Active Brownian Motion with Directional Reversals
AActive Brownian Motion with Directional Reversals
Ion Santra, Urna Basu,
1, 2 and Sanjib Sabhapandit Raman Research Institute, Bengaluru 560080, India S. N. Bose National Centre for Basic Sciences, Kolkata 700106, India
Active Brownian motion with intermittent direction reversals are common in a class of bacteria like
Myxo-coccus Xanthus and
Pseudomonas putida . We show that, for such a motion in two dimensions, the presence ofthe two time scales set by the rotational diffusion constant D R and the reversal rate γ give rise to four distinctdynamical regimes showing distinct behaviors. We analytically compute the position distribution which shows acrossover from a strongly non-diffusive and anisotropic behavior at short-times to a diffusive isotropic behaviorvia an intermediate regime. We find that the marginal distribution in the intermediate regime shows an expo-nential or Gaussian behavior depending on whether γ is larger or smaller than D R . We also find the persistenceexponents in the four regimes. In particular, we show that a novel persistence exponent α = Understanding the dynamics of active particles like mi-croorganisms, Janus colloids, and nanomotors has both the-oretical and practical importance [1–5]. Active motions arepersistent, break time-reversal symmetry and are intrinsicallynonequilibrium. Theoretical attempts to study such active mo-tion can be broadly classified into two complementary cate-gories: hydrodynamic approach using field theory [2, 6, 7] andmicroscopic approach using simple stochastic particle mod-els [8, 9]. Last decade has seen immense theoretical activitieson both fronts to explain and predict the novel behavior ofactive matter [10–18].Run-and-tumble particles (RTP) [19, 20] and active Brow-nian particles (ABP) [21–23] are the two widely used modelsfor the motion of individual active particles. They describe theoverdamped motion of a particle with constant speed v alonga stochastically evolving internal orientation. For RTP, the in-ternal orientation changes by a finite amount via an intermit-tent ‘tumbling’ event [24, 25]. On the other hand, the orienta-tion undergoes a rotational diffusion motion for ABP [26, 27].Actual bacterial motion, however, is more involved thaneither of these two minimal models. In reality, a bacteriummoves like an ABP with tumbling events resulting in addi-tional finite but random changes of the orientation [28]. A spe-cial scenario is when the tumbling angle distribution is peakedaround 180 ◦ —i.e., most of the tumbling events result in a di-rectional reversal. Such motion has been observed in manydifferent classes of bacteria where the direction reversal isbrought about by some internal mechanism. One well-knownexample is Myxococcus xanthus , a non-flagellate soil bacteriawhich glide on slime trails. In this case, the internal proteinoscillations reverse the cell polarity which causes the direc-tional reversal [29–32].
Pseudomonas putida is another ex-ample, where a reversal of swimming direction occurs due tothe reversal in the rotation direction of polar flagella [33, 34].Similar dynamics is observed in many other marine [35, 36],monotrichous and peritrichous bacteria [37].The minimal model that mimics the active motion withdirectional reversals is a direction reversing active Brown-ian particle (DRABP). In two dimensions, the position xxx =( x , y ) and orientation θ of a DRABP evolve according to the FIG. 1. A typical trajectory of a DRABP generated by discretizing(1), where in a small interval ∆ t , the particle reverses the directionwith probability γ ∆ t and with probability 1 − γ ∆ t performs an ABP: { ∆ x ( t ) , ∆ y ( t ) } = v σ ( t ) ∆ t { cos θ ( t ) , sin θ ( t ) } ; ∆ θ ( t ) = √ D R ∆ t χ ,where χ is drawn from a standard normal distribution. The arrowsindicate the instantaneous velocity vectors. The inset shows a long-time trajectory [which resembles a Brownian trajectory] with the twoend-points marked. See [42] for an animation. Langevin equations,˙ x ( t ) = ζ x ( t ) ≡ v σ ( t ) cos θ ( t ) , (1a)˙ y ( t ) = ζ y ( t ) ≡ v σ ( t ) sin θ ( t ) , (1b)˙ θ ( t ) = p D R η ( t ) , (1c)where D R is the rotational diffusion constant and η ( t ) is aGaussian white noise with h η ( t ) i = h η ( t ) η ( t ) = δ ( t − t ) . The dichotomous noise σ ( t ) alternates between ± γ , triggering the direction reversal [see Fig. 1 fora typical trajectory]. This model has been used to explain theexperimentally measured mean-squared displacement [32, 34]and compute optimal diffusion coefficient [43, 44]. Beyondthat there has been no theoretical progress for the DRABP,even for the single particle position distribution —which de-scribes the density profile for dilute multiparticle systems.For active particles like bacteria the first-passage time toreach a particular target such as food source, weak spot ofthe host or toxins is particularly relevant [38]. For example,certain starvation induced complex processes have been seenin Myxococcus xanthus [30] and
Pseudomonas putida [39], a r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n (a) (b) (c) FIG. 2. Dynamical evolution of the position distribution P ( x , y , t ) for the case γ > D R obtained from numerical simulations. Here we havetaken γ = . D R = .
