eff = v D R (2 D R + 5 r )(4 D R + r )( D R + r ) ,D y eff = 2 v D R ( D R + r )(4 D R + r ) . (46)Figure 11 shows plots of D x eff and D y eff as functions of D R , for a set of values of r. It is interesting to note that theseeffective diffusion constants are non-monotonic in D R –for a fixed r, D x,y eff reach their corresponding maximumvalues for some intermediate values of D R which increasesas r is increased. B. Marginal position distributions
To understand the behaviour of the position distribu-tion, let us first look at a trajectory with n resetting events during the interval [0 , t ]. Let us also assume that t i denotes the interval between the i and ( i − t, the position ( x ( t ) , y ( t )) canbe expressed as a sum of position increments over theintervals t i , x ( t ) = n +1 (cid:88) i =1 x ( t i ) , (47) y ( t ) = n +1 (cid:88) i =1 y ( t i ) . (48)Let us remember that, in between the resetting eventsthe system evolves as an ordinary ABP and hence, thefluctuations of x ( t i ) and y ( t i ) follow the distribution P ( x i , y i , t i ) , where we have used the notation x i ≡ x ( t i )and y i ≡ y ( t i )As before, we focus on the marginal distributions of x and y -components separately. From Eq. (47), the x -distribution in the presence of orientation resetting canbe formally written as, P ( x, t ) = ∞ (cid:88) n =0 r n e − rt (cid:90) n +1 (cid:89) i =1 d t i d x i P ( x i , t i ) × δ (cid:18) x − n +1 (cid:88) i =1 x i (cid:19) δ (cid:18) t − n +1 (cid:88) i =1 t i (cid:19) , (49)where P ( x i , t i ) denotes the probability that, in the ab-sence of resetting, the ABP has a displacement x i duringthe time-interval t i , starting from θ = 0 . The y -marginaldistribution also has a similar form, P ( y, t ) = ∞ (cid:88) n =0 r n e − rt (cid:90) n +1 (cid:89) i =1 d t i d y i P ( y i , t i ) × δ (cid:18) y − n +1 (cid:88) i =1 y i (cid:19) δ (cid:18) t − n +1 (cid:88) i =1 t i (cid:19) , (50)where P ( y i , t i ) denotes the probability that the y -component of the position of the free ABP has a dis-placement y i during the interval t i , staring from θ = 0 . Let us note that P ( x i , t i ) and P ( y i , t i ) have differentfunctional forms, in particular for small t i , even thoughwe have used the same letter for notational simplicity.It is hard to compute the marginal distributions fromthe above equations exactly, as explicit form for the po-sition distributions in the absence of resetting are notknown. However, as we will see below, we can still under-stand the different behaviours in the short and long-timeregimes.Let us first focus at the long-time regime. As indicatedby the moments, we expect a diffusive motion for both x and y -components in this regime. For simplicity, let usfirst consider the x -component. From Eq. (47), we seethat the net displacement along x -direction is given bya sum of n + 1 random variables, namely, the displace-ments during the intervals t i . Since, after each reset, the3 -4 -2 0 2 4 ( x- µ x )/ σ x σ x P ( x,t ) D R =0.5 D R =1.0 D R =2.0 FIG. 12: Orientation resetting: Plot of the scaled marginal x -distribution at a long-time t = 500 for r = 1 and differentvalues of D R . The solid black line shows the standard normaldistribution. v = 1 here. orientation θ is brought back to its initial value, and thetime-evolution starts afresh, the variables x ( t i ) are inde-pendent and identically distributed (of course, the dura-tion t i are different). Even though the distribution of x i is not known explicitly, its moments are all finite. Over alarge time interval t, the number n of the resetting eventsis typically large, with (cid:104) n (cid:105) = rt. For t (cid:29) r − then x ( t )is a sum of a large number n of independent and iden-tically distributed random variables. From central limittheorem, we can then expect that x ( t ) has a Gaussiandistribution, P ( x, t ) = 1 (cid:112) πσ x ( t ) exp (cid:20) − ( x − µ x ( t )) σ x ( t ) (cid:21) , (51)where µ x ( t ) = (cid:104) x ( t ) (cid:105) and σ x ( t ) are the mean and vari-ance given by Eqs. (43) and (45) (with large t ). Note thatthis prediction is independent of the value of r ; for each r, there exists