01 and initial orientation θ = π / . The panels (a), (b) and (c) correspond to t = t =
50 (II) and t = γ < D R the intermediate regime (b) is replaced by regime (III) which looks similar to (c). which in turn would depend on the first-passage properties.However, no theoretical results are available for the first-passage statistics of DRABP. In this context computing thepersistence exponent α which characterizes the power-law de-cay of the probability distribution of the first-passage time, F ( t ) ∼ t − ( + α ) is of broad interest [40, 41].In this Letter, we obtain exact analytical results for the po-sition distribution and persistence exponent of the DRABP.We show that, as time t increases, the presence of the twotime-scales D − R and γ − gives rise to four distinct dynamicalregimes characterized by different dynamical behaviors: (I) t (cid:28) min ( γ − , D − R ) , (II) γ − (cid:28) t (cid:28) D − R , (III) D − R (cid:28) t (cid:28) γ − ,and (IV) t (cid:29) max ( γ − , D − R ) . Note that, only one of the twointermediate regimes (II) and (III) is available depending onthe values of γ and D R .We find that the position distribution shows a crossoverfrom a strongly non-diffusive and anisotropic behavior atshort-times to an eventual diffusive isotropic behavior via anintermediate regime (II) or (III) whose behaviors are very dif-ferent. In regime (I), starting from the origin with a fixed ori-entation θ , the position distribution of a DRABP is stronglyanisotropic and shows a plateau-like structure around the ori-gin accompanied by a single peak near v t along θ [Figs. 2(a)and 3(a)]. For γ > D R the anisotropy persists [Fig. 2(b)] in theintermediate time regime (II), however, the marginal positiondistribution has a single peak at the origin which decays expo-nentially [Fig. 3(b)]. On the other hand, for D R > γ the distri-bution in the intermediate regime (III) is Gaussian [Fig. 3(c)]and similar to ABP. At late times (IV), the dynamics becomesdiffusive with an effective diffusion constant and the positiondistribution is Gaussian [Figs. 2(c) and 3(d)].The persistence property also shows distinct behaviorsin the dynamical regimes (I)–(IV). We show that the x and y components are characterized by different persistenceexponents, α x and α y respectively, in these regimes—whichis summarized in Table I. The most noteworthy is a newpersistence exponent α y = γ > D R , starting from θ = , we find that, α y shows a non-monotonic behavior— α y = / α y = α y = / Position distribution.—
The correlated nature of the ef-fective noises ζ x , y ( t ) in Eq. (1) make the dynamics non-diffusive and anisotropic at short-times [see SupplementalMaterial [42] for details]. In the following we first considerthe two extreme regimes (I) and (IV) before coming to theintermediate regimes (II) and (III). We set x ( ) = y ( ) = Short-time regime (I): Starting from the initial orientation θ ( ) = θ and σ ( ) = , for t (cid:28) D − R the effective noises canbe approximated as, ζ x ( t ) ≈ v σ ( t )( cos θ − φ ( t ) sin θ ) , (2a) ζ y ( t ) ≈ v σ ( t )( sin θ + φ ( t ) cos θ ) , (2b)where φ ( t ) = √ D R R t ds η ( s ) denotes a standard Brown-ian motion. Here we have approximated cos φ ( t ) ’ φ ( t ) ’ φ ( t ) for t (cid:28) D − R as φ ( t ) ∼ √ D R t (cid:28) P ( x , t ) and P ( y , t ) corresponding to Eqs. (2), we adopt a trajectory basedapproach. The trajectory of the DRABP over a time-interval [ , t ] can be divided into ( n + ) intervals, punctuated by n direction reversals; σ remains constant between two consecu-tive reversals. We show that, for a specific sequence of inter-vals with duration { s i , i = , , ..., n + } , the distribution of thefinal position x is a Gaussian with mean cos θ ∑ n + i = ( − ) i s i and variance b n sin θ , where b n = D R n + ∑ i = " i − ∑ j = ( − ) i + j s i s j ( t j + t j − ) + s i ( t i + t i − ) . Here t i = ∑ ij = s j with t = t n + = t . The position distri-bution is then obtained by taking weighted contributions fromall such trajectories. Skipping details [42], we get, P ( x , t ) = e − γ t sin θ √ π ∞ ∑ n = γ n Z t n + ∏ i = ds i δ ( t − ∑ n + i = s i ) √ b n × exp " − ( x + cos θ ∑ n + i = ( − ) i s i ) θ b n , (3)where each term corresponds to trajectories with a fixed num-ber n of reversals. We also obtain the y -marginal distribu-tion P ( y , t ) in the same manner, whose explicit form is givenin [42].Equation (3) provides the exact time-dependent marginaldistribution for the process (2). Even though the infinite seriescannot be summed explicitly to obtain a closed-form expres-sion, it can be systematically evaluated numerically to obtain P ( x , t ) for arbitrary γ and t . In fact, for t . γ − , it sufficesto consider the first few terms to get a reasonably good esti-mate of the marginal distributions [42]. Figure 3(a) comparesthis estimate, evaluated up to n = P ( x , t ) ob-tained from numerical simulations. Clearly, this perturbativeapproach is extremely successful in accurately predicting thecharacteristic shape of the distribution, with a wide plateaunear the origin and a peak near x = v t , in this short-timeregime (I).Physically, the peak in the distribution is a manifestationof the ABP nature of the motion—the n = t (cid:28) D − R , the orientation θ evolves slowly, and the dynamics can be thought of as a one-dimensional RTP with an effective velocity v cos θ . Now, forsmall values of γ , the trajectories with a single flip contributea constant value (the plateau) γ e − γ t / ( v cos θ ) . This agreeswell with the exact result [Eq. (3)] to leading order in γ . In-terestingly, such a plateau with a boundary-peak has been ob-served for motile bacteria in emulsion droplets [45].The anisotropic nature of the distribution in this short-timeregime is a direct artifact of the fixed initial orientation. If,instead, the initial orientation is chosen uniformly, the positiondistribution becomes isotropic and an additional peak emergesat the origin [42].
Long-time regime (IV): For a given time t , mathematically,this regime can be accessed by taking both D R and γ large( (cid:29) t − ). For large D R and arbitrary γ , the effective-noiseautocorrelation becomes [42], h ζ a ( t ) ζ b ( t ) i ≈ D eff δ a , b [ D R + γ ] exp (cid:0) − [ D R + γ ] | t − t | (cid:1) where 2 D eff = v / ( D R + γ ) . Thusin the limit D R , γ → ∞ , it tends to 2 D eff δ a , b δ ( t − t ) which re-sults in the isotropic Gaussian distribution, P ( x , y , t ) ≈ D eff t G (cid:18) x √ D eff t , y √ D eff t (cid:19) , (4)with G ( ˜ x , ˜ y ) = e − ( ˜ x + ˜ y ) / / ( π ) . The corresponding x -marginal distribution (which, obviously, is also a Gaussian)is plotted in Fig. 3(d) along with the data from numerical sim-ulations; an excellent agreement validates our prediction. -16 -8 0 8 16 x -6 -3 P ( x , t ) D R = 5 D R = 10 D R = 20 -0.5 0 0.5 x -3 P ( x , t ) γ = 0.01γ = 0.1γ = 0.5 -16 -8 0 8 16 x -6 -3 P ( x , t ) D R = 5 D R = 10 D R = 20 -0.2 0 0.2 x -3 P ( x , t ) γ =10γ =25γ = 50 γ =10 −3 (a) (b)(d)(c) t =50 t =100 γ =1 t =10 D R =10 -4 D R =10 -3 t =1 FIG. 3. Marginal position distribution P ( x , t ) in the different dy-namical regimes : t (cid:28) min ( D − R , γ − ) (a), γ − (cid:28) t (cid:28) D − R (b), D − R (cid:28) t (cid:28) γ − (c) and t (cid:29) max ( D − R , γ − ) (d). The symbols in-dicate the data obtained from the numerical simulations while solidblack lines show the corresponding analytical predictions. Here v = θ = π / θ = π / Intermediate-time regime (III): This regime corresponds to D R (cid:29) t − (cid:29) γ , where h ζ a ( t ) ζ b ( t ) i → ( v / D R ) δ a , b δ ( t − t ) . Therefore, the typical position distribution is again Gaussianwith the width v p t / D R , P ( x , y , t ) ≈ D R v t G xv p t / D R , yv p t / D R ! , (5)with G ( ˜ x , ˜ y ) = e − ( ˜ x + ˜ y ) / / ( π ) . Note that this result is sameas in the case of ABP for t (cid:29) D − R [27] —adding direc-tional reversal does not change the physical scenario in thisregime. We validate this prediction with numerical simula-tions in Fig. 3(c). Intermediate-time regime (II): The correlated noise leadsto an intriguing behavior in this regime γ − (cid:28) t (cid:28) D − R . For t (cid:29) γ − , the frequent reversals lead to a Gaussian whitenoise ξ ( t ) with zero-mean and the correlator h ξ ( t ) ξ ( t ) i = γ − δ ( t − t ) [42]. Thus, for t (cid:28) D − R , from Eqs. (2), the effec-tive noises can be approximated as, ζ x ( t ) ≈ v ξ ( t )( cos θ − φ ( t ) sin θ ) , (6a) ζ y ( t ) ≈ v ξ ( t )( cos θ + φ ( t ) sin θ ) . (6b)Equations (6) describe a Brownian motion with stochasticallyevolving diffusion coefficients. Some specific versions ofsuch models have been studied recently [46] in a different con-text.The Gaussian nature of ξ ( t ) , for a fixed { φ ( s ) } trajec-tory, allows us to evaluate the characteristic function, h e ikkk · xxx i = D exp h − kkk T ΣΣΣ ( t ) kkk iE φ , where kkk = ( k x , k y ) T and ΣΣΣ ( t ) is thecorrelation matrix whose explicit form is given in [42].The subscript φ denotes averaging over the Brownian paths { φ ( s ) } , which can be performed using path integral approach.This yields [42], h e ikkk · xxx i = √ cosh ω t exp (cid:20) − ω tanh ω t D R ( k x tan θ + k y ) ( k x − k y tan θ ) (cid:21) , (7)where ω = v cos θ ( k x − k y tan θ ) p D R / γ . The characteris-tic functions corresponding to the marginal x and y distribu-tions are obtained by putting k y = k x = θ = π / π / θ = π ) thecharacteristic function for the x -marginal (respectively y -marginal) becomes h e ikx i = h cosh (cid:16) v kt p D R / γ (cid:17)i − . Thisleads to a scaling form P ( x , t ) = v t q γ D R f ( xv t q γ D R ) . Wefind the exact scaling function, i.e., the probability density forthe scaled variable ˜ x = xv t q γ D R as, f ( ˜ x ) = √ π Γ (cid:18) + i ˜ x (cid:19) Γ (cid:18) − i ˜ x (cid:19) , (8)where Γ ( z ) is the gamma function. Figure 3(b) shows an ex-cellent agreement between Eq. (8) and the numerical simula-tions. Using the asymptotic expression for Γ ( z ) we find thatthe tails of the distribution decay as f ( ˜ x ) ∼ p / ( π | ˜ x | ) e − π | ˜ x | . From the scaling form it is evident that the variance h x ( t ) i c ∝ t for these specific values of θ . For arbitrary initial orientation θ , one can numerically in-vert Eq. (7), which also yields a similar exponential tail. Notethat, in this case, h x ( t ) i c ∝ t [42] which indicates a diffusivemotion, but the distribution is strongly non-Gaussian with ex-ponential tails. This Brownian but non-Gaussian feature isseen in a large range of systems like beads diffusing on lipidtubes, colloidal suspensions, cells and nematodes [47–50]. First-passage properties.—
We next consider the survivalprobability which is the cumulative distribution of the first-passage time. We set θ = x and y denote directionsparallel and orthogonal to the initial orientation, respectively.Let S y ( t ; y ) denote the probability that, starting from somearbitrary position y ( ) = y the y -component of the positionhas not crossed the y = t ; S x ( t ; x ) is definedsimilarly.The most interesting scenario appears for γ > D R where S y ( t ) shows three distinct persistence behaviors in the threedifferent dynamical regimes. In the short-time regime (I),Eq. (2b) leads to ˙ y = v σ ( t ) φ ( t ) for θ =
0. This correspondsto a random acceleration process between two consecutive re-versals. Since, there are very few reversals in this regime, weexpect the persistence exponent α y = /
4, as in the randomacceleration process [51, 52]. This is verified in Fig. 4(a) us-ing numerical simulations.In the intermediate time regime (II) the effective y -dynamics [see (6b)] becomes ˙ y = v ξ ( t ) φ ( t ) for θ = . Wecompute the survival probability by solving the correspond-ing Fokker Planck equation with an absorbing boundary con-dition at y = t -3 -2 -1 S y ( t ) γ =0.0005 γ =0.001 γ =0.002 γ =0.004 γ =0.008 t -3 -2 -1 S y ( t ) D R =0.001 D R =0.002 D R =0.004 D R =0.008 t -1/4 t -1 t -1/2 t -1 (a) (b) FIG. 4. S y ( t ) vs t for γ > D R : (a) shows the crossover from α y = / α y = D R = − and y = . . (b) shows thecrossover from α y = α y = / γ = y = . α y / / / α x / / / x and y components of theDRABP as defined by decay of the survival probability S ( t ) ∼ t − α inthe different dynamical regimes: (I) t (cid:28) min ( D − R , γ − ) , (II) γ − (cid:28) t (cid:28) D − R , (III) D − R (cid:28) t (cid:28) γ − , and (IV) t (cid:29) max ( γ − , D − R ) . α y = α y = / α y = t ∼ γ − using numerical simulationsin Fig. 4(a).In the large time regime (IV), as discussed earlier, the par-ticle behaves like an ordinary diffusion process with an effec-tive diffusion constant. Consequently, the survival probabilitydecays with the Brownian exponent α y = / α y = α y = / t ∼ D − R .If D R > γ , we effectively see two distinct exponents α y = / α y = / S x ( t ) which shows Brownian behavior at alltimes except in regime (I) where α x = . A summary of theexponents in all the regimes is provided in Table I.In conclusion, we provide a comprehensive analytical un-derstanding of the DRABP motion that models a wide rangeof bacterial motion. The DRABP shows many novel featurescompared to the known active particle models in 2D — thepresence of direction reversal along with rotational diffusiongives rise to four distinct dynamical regimes each of whichcorresponds to a different position distribution and persistenceexponent. In particular, we find that, the position distribu-tion in the short-time and intermediate-time regimes have cer-tain unique features, very different from ordinary ABP andRTP. Moreover, we find a novel persistence exponent α = [1] P. Romanczuk, M. Bar, W. Ebeling, B. Lindner, and L.Schimansky-Geier, Active Brownian Particles. From Individualto Collective Stochastic Dynamics , Eur. Phys. J. Special Topics , 1 (2012).[2] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool,J. Prost, Madan Rao, and R. Aditi Simha,
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In this Supplemental Material we provide some details of the computations of the moments and distributions ofposition, and the survival probability.