some t (cid:29) r − above which we expect a Gaus-sian distribution, albeit with different r -dependent meansand variances. Figure 12 shows a plot of σ x ( t ) P ( x, t ) vs( x − µ x ( t )) /σ x ( t ) for r = 1 , t = 500 and different valuesof D R ; a perfect collapse verifies the prediction.The same argument can be applied to y ( t ) , fromEq. (48), and we expect, P ( y, t ) = 1 (cid:113) πσ y ( t ) exp (cid:20) − y σ y ( t ) (cid:21) , (52)where σ y ( t ) is the large t -behaviour obtained fromEq. (45). We also observe a perfect collapse for P ( y, t ),as depicted in Figure 13 which verifies our prediction.In the short-time regime, the average number of reset-ting events is small and we can expect small r contribu-tions to dominate. From Eq. (49), one can adopt a per-turbative approach, that is, for small r we compute thedistribution at short-times. In fact, to obtain the leadingorder correction introduced by the resetting, we truncatethe sum after n = 1 , which is equivalent to keeping linearorder in r (apart from the e − rt factor). We then get, P ( x, t ) = e − rt [ P ( x, t ) + rP ( x, t ) + O ( r )] . (53) -4 -2 0 2 4 y / σ y σ y P ( y,t ) D R =0.1 D R =1.0 D R =2.0 FIG. 13: Orientation resetting: Plot of the scaled marginal y -distribution at a long-time t = 500 for r = 1 and differentvalues of D R obtained from numerical simulations. The solidblack line shows the standard normal distribution. v = 1 here. Here P ( x, t ) is the short-time marginal x -distribution forthe active Brownian particle without resetting given inEq. (A13) and P ( x, t ) is the leading order correction dueto resetting, P ( x, t ) = (cid:90) t d t (cid:90) b ( t ) a ( t ) d x P ( x , t ) P ( x − x , t − t ) . (54)The limits on the x -integral are determined from thecondition that P ( x, t ) is non-zero only in the region − t ≤ x ≤ t and are given by, a ( t ) = max( − t , x − t + t ) ,b ( t ) = min( t , x + t − t ) . (55)Using Eq. (A13) the integrals in Eq. (54) can be eval-uated numerically with arbitrary accuracy. The result-ing P ( x, t ) , which is expected to be valid in the regime t (cid:28) D − R , is plotted in Fig. 14 for different (small) valuesof D R and a fixed (small) values of r and t along withthe same obtained from numerical simulations. The ana-lytical prediction matches well with the results from sim-ulation indicating that the perturbative approach worksfairly well in this regime. The position distribution ap-pears similar in shape to that in the absence of resetting,with a peak near x = v t. However, quantitatively theyare different, as can be seen from the plot — we haveincluded the corresponding curves for r = 0 as dashedlines for easy comparison. Clearly, the effect of resettingbecomes more pronounced away from the peak.We follow the same perturbative procedure to com-pute the y -marginal distribution also. From Eq. (50), wewrite, to the leading order in r,P ( y, t ) = e − rt [ P ( y, t ) + rP ( y, t ) + O ( r )] , (56)with, P ( y, t ) = (cid:90) t d t (cid:90) ˜ b ( t )˜ a ( t ) d y P ( y , t ) P ( y − y , t − t ) . (57)4 x -2 -1 P ( x,t ) D R = 0.01 D R = 0.02 D R = 0.03 FIG. 14: Orientation resetting: Plot of the marginal x -distribution at a short-time t = 2 for r = 0 . D R . We have taken v = 1 . As before, the integration limits are obtained from thecondition that − t ≤ y ≤ t and t − t ≤ y − y ≤ t − t , ˜ a ( t ) = max( − t , y − t + t )˜ b ( t ) = min( t , y + t − t ) . (58)We obtain P ( y, t ) by numerically evaluating the in-tegral in Eq. (57). As before, we restrict ourselves inthe regime t (cid:28) D − R , so that the short-time expressionof P ( y, t ) [see Eq. (A15)] is applicable. The resultingmarginal distribution P ( y, t ) is plotted in Fig. 15 for aset of values of D R with a fixed (small) r = 0 . t = 1 along with the same obtained from numerical sim-ulations. The distribution has a single peak at the origin,similar to the r = 0 case (indicated by dashed lines) inshape. However, the correction due to resetting makesit non-Gaussian, the difference with r = 0 case is clearlyvisible near the peaks. VI. CONCLUSIONS