NOISE CORRELATION AND MOMENTS
The DRABP dynamics follows the overdamped Langevin equations (1) in the main text,˙ x ( t ) = ζ x ( t ) ≡ v σ ( t ) cos θ ( t ) , (1a)˙ y ( t ) = ζ y ( t ) ≡ v σ ( t ) sin θ ( t ) , (1b)where σ ( t ) = ± γ and θ follows a Brownian motion with diffusion constant D R . To compute the moments of the position we need the auto-correlations of these effective noises. Since the σ and θ processes are independent, it suffices to compute the correlations of σ and cosθ, sin θ separately.We begin with the σ -process. In this case, the propagator ψ ( σ, t | σ , , i.e., the probability that σ ( t ) = σ given that σ (0) = σ , is given by, ψ ( σ, t | σ ,
0) = 12 (cid:16) σσ e − γt (cid:17) . (2)The mean and the autocorrelation functions for the σ -process can be immediately obtained using the above equation, h σ ( s ) i = σ e − γs , and h σ ( s ) σ ( s ) i = e − γ | s − s | . (3)The propagator for the Brownian motion θ is a Gaussian — the probability that θ ( t ) = θ given that θ (0) = θ is givenby, P ( θ, t | θ ,
0) = 1 √ πD R t exp (cid:20) − ( θ − θ ) D R t (cid:21) . (4)Consequently, h cos θ ( s ) i = cos θ e − D R s , h sin θ ( s ) i = sin θ e − D R s , (5)and the auto-correlation functions, h cos θ ( s ) cos θ ( s ) i = 12 (cid:20) e − D R | s − s | + e − D R ( s + s +2 min[ s,s ]) cos 2 θ (cid:21) , (6a) h sin θ ( s ) sin θ ( s ) i = 12 (cid:20) e − D R | s − s | − e − D R ( s + s +2 min[ s,s ]) cos 2 θ (cid:21) , (6b) h cos θ ( s ) sin θ ( s ) i = sin(2 θ )2 e − D R ( s + s +2 min[ s,s ]) . (6c)Combining Eqs. (3)-(6c), we have the mean of the effective noises, h ζ x ( s ) i = v h σ ( s ) ih cos θ ( s ) i = v σ cos θ e − ( D R +2 γ ) s , (7a) h ζ y ( s ) i = v h σ ( s ) ih sin θ ( s ) i = v σ sin θ e − ( D R +2 γ ) s , (7b) a r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n and the auto-correlations, h ζ x ( s ) ζ x ( s ) i = v h σ ( s ) σ ( s ) ih cos θ ( s ) cos θ ( s ) i = v (cid:20) e − ( D R +2 γ ) | s − s | + e − γ | s − s |− D R ( s + s +2 min[ s,s ]) cos 2 θ (cid:21) , (8a) h ζ y ( s ) ζ y ( s ) i = v h σ ( s ) σ ( s ) ih sin θ ( s ) sin θ ( s ) i = v (cid:20) e − ( D R +2 γ ) | s − s | − e − γ | s − s |− D R ( s + s +2 min[ s,s ]) cos 2 θ (cid:21) , (8b) h ζ x ( s ) ζ y ( s ) i = v h σ ( s ) σ ( s ) ih cos θ ( s ) sin θ ( s ) i = v sin 2 θ e − γ | s − s |− D R ( s + s +2 min[ s,s ]) . (8c)Note that from now on we always consider σ = 1 as that is the initial condition used in the main text. Position Moments
We compute the first two moments, namely, mean and variance of the position distribution of DRABP exactlyusing the effective noise auto-correlations, h x ( t ) i = Z t ds h ζ x ( s ) i , h x ( t ) i = Z t Z t ds ds h ζ x ( s ) ζ x ( s ) i (9) h y ( t ) i = Z t ds h ζ x ( s ) i , h x ( t ) i = Z t Z t ds ds h ζ x ( s ) ζ x ( s ) i . (10)Using Eqs. (7b)-(8) we obtain the mean, h x ( t ) i = v cos θ γ + D R (cid:16) − e − t (2 γ + D R ) (cid:17) ; h y ( t ) i = v sin θ γ + D R (cid:16) − e − t (2 γ + D R ) (cid:17) , (11)and the second moments, h x ( t ) i = v t (2 γ + D R ) + v (2 γ + D R ) ( e − (2 γ + D R ) t −
1) + v cos 2 θ (3 D R − γ ) (cid:20) e − D R t − D R + 1 − e − ( D R +2 γ ) t ( D R + 2 γ ) (cid:21) , (12a) h y ( t ) i = v t (2 γ + D R ) + v (2 γ + D R ) ( e − (2 γ + D R ) t − − v cos 2 θ (3 D R − γ ) (cid:20) e − D R t − D R + 1 − e − ( D R +2 γ ) t ( D R + 2 γ ) (cid:21) . (12b)The presence of the two time-scales D − R and γ − gives rise to four distinct dynamical regimes characterizedby different dynamical behaviors: a short-time regime (I) t (cid:28) min( γ − , D − R ), two intermediate-time regimes (II) γ − (cid:28) t (cid:28) D − R (accessible for γ > D R ) and (III) D − R (cid:28) t (cid:28) γ − (accessible for γ < D R ), and the long-time regime(IV) t (cid:29) max( γ − , D − R ) . In the following we look at how the variance h x ( t ) i c = h x ( t ) i − h x ( t ) i and h y ( t ) i c = h y ( t ) i − h y ( t ) i behave inthese regimes. Short-time regime (I)
In this regime, i.e., for t (cid:28) min( γ − , D − R ) , the variance behavior can be obtained by simply expanding Eqs. (12)in a Taylor series in t, h x ( t ) i c = v t D R + 2 γ − ( D R − γ ) cos 2 θ ) + O( t ) , (13a) h y ( t ) i c = v t D R + 2 γ + ( D R − γ ) cos 2 θ ) + O( t ) . (13b)Clearly, there is an anisotropy in the system if we begin from arbitrary θ (except when cos 2 θ = 0), on the otherhand, if the initial orientation is chosen uniformly in [0 , π ], then h x ( t ) i c = h y ( t ) i c = v t ( D R + 2 γ ) + O ( t ). Intermediate regime (II)
In this regime, γt (cid:29) D R t (cid:28) , hence, the behavior of the variance is obtained by neglecting terms ∼ e − γt and then expanding the resulting expression in a series in D R t, h x ( t ) i c ≈ v t γ (1 + cos 2 θ ) − v t D R γ cos 2 θ + O( t ) , (14a) h y ( t ) i c ≈ v t γ (1 − cos 2 θ ) + v t D R γ cos 2 θ + O( t ) . (14b)Thus in this regime, the anisotropy persists and the variance ∝ t , except for θ = π/ , π/ , π ) for x (respectively, y ) direction the coefficient of t vanishes and the variance ∝ t . Intermediate regime (III)
In this regime D R t (cid:29) γt (cid:28) ∼ e − D R t . Expanding the resulting expression in aseries of γt we get the variance behavior, In the regime (III), i.e., for D − R (cid:28) t (cid:28) γ − , we get, h x ( t ) i c = h y ( t ) i c ≈ v tD R (15)which indicates that if D R > γ , the anisotropy vanishes already in the intermediate regime and the motion becomesdiffusive with an effective diffusion coefficient v / (2 D R ) . Long-time regime (IV)
Finally, for t (cid:29) max( γ − , D − R ) , we have, h x ( t ) i c = h y ( t ) i c ≈ D eff t. (16)The dynamics is isotropic and diffusive with an effective diffusion constant D eff = v D +2 γ ) . EFFECTIVE DYNAMICS IN THE DIFFERENT REGIMES
The DRABP dynamics in the four different temporal regimes can be effectively described by some simpler versionsof Eq. (1b). In this section we summarize these effective dynamics in the different regimes.
Short-time regime (I)
In this regime t is much smaller than both time-scales D − R and γ − . Let us suppose that the particle starts froman initial orientation θ , then the effective noises in Eq. (1b) can be written as, ζ x ( t ) = v σ ( t )(cos θ cos φ ( t ) − sin θ sin φ ( t )) (17a) ζ y ( t ) = v σ ( t )(sin θ cos φ ( t ) + cos θ sin φ ( t )) (17b)where φ ( t ) = √ D R R t ds η ( s ) is a standard Brownian motion. At times t (cid:28) D − R , φ ( t ) ∼ √ D R t (cid:28)
1, so we canuse the approximation cos φ ’ φ ’ φ to the leading order in φ. Equations (17) then reduce to, ζ x ( t ) ≈ σ ( t )( A − Bφ ( t )) (18a) ζ y ( t ) ≈ σ ( t )( B + Aφ ( t )) . (18b)where we have used A = v cos θ and B = v sin θ for notational simplicity. We use this form of the effective noisecorrelation to compute the position distribution in this regime in Sec. . Intermediate-time regime (II)
In this regime also t (cid:28) D − R , so the approximations cos φ ’ φ ’ φ remain still valid. The φ changesappreciably at the time scale of D − R , during which there is a large number of σ -reversal events as γ (cid:29) D R . Thus, inthis regime, the effective noise can be approximated as, ζ x ( t ) = ξ ( t )( A − Bφ ( t )) , (19a) ζ y ( t ) = ξ ( t )( B + Aφ ( t )) (19b)where ξ ( t ) is a Gaussian white noise with the following properties, h ξ ( t ) i = 0 , h ξ ( t ) ξ ( t ) i = 1 γ δ ( t − t ) . (20)We use this approximation of the effective noise to calculate the position distribution (Sec. ) and persistence exponent(Sec. ) of the DRABP in regime (II). Intermediate-time regime (III) and long-time regime (IV)
These regimes can be accessed using D R → ∞ . For large D R , effective noise correlations Eqs. (8) reduce to, h ζ a ( s ) ζ b ( s ) i ≈ δ a,b v − ( D R + 2 γ ) | s − s | ) . (21)In regime (IV), taking the limits D R → ∞ and γ → ∞ , while keeping D R /γ and 2 D eff = v ( D R +2 γ ) finite, we get, h ζ a ( s ) ζ b ( s ) i = 2 D eff δ a,b δ ( s − s ) . (22)This is the effective noise correlation used in regime (IV) in the main-text. On the other hand, in the limit D R → ∞ and γ/D R (cid:28)
1, with v /D R finite, we get, h ζ a ( s ) ζ b ( s ) i = δ a,b v D R δ ( s − s ) , (23)which is used to calculate the position distribution of the DRABP in regime (III) in the main-text. POSITION DISTRIBUTION: SHORT-TIME REGIME
In this section we provide the details of the computation leading to the marginal position distribution in the short-time regime (I) t (cid:28) min( γ − , D − R ), i.e., when the time is much smaller than both the characteristic time scales ofthe system. The dynamics in this regime is governed by the Langevin equations ˙ x = ζ x ( t ) and ˙ y = ζ y ( t ) where theeffective noise ζ x ( t ) and ζ y ( t ) are given by Eq. (18).Now, let us assume that during time t there are n orientational reversals. We can thus divide the duration [0 , t ]into n + 1 intervals, such that σ changes sign at the beginning of each interval and remains constant throughout theinterval. Let s i be the duration of the i -th interval as shown in Fig. 1 and σ i = ( − i − denotes the value of σ in thisinterval. For the sake of convenience we also define t i = P ij =1 s j which is the total time elapsed before the start ofthe ( i + 1)-th interval. Obviously, t = 0 and t n +1 = t. σ = 1 σ = − σ = 1 σ n +1 = ( − n s s s s n +1 t FIG. 1: Schematic representation of the reversal process: s i denotes the interval between i th and ( i + 1) th reversalevents during which σ i = ( − i − remains constant.For a given trajectory { σ i , s i } , the final position of the particle can then be expressed as, x ( t ) = A n +1 X i =1 σ i s i − B n +1 X i =1 σ i z i , (24a) y ( t ) = B n +1 X i =1 σ i s i + A n +1 X i =1 σ i z i (24b)where we have denoted z i = R t i t i − ds φ ( s ) . Since φ ( s ) is an ordinary Brownian motion, its integral should follow aGaussian distribution— in fact, { z i ; i = 1 , · · · n + 1 } form a set