We study the position distribution of an active Brown-ian particle in 2D under stochastic resetting. An ABP ischaracterized by its position as well as an internal orien-tation. We show that depending on whether the resettingprotocol affects the position degrees of freedom or theorientational degree, the ABP shows a wide range of richbehaviour. In particular, we study three different reset-ting protocols, namely, resetting both position and ori-entation to their initial value, resetting only the position,and resetting only the orientation. We find that in thefirst two cases the position reaches stationary states. Weshow that the interplay between the time-scales due toresetting and the rotational diffusion leads to a set of dif-ferent regimes – depending on whether the resetting rate r is smaller or larger than the rotational diffusion con-stant D R , the stationary distributions take very differentshape. Using renewal approach, we compute exactly themarginal distributions of the x and y -components in thelimiting cases r (cid:28) D R and r (cid:29) D R . -0.4 -0.2 0 0.2 0.4 y P ( y,t ) D R =0.01 D R =0.02 D R =0.05 FIG. 15: Orientation resetting: Plot of the marginal y -distribution at a short-time t = 1 for r = 0 . D R . The solid black lines correspond to the analyt-ical prediction Eq. (56) and the dashed red lines correspondto Gaussian ( r = 0) curves. We have taken v = 1 . In the first case, i.e. , when both the position and ori-entation are reset to their initial value, we find that, forsmall resetting rates r (cid:28) D R , the marginal distributions P st ( x ) and P st ( y ) are exponential in nature, with thesame decay exponent. On the other hand, for r (cid:29) D R , the position distribution becomes strongly anisotropic.The marginal x -distribution is non-zero only for x > x → + . The y -distribution, which is symmetric, shows a very differentbehavior, with an algebraic divergence near the origin( | y | →
0) and a compressed exponential decay at thetails.For the position-resetting case, the position distribu-tion is isotropic for all values of r and D R . For r (cid:28) D R the distribution turns out to be exponential in nature.For large values of r (cid:29) D R , the position distributionshows a logarithmic divergence near the origin, while de-caying exponentially at the tails.In the third case, i.e. , when the resetting protocol af-fects only the orientation of the ABP, the position ofthe particle does not reach a stationary state, but con-tinues to increase along the x -direction with an effec-tive velocity. However, the nature of the motion changesfrom ballistic, at short-times, to diffusive, at late times( t (cid:29) ( r + D R ) − ). We show that, at late times, the typi-cal position fluctuations around the mean are character-ized by Gaussian distributions for both x and y compo-nents, albeit with different effective diffusion constants.At short-times, the position distribution remains stronglynon-Gaussian, which we characterize using a perturba-tive approach for small resetting rates.The resetting of a particle in position space can bethought of as the effect of switching on and off an ex-ternal trap, in the spirit of Ref. [23, 28]. On the otherhand, the orientation resetting can be envisaged as theeffect of an external magnetic field on magnetic activeparticles [62–65], which is switched on at random times.In general, it is interesting to study what happens when5the position and orientation resetting can occur indepen-dently of each other. Another obvious open question ishow the persistence properties of the ABP are affectedin the presence of such resetting mechanisms. It wouldbe also interesting to see how introduction of resettingmechanism affects other active particle models and if ageneral picture emerges. The behaviour of active parti-cles under different resetting protocols present anotherintriguing set of questions. Acknowledgments
V. K. acknowledges support from Raman Research In-stitute where he worked as a visiting student and wherethis work was carried out. O. S. acknowledges the In-spire grant from DST, India. U. B. acknowledges sup-port from Science and Engineering Research Board, Indiaunder Ramanujan Fellowship (Grant No. SB/S2/RJN-077/2018).