of ( n + 1) correlated Gaussian variables with thecorrelation matrix C ij = h z i z j i . The linear combination P n +1 i =1 σ i z i then also follows a Gaussian distribution with thevariance b n = n +1 X i,j =1 σ i σ j C ij . (25)From Eq. (24), we can then write the marginal position distributions for a given trajectory { σ i , τ i } , P ( x, { σ i , s i } ) = 1 B √ πb n exp " − ( x − A P n +1 i =1 σ i s i ) b n B , (26a) P ( y, { σ i , s i } ) = 1 A √ πb n exp " − ( y − B P n +1 i =1 σ i s i ) b n A . (26b)Note that, for notational simplicity we have used the same letter P to denote both x and y distributions. Thevariance b n is obtained from Eq. (25) using the correlation matrix C ij which can be computed explicitly using theauto-correlation of the Brownian motion h φ ( s ) φ ( s ) i = 2 D R min( s, s ) ,C ij = D R ( t i − t i − )( t j − t j − ) for i < jD R ( t j − t j − )( t i − t i − ) for j < i D R t i − t i − ) ( t i + 2 t i − ) for i = j. (27)To obtain the actual position distribution P ( x, t ) [1] we need to take into account the contributions from all possibletrajectories { σ i , s i } and all possible number of flipping events, P ( x, t ) = ∞ X n =0 γ n e − γt P n ( x, t ) (28)where P n ( x, t ) denotes the contribution from the trajectories with n number of reversals, P n ( x, t ) = Z t n +1 Y i =1 ds i δ ( t − X i s i ) P ( x, { σ i , s i } ) . (29)This result is quoted as Eq. (3) in the main text. Similarly, for the y -component, P ( y, t ) = ∞ X n =0 γ n e − γt P n ( y, t ) = ∞ X n =0 γ n e − γt Z t n +1 Y i =1 ds i δ ( t − X i s i ) P ( y, { σ i , s i } ) . (30)As discussed in the main text, for t (cid:28) γ − , it suffices to look only at the first few terms as the average numberof reversal is given by h n i = γt during a time-interval t. In fact, this can be thought of as a perturbative approachin γ, which should work well for the DRABP in the regime t (cid:28) min( γ − , D − R ) . In the following we compute thefirst few terms explicitly and illustrate, by comparing the corresponding analytical prediction with the data fromnumerical simulations, that the series converges reasonably fast. We restrict ourselves to marginal x -distribution only,the y -distribution can be obtained following the same procedure. ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● n = = = - - - - - x P ( x , t ) FIG. 2: Convergence of the infinite series (28): Plot of P ( x, t = 1) for γ = 0 . D R = 10 − and θ = π/
4. Themagenta symbols represent the result obtained from simulations, while the dashed lines indicate P ( x, t ) in Eq. (28)calculated upto specified terms as mentioned in the legends.For n = 0 , there are no reversal events. In this case b = 2 D R t / P ( x, t ) = √ √ πD R Bt / exp (cid:18) − x − At ) B D R t (cid:19) . (31)For n = 1 , the velocity reverses once, after some time s ∈ [0 , t ] . In this case, we have, from Eqs. (25) and (27), b = 2 D R s + s ) − s s ] . (32)Corresponding contribution to the position distribution is given by, P ( x, t ) = r πD R B t Z t ds exp (cid:16) − x − A (2 s − t )) B D R ( t − s ( t − s )) (cid:17)p t − s ( t − s )Similarly, for n = 2 , i.e., two reversals, b = 2 D R " ( s + s + s ) − s ( s + τ ) ( s + s + s ) + 6 s (2 s + s ) (33)and P ( x, t ) = r πD R B t Z t ds Z t − s ds exp (cid:18) − y − B ( t − s )) A D ( s +( t − s − s ) + s ) (cid:19)p s + ( t − s − s ) + s (34)and so on. Clearly, P n ( x, t ) can be obtained systematically by evaluating the integrals numerically.To illustrate the convergence of P ( x, t ) in Eq. (28), in Fig. 2 we plot the contributions from the first few termsseparately for γ = 0 . t = 1 (the uppermost curve in Fig. 2(a) in the main text). Clearly, the effect of reversalis immediately visible from the n = 1 term, which changes the shape of the distribution drastically, adding a plateauaround the origin, in addition to the Gaussian peak near x ∼ v t cos θ ; the higher order terms only add quantitativecorrections. As expected, when γ is increased, we obtain better quantitative match by including higher order terms(see Fig. 2).We conclude the discussion about the short-time regime with a brief comment. If the particle starts from a uniforminitial orientation in [0 , π ], then the resulting distribution is always isotropic. The corresponding marginal distributioncan be obtained by integrating Eq. (28) over θ ,¯ P ( x, t ) = 12 π Z π dθ P ( x, t ) . (35)There is a central peak in this case which increases with the increase in γ . Using Eq. (35) and the first few termsof Eq. (28) one can get a good approximation of this distribution as shown in Fig. 3. ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■◆ γ = ● γ = ■ γ = - P ( y , t ) FIG. 3: Left: Comparison of probability distribution for uniform initial orientation obtained from numericalsimulation and calculating upto n = 2 term of the series in Eq. (35). Right: P ( x, y, t ) and the corresponding contourplot for t = 1 and γ = 0 . , D R = 0 .
01 obtained from numerical simulation of Eq. (1) in the main text with uniforminitial orientation . POSITION DISTRIBUTION: INTERMEDIATE-TIME REGIME (II)
In this section we focus on the position distribution in the regime II and provide a detailed derivation of the Eqs. (8)and (9) in the main text. As discussed in the main text and Sec. , the effective noises governing the DRABP dynamicsin this regime, i.e., for γ − (cid:28) t (cid:28) D − R can be approximated by Eq. (19), We can thus write the characteristic functionfor the joint distribution as, h exp ( i k · x ) i = (cid:28) exp (cid:18) i Z t ds ξ ( s ) [ k x ( A − Bφ ( s )) + k y ( B + Aφ ( s ))] (cid:19)(cid:29) ( ξ,φ ) , where the averaging is over both { ξ ( t ) } and { φ ( t ) } trajectories; k = (cid:18) k x k y (cid:19) and x = (cid:18) xy (cid:19) . Now, for a given trajectory { φ ( t ) } , the averaging over { ξ ( t ) } can be done immediately to yield, h exp ( i k · x ) i = (cid:28) exp (cid:20) − k T Σ ( t ) k (cid:21)(cid:29) φ , (36)where the covariance matrix is given by, Σ ( t ) = (cid:20) h x ( t ) i φ h x ( t ) y ( t ) i φ h x ( t ) y ( t ) i φ h y ( t ) i φ (cid:21) = 1 γ R t ds ( A − Bφ ( s )) R t ds ( A − Bφ ( s ))( B + Aφ ( s )) R t ds ( A − Bφ ( s ))( B + Aφ ( s )) R t ds ( B + Aφ ( s )) , (37)where the average is now over the φ -process only. Remembering that φ is a standard Brownian motion, we can dothe averaging over φ using path integral, h exp ( i k · x ) i = Z ∞−∞ dX Z X D φ exp " − Z t ds ˙ φ D R − γ ( k x ( A − Bφ ) + k y ( B + Aφ )) ! = Z ∞−∞ dX Z X D φ exp " − Z t ds Z γ ( φ + Z /Z ) + ˙ φ D R ! , (38)where Z = ( k y A − k x B ) and Z = ( k x A + k y B ). Using the variable shift φ → φ + Z Z and X → X + Z Z , Eq. (38)reduces to, h exp ( i k . x ) i = Z ∞−∞ dX Z XZ /Z D φ exp " − Z t ds ˙ φ D R + Z γ φ ! . (39)The form of the path integral in the above equation corresponds to the imaginary time propagator of a quantumharmonic oscillator with Hamiltonian H = − ~ m d dx + mω x , upon setting ~ = 1, m = D R and ω = Z D R γ . Itpropagates from initial position Z Z to the final position X in time t . Thus, we have, h exp ( i k . x ) i = Z ∞−∞ dX U ( X, Z /Z , t ) , (40)where, U ( X f , X i , t ) is the propagator of a quantum harmonic oscillator with initial and final points X i and X f respectively in imaginary time t . This is well known in literature [2] and with the mappings mentioned earlier wehave, U ( X f , X i , t ) = r ω πD R sinh( ωt ) exp (cid:20) − ω D R sinh( ωt ) (cid:18) (cid:0) X f + X i (cid:1) cosh( ωt ) − X f X i (cid:19)(cid:21) . (41)Using the above expression in Eq. (40) and performing the integral over X , we obtain, h exp ( i k . x ) i = 1 √ cosh ωt exp (cid:20) − ωZ tanh ωt D R Z (cid:21) . (42)The characteristic functions for x and y marginal distributions are obtained by taking k y = 0 and k x = 0 respectively, h e ikx i = √ cosh ω x t exp h − ω x D R cot θ tanh ω x t i , (43a) h e iky i = √ cosh ω y t exp h − ω y D R tan θ tanh ω y t i . (43b)where ω x = kB q D R γ and ω y = kA q D R γ . Although these characteristic functions are hard to invert for arbitrary θ , for θ = π/ π/ h e ikx i = h cosh (cid:16) v kt q D R γ (cid:17)i − / and h e iky i = h cosh (cid:16) v kt q D R γ (cid:17)i − / for θ = 0 and π .This can be inverted exactly to yield P ( x, t ) = 14 v t r γπ D R Γ (cid:18)
14 + i xv t r γ D R (cid:19) Γ (cid:18) − i xv t r γ D R (cid:19) , [for θ = π/ , π/
2] (44) P ( y, t ) = 14 v t r γπ D R Γ (cid:18)
14 + i yv t r γ D R (cid:19) Γ (cid:18) − i yv t r γ D R (cid:19) , [for θ = 0 , π ] , (45)where Γ( z ) denotes the gamma functions. SURVIVAL PROBABILITY IN THE INTERMEDIATE REGIME II
In this section we work out in detail the survival probability of a DRABP in the intermediate time regime ( γ − (cid:28) t (cid:28) D − R ). We are interested in the y -marginal survival probability S y ( t ; y ) i.e., the probability that a particlestarting from (0 , y ) with an initial orientation θ = 0 has not crossed the y = 0 line till time t . Mathematically, S y ( t ; y ) = Z ∞ d y P ( y, t ; y ) (46)where P ( y, t ; y ) is the marginal probability distribution in the presence of an absorbing wall at y = 0, startingfrom the initial position y . The survival probability in this regime is actually determined by trajectories which havealready survived regime (I). Thus, in principle, one should take into account dynamics of both the regimes (I) and(II). However, the regime (I) almost vanishes for γ (cid:29) y -direction [Eq. (19) with θ = 0] in the intermediateregime (II), ˙ y = v ξ ( t ) φ ( t ) (47)where φ ( t ) is a Brownian motion. We can write the corresponding forward Fokker-Planck (FP) equation for P ( y, φ, t ) , i.e., the probability that y ( t ) = y and φ ( t ) = φ,∂∂t P ( y, φ, t ) = v φ γ ∂ ∂y P ( y, φ, t ) + D R ∂ ∂φ P ( y, φ, t ) . (48)Note that, for notational simplicity we have suppressed the initial position dependence [1]. We need to solve this FPequation with the initial condition P ( y, φ,