Appendix A: Brief review of active Brownianmotion in 2D
For the sake of completeness we provide a brief re-view of the free ABP dynamics in this Appendix. In theabsence of resetting the position and orientation of theABP evolves following the Langevin equation (1). Weassume that at time t = 0 the particle starts from theorigin x = y = 0 , oriented along some arbitrary direction θ = θ . As the orientation evolves following an ordinaryBrownian motion the probability that the orientation is θ at time t, given that it was θ (cid:48) at some earlier time t (cid:48) is, P ( θ, t | θ (cid:48) , t (cid:48) ) = 1 (cid:112) πD R ( t − t (cid:48) ) exp (cid:20) − ( θ − θ (cid:48) ) D R ( t − t (cid:48) ) (cid:21) . (A1)In the following we quote the results for the momentsand distribution of the position components x and y ofthe ABP.
1. Moments of the position components
The moments of the position coordinate x, y can be ob-tained in a straightforward manner [44] using the Brow-nian propagator for the orientation θ given in Eq. (A1).Here we compute the explicit expressions for the first twomoments for arbitrary values of the initial orientation θ . Integrating the Langevin equation (1) and taking averageover all possible trajectories, we have, (cid:104) x ( t ) (cid:105) θ = v (cid:90) t d s (cid:104) cos θ ( s ) (cid:105) θ , (cid:104) y ( t ) (cid:105) θ = v (cid:90) t d s (cid:104) sin θ ( s ) (cid:105) θ , (A2) where we have used the superscript θ to denote the ini-tial orientation. From Eq. (A1) we have, (cid:104) cos θ ( s ) (cid:105) θ = (cid:90) d θ cos θ ( s ) e − ( θ − θ DRs √ πD R s = cos θ e − D R s , (A3)and similarly, (cid:104) sin θ ( s ) (cid:105) θ = sin θ e − D R s . (A4)The average positions can be computed using the aboveequations in Eqs. (A2), (cid:104) x ( t ) (cid:105) θ = v D R cos θ (cid:0) − e − D R t (cid:1) , (cid:104) y ( t ) (cid:105) θ = v D R sin θ (cid:0) − e − D R t (cid:1) . (A5)Equations (A5) are used in Sec. IV A to compute thefirst position moments in the presence of the positionresetting.For computing the moments in the presence of theposition-orientation resetting, we need the ABP momentsfor θ = 0 . In this case Eqs. (A5) reduce to, (cid:104) x ( t ) (cid:105) = v D R (cid:0) − e − D R t (cid:1) , (cid:104) y ( t ) (cid:105) = 0 . (A6)which have been used in Sec. III A to compute the averagepositions.Next, we look at the second moments. From Eq. (1),we can write, (cid:104) x ( t ) (cid:105) θ = 2 v (cid:90) t d s (cid:90) s d s (cid:48) (cid:104) cos θ ( s ) cos θ ( s (cid:48) ) (cid:105) θ , (cid:104) y ( t ) (cid:105) θ = 2 v (cid:90) t d s (cid:90) s d s (cid:48) (cid:104) sin θ ( s ) sin θ ( s (cid:48) ) (cid:105) θ . (A7)The two-point correlations (cid:104) cos θ ( s ) cos θ ( s (cid:48) ) (cid:105) θ and (cid:104) sin θ ( s ) sin θ ( s (cid:48) ) (cid:105) θ can be calculated exactly usingEq. (A1); for s > s (cid:48) we get, (cid:104) cos θ ( s ) cos θ ( s (cid:48) ) (cid:105) θ = 12 e − D R ( s − s (cid:48) ) (cid:20) e − D R s (cid:48) cos 2 θ (cid:21) , (cid:104) sin θ ( s ) sin θ ( s (cid:48) ) (cid:105) θ = 12 e − D R ( s − s (cid:48) ) (cid:20) − e − D R s (cid:48) cos 2 θ (cid:21) . (A8)Now, substituting Eq. (A8) in Eq. (A7) and evaluatingthe integrals, we get, (cid:104) x ( t ) (cid:105) θ = v D R (cid:20) D R t + e − D R t − e − D R t − e − D R t ) cos 2 θ (cid:21) , (cid:104) y ( t ) (cid:105) θ = v D R (cid:20) D R t + e − D R t − − (3 + e − D R t − e − D R t ) cos 2 θ (cid:21) . (A9)These expressions have been used in Sec. IV A to computethe variances in the presence of position resetting.Once again, for calculating the variances in theposition-orientation case we need the expressions for θ = 0, (cid:104) x ( t ) (cid:105) = v tD R + v D R (cid:2) e − D R t + 8 e − D R t − (cid:3) , (cid:104) y ( t ) (cid:105) = v tD R − v D R (cid:2) e − D R t − e − D R t + 15 (cid:3) . (A10)which were obtained in [44]. The above results are usedin Appendix. B to obtain Eqs. (B1) and (B3). It is alsostraightforward to calculate the third moment of x ( t ) us-ing Eq. (A1). For θ = 0 it turns out to be, (cid:104) x ( t ) (cid:105) = 1240 D R (cid:2) e − D R t (120 D R t + 169) − e − D R t − e − D R t + 720 D R t − (cid:3) . (A11)This expression is used in Eq. (17) to compute the skew-ness; see also Appendix B.
2. Position Distribution
In the absence of resetting, the position distributionof the ABP is given by P ( x, y, t ) = (cid:82) d θ P ( x, y, θ, t )where P ( x, y, θ, t ) denotes the probability that theABP has the position ( x, y ) and orientation θ at time t. P ( x, y, θ, t ) evolves according to the Fokker-Planckequation, ∂ P ∂t = − v (cid:20) cos θ ∂ P ∂x + sin θ ∂ P ∂y (cid:21) + D R ∂ P ∂θ . (A12)Formally, the above equation can be solved using Fouriertransformation with respect to position coordinates andthe Fourier transform of P ( x, y, θ, t ) can be expressedin terms of an infinite series of Matthieu functions [66].Unfortunately the Fourier transform cannot be invertedanalytically, and no closed form expression for the posi-tion distribution is available. However, marginal positiondistributions for the x and y components, starting from θ = 0 , in short-time and long-time regimes are knownseparately. For the sake of completeness, we quote theseexpressions here.In the short-time regime ( t (cid:28) D − R ), the marginal x -distribution can be expressed in a scaling form, P ( x, t ) = 1 v D R t f x (cid:18) v t − xv D R t (cid:19) , (A13) where the scaling function is given by, f x ( u ) = 12 √ πu ∞ (cid:88) k =0 ( − k (4 k + 1)2 k (cid:18) kk (cid:19) e − (4 k +1)28 u . (A14)The y -marginal distribution, on the other hand, has aGaussian form in this short-time regime, P ( y, t ) = √ v √ πD R t exp (cid:20) − y v D R t (cid:21) . (A15)At late times t (cid:29) D − R the anisotropy goes away, andit has been shown in Ref. [52] that in this regime both x and y marginal distribution admits a large deviationform, which is quoted in Eq. (10). Initial orientation θ (cid:54) = 0 : Next we look at marginalposition distribution starting from any arbitrary θ (cid:54) = 0 . In this case, we can substitute θ ( t ) = θ + φ ( t ) in Eq. (1)where φ ( t ) undergoes a standard Brownian motion with φ (0) = 0 . At short-times φ ( t ) ∼ √ t is small, and to the leadingorder we can approximate sin φ ( t ) (cid:39) φ ( t ) and cos φ ( t ) (cid:39) . In this regime, the Langevin equations (1) reduce to,˙ x ( t ) (cid:39) v [cos θ − φ ( t ) sin θ ] , ˙ y ( t ) (cid:39) v [sin θ + φ ( t ) cos θ ] . (A16)Clearly, for non-zero θ both x and y -components havesystematic drifts. To a first approximation, the positiondistribution can then be written as, P θ ( x, y, t ) = δ ( x − v t cos θ ) δ ( y − v t sin θ ) , (A17)where we have used the superscript θ to denote theinitial orientation. The above expression, when inte-grated over y, gives the x -marginal distribution quotedin Eq. (36). Note that here the fluctuation of the orienta-tion is completely neglected. A better approximation is,of course, when the effect of D R is included, in which casethe marginal distributions would be Gaussian. However,as shown in the Sec. IV B, Eq. (A17) suffices for comput-ing the stationary distribution in the r (cid:29) D R limit forthe position resetting.In the long-time limit t (cid:29) D − R , on the other hand, theposition distribution does not depend on the initial valueof orientation and we expect the typical fluctuations tobe Gaussian in nature, as given by Eq. (11), P θ ( x, t ) = (cid:115) D R πv t exp (cid:20) − D R x v t (cid:21) . (A18) Appendix B: Exact computation of moments forposition-orientation resetting
In this Appendix we present the exact analytical ex-pressions for the higher moments of the x and y compo-nents of position in the presence of position-orientation7resetting. We can calculate the second moment of x ( t )from Eq. (3) as, (cid:104) x ( t ) (cid:105) = 2 v (2 D R + r ) r ( D R + r )(4 D R + r )+ v D R e − rt (cid:20) e − D R t D R + r + 2 e − D R t D R + r − r (cid:21) . (B1)Using Eqs. (4) and (B1) we obtain the variance σ x = (cid:104) x ( t ) (cid:105) − (cid:104) x ( t ) (cid:105) , σ x = v (4 D R + 2 rD R + r ) r (4 D R + r )( D R + r ) + v e − rt (cid:34) e − D R t D R (4 D R + r ) − e − (2 D R + r ) t ( D R + r ) + 2(4 D R + r ) e − D R t D R ( D R + r ) − rD R (cid:35) . (B2)The short-time and long-time limiting behaviour ob-tained from the above equation are quoted in the maintext.Similarly, we also calculate the variance σ y , which isnothing but the second moment for the y -component.Using Eq. (A10) in the renewal equation, we get, σ y = (cid:104) y ( t ) (cid:105) = 4 v D R r ( D R + r )(4 D R + r ) − v D R e − rt (cid:20) e − D R t D R + r − e − D R t D R + r + 3 r (cid:21) . (B3)To calculate the skewness of P ( x, t ) we need the thirdmoment. Using the expression of (cid:104) x ( t ) (cid:105) given byEq. (A11) along with Eq. (3), we get, (cid:104) x ( t ) (cid:105) = v D R (cid:34) D R (3 D R + r )(6 D R + r ) r ( D R + r ) (4 D R + r )(9 D R + r )+ 5 e − ( D R + r ) t ( D R + r ) (cid:18) D R + 149 r + 120 D R ( D R + r ) t (cid:19) − e − rt (cid:18) r + 16 e − D R t D R + r + 9 e − D R t D R + r (cid:19)(cid:35) . (B4)The exact time-dependent expression for skewness γ canbe obtained using Eqs. (4), (B2) and (B4); we omit therather long expression and quote the stationary value γ st in Eq. (18) obtained by taking the limit t → ∞ . Appendix C: Asymptotic behaviour of P st ( y ) forposition-orientation resetting To find the behaviour of P st ( y ) for small and large val-ues of y, we use the asymptotic expansion of the Kelvinfunctions appearing in Eq. (24). From the series expan-sion near w = 0 , we have,ker / ( w ) = Γ(1 / / w − / + O ( w / ) , kei / ( w ) = − Γ(1 / / w − / + O ( w / ) . (C1)Inserting the above expressions in Eq. (24) along withEq. (23) we get an algebraic divergence of P st ( y ) near y = 0 which is quoted in Eq. (25).On the other hand, for large values of the argument w, we have (see Sec. 10.67 in Ref. [61]),ker / ( w ) = e − w/ √ (cid:114) π w cos (cid:18) w √ π (cid:19) + O (cid:18) w / (cid:19) , kei / ( w ) = e − w/ √ (cid:114) π w sin (cid:18) w √ π (cid:19) + O (cid:18) w / (cid:19) . Using the above expressions along with Eqs. (24) and(23), we get the large z -behaviour of the scaling functionquoted in Eq. (26). Appendix D: Exact computation of moments forposition resetting
In this Appendix we provide the explicit expressionsfor the second moments of the position in presence ofthe resetting protocol II, i.e. , for only position resetting.Using the renewal equation (28) for n = 2 , along withEqs. (A9) and (A10), we get, (cid:104) x ( t ) (cid:105) = v D R (cid:20) D R − r ) e − ( D R + r ) t (3 D R − r )( D R + r ) − D R − r ) e − rt r (4 D R − r )+ D R e − D R t (4 D R − r )(3 D R − r ) + D R r ( D R + r ) (cid:21) . (D1)The variance can be calculated using the above equationalong with Eq. (29) and is given by, σ x ( t ) = v (cid:20) e − ( D R + r ) t (3 D R − r )( D R + r ) (cid:18) D R ( D R − r ) − rD R (cid:19) − e − rt (2 D R − r ) rD R (4 D R − r ) − ( e − rt + e − D R t )( D R − r ) + e − D R t (4 D R − r )(3 D R − r ) + 1 r ( D R + r ) (cid:21) . (D2)To get the behavior in the short-time regime, i.e. , for t (cid:28) min( r − , D − R ) we can use the Taylor series expan-sion around t = 0 . The resulting expansion is quoted inEq. (31). On the other hand, in the t → ∞ limit thevariance reaches the stationary value quoted in Eq. (32).The variance of y ( t ) also satisfies the renewal equation(27), (cid:104) y ( t ) (cid:105) = e − rt (cid:104) y ( t ) (cid:105) + r (cid:90) t d s e − rs × (cid:90) ∞−∞ d θ (cid:104) y ( s ) (cid:105) θ e − θ DR ( t − s ) (cid:112) πD R ( t − s ) , (D3)where (cid:104) y ( t ) (cid:105) , the second moment of ABP starting with θ = 0 is given by Eq. (A10) and (cid:104) y ( t ) (cid:105) θ , the second8moment of ABP starting with arbitrary orientation θ isgiven in Eq. (A9). Using these expressions in Eq. (D3)we have, (cid:104) y ( t ) (cid:105) = v r ( D R + r ) + v (cid:20) e − ( D R + r ) t (3 D R − r )( D R + r ) − e − rt r (4 D R − r ) − e − D R t (4 D R − r )(3 D R − r ) (cid:21) . (D4)The short-time behaviour is quoted in Eq. (31) in themain text. Appendix E: Position resetting: marginaldistribution for r (cid:29) D R In this Appendix we provide the details of the calcu-lation leading to Eq. (37). Substituting P θ ( x, s ) fromEq. (36) in Eq. (34), we get the stationary distribution, P st ( x ) = r π (cid:90) ∞ d s e − rs (cid:90) π − π d θ δ ( x − v s cos θ )= rv π (cid:90) ∞ d s e − rs (cid:90) π d θ | cos θ | δ (cid:18) xv cos θ − s (cid:19) . (E1)Here, in the second step, we have used the fact that cos θ is an even function of θ. Now, for x > , the δ -functioncontributes only when cos θ > , i.e. , 0 ≤ θ ≤ π . Thus,evaluating the s -integral, we have, for x > ,P st ( x ) = rv π (cid:90) π/ d θ | cos θ | exp (cid:20) − rxv cos θ (cid:21) = rv π K (cid:18) rxv (cid:19) . (E2)Here K ( z ) is the modified Bessel function of the secondkind. For x < , on the other hand, the s -integral inEq. (E1) is non-zero only when π/ ≤ θ ≤ π. In this casewe have, P st ( x ) = rv π (cid:90) ππ/ d θ | cos θ | exp (cid:20) − rxv cos θ (cid:21) = rv π K (cid:18) − rxv (cid:19) . (E3)Combining Eqs. (E2) and (E3) we get the completemarginal distribution quoted in Eq. (37). Appendix F: Exact computation of the moments forthe orientation resetting
To compute the moments of the position coordinatesin the presence of the orientation resetting we start fromEq. (41). Taking statistical average over all possible tra-jectories of θ , we get, (cid:104) x ( t ) (cid:105) = v (cid:90) t d s (cid:104) cos θ ( s ) (cid:105) , (cid:104) y ( t ) (cid:105) = v (cid:90) t d s (cid:104) sin θ ( s ) (cid:105) . (F1)The averages appearing on the right hand side can becomputed using the renewal equation (40) for P ( θ, t ) . We have, (cid:104) cos θ ( s ) (cid:105) = (cid:90) ∞−∞ d θ cos θ P ( θ, s )= D R D R + r e − ( D R + r ) s + rD R + r , and (cid:104) sin θ ( s ) (cid:105) = 0 . Using the above expression inEq. (F1) we get the mean x -position which is quoted inEq. (43). Obviously, (cid:104) y ( t ) (cid:105) = 0 . Variance of x ( t ) and y ( t ) : From Eq. (41) we have, (cid:104) x ( t ) (cid:105) = v (cid:90) t d s (cid:90) t d s (cid:48) (cid:104) cos θ ( s ) cos θ ( s (cid:48) ) (cid:105) , (cid:104) y ( t ) (cid:105) = v (cid:90) t d s (cid:90) t d s (cid:48) (cid:104) sin θ ( s ) sin θ ( s (cid:48) ) (cid:105) . (F2)To compute the position moments we first need to calcu-late the auto-correlations appearing in the above equa-tions. Let us first consider the two-time correlation ofcos θ ; for s > s (cid:48) we have, C ( s, s (cid:48) ) ≡ (cid:104) cos θ ( s ) cos θ ( s (cid:48) ) (cid:105) = (cid:90) d θ d θ (cid:48) cos θ cos θ (cid:48) P ( θ, s | θ (cid:48) , s (cid:48) ) P ( θ, s (cid:48) | , , where the propagator P ( θ, s | θ (cid:48) , s (cid:48) ) satisfies the renewalequation (40). Using Eq. (40) in the above equation andperforming the integrals, we get, for s > s (cid:48) ,C ( s, s (cid:48) ) = r ( D R + r ) + 2 D R D R + r e − ( D R + r ) s − D R s (cid:48) + rD R ( D R + r ) (cid:18) e − ( D R + r ) s (cid:48) − e − ( D R + r ) s (cid:19) + D R (2 D R + 5 r )(4 D R + r )( D R + r ) e − ( D R + r )( s − s (cid:48) ) . (F3)Repeating the same exercise for sin θ, we get, (cid:104) sin θ ( s ) sin θ ( s (cid:48) ) (cid:105) = 2 D R D R + r e − ( D R + r )( s − s (cid:48) ) × [1 − e − (4 D R + r ) s (cid:48) ] . (F4)Using the above expressions, it is straightforward to cal-culate the second moments. For the x -component we get, (cid:104) x ( t ) (cid:105) = v ( r + D R ) (cid:20) r t + 2 D R t (2 D R + 9 rD R + r )( D R + r )(4 D R + r ) (cid:21) + 4 v e − (4 D R + r ) t D R + r ) − v D R (2 D R + 16 rD R + 5 r )( D R + r ) (4 D R + r ) + 2 v e − ( D R + r ) t ( D R + r ) (cid:18) rD R t + 4 D R + 33 D R r − r D R + r )(4 D R + r ) (cid:19) . 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