0) = δ ( y − y ) δ ( φ ) and boundary conditions P ( y, φ, t ) → φ ( t ) → ±∞ and P (0 , φ, t ) = P ( ∞ , φ, t ) = 0 . For simplicity, we make a change of variable v t γ → t and define Λ = 2 γD R /v .Equation (48) then becomes ∂∂t P ( y, φ, t ) = φ ∂ ∂y P ( y, φ, t ) + Λ ∂ ∂φ P ( y, φ, t ) (49)The absorbing boundary condition at y = 0 can be taken care of by using the sin-eigenbasis sin( ky ) with k ≥ . It isalso convenient to take a Laplace transform w.r.t. time t ,˜ P ( k, φ, s ) = Z ∞ dt e − st Z ∞ dy sin( ky ) P ( y, φ, t ) . (50)Equation (49) reduces to an ordinary second order differential equation in terms of ˜ P ( k, φ, s ),Λ d d φ ˜ P ( k, φ, s ) − ( s + φ k ) ˜ P ( k, φ, s ) = − sin( ky ) δ ( φ ) . (51)with the boundary condition ˜ P ( k, φ, s ) → φ → ±∞ . For φ = 0 , The general solution of Eq. (51) is given by˜ P ( k, φ, s ) = a D − q φ r k Λ ! + b D − q − φ r k Λ ! (52)where q = (1+ sk Λ ), D ν ( z ) denotes the parabolic cylinder function [3] and a, b are two arbitrary constants independentof φ . Using the boundary conditions for φ → ±∞ , and the fact that ˜ P ( k, φ, s ) is continuous at φ = 0 we have,˜ P ( k, φ, s ) = a D − q (cid:18) φ q k Λ (cid:19) , for φ > a D − q (cid:18) − φ q k Λ (cid:19) , for φ < . (53)Integrating Eq. (51) across φ = 0 , we get,d ˜ P d φ (cid:12)(cid:12)(cid:12)(cid:12) φ =0 + − d ˜ P d φ (cid:12)(cid:12)(cid:12)(cid:12) φ =0 − = − sin( ky )Λ . Using this equation with Eq. (53) we get, a = 2 q sin( ky ) √ πk Λ Γ (cid:16) q (cid:17) . (54)Finally, combining Eq. (54) with Eq. (53) we get,˜ P ( k, φ, s ) = 2 q sin( ky ) √ πk Λ Γ (cid:16) q (cid:17) D − q | φ | r k Λ ! , (55)where, as before, we have denoted q = (1 + sk Λ ) . Since we are interested in the y -marginal distribution, we integrateover φ to get, ˆ P ( k, s ) = 2 sin( ky ) s + k Λ F (cid:18) , q + 12 , q + 22 , − (cid:19) P ( y, t ) . Here F ( a, b, c, z ) denotes the Hypergeometric function [3].To find the position distribution we need to invert the Laplace and sin transformations. The inverse Laplacetransform is defined by the integral, ˆ P ( k, t ) = Z c + i ∞ c − i ∞ d s e st ˜ P ( k, s ) (56)where c is chosen such that all the singularities of the integrand lie to the left of the Re ( s ) = c line. To computethe above integral let us first recast ˜ P ( k, s ) as,˜ P ( k, s ) = 2 sin( ky ) s + k Λ ˜ F (cid:18) , q + 12 , q + 22 , − (cid:19) Γ (cid:18) q + 22 (cid:19) , (57)where ˜ F ( a, b, c, z ) = F ( a, b, c, z ) / Γ( c ) denotes the regularized Hypergeometric function which is analytic for allvalues of a, b, c and z. From Eq. (57), it is straightforward to identify the singularities of ˜ P ( k, s ) , on the complex s -plane all of which lie on the negative real s -axis: s n = − k Λ(4 n + 5) with n = − , , , , · · · where s − comes fromthe prefactor ( s + Λ k ) − while s n ≥ are obtained from the singularities q n = − n + 1) of Γ (cid:0) q +22 (cid:1) . The inverse Laplace transform of Eq. (57) can then be expressed asˆ P ( k, t ) = ∞ X n = − e s n t R n (58)where R n denotes the residue of ˜ P ( k, s ) at s = s n . These residues can be computed exactly and turn out to be R n = 2 sin( ky ) ( − n +1 ( n + 1)! ˜ F (cid:18) , − n − , − n, − (cid:19) . Using the above expression in Eq. (58) and shifting n → n −
1, we get,ˆ P ( k, t ) = 2 sin( ky ) ∞ X n =0 ( − n n ! e − (1+4 n ) k Λ t F (cid:18) , − n + 12 , − n + 1 , − (cid:19) . (59)Using properties of Hypergeometric functions, it can be shown that ˜ F (cid:18) , − n + 12 , − n + 1 , − (cid:19) = ( − n √ (cid:18) − / n (cid:19) n !Substituting the above identity in Eq. (59) we finally get,ˆ P ( k, t ) = √ ky ) e − k Λ t ∞ X n =0 (cid:18) − n (cid:19) e − nk Λ t = sin( ky ) p cosh (2 k Λ t ) (60)The position distribution is given by the inverse sin-transform, P ( y, t ; y ) = 2 π Z ∞ dk sin( ky ) ˆ P ( k, t ) = 1 π Z ∞ dk ky ) sin( ky ) p cosh (2 k Λ t )= 1 π Z ∞ dk [cos( k ( y − y )) − cos( k ( y + y ))] p cosh (2 k Λ t ) . Clearly, P ( y, t ) has a scaling form, P ( y, t ; y ) = 14Λ t " f (cid:16) y − y t (cid:17) − f (cid:16) y + y t (cid:17) , where, the scaling function f ( z ) can be evaluated exactly, f ( z ) = 1 π Z ∞ dκ cos( κz ) p cosh( κ/
2) = 1 √ π Γ (cid:16)
14 + iz (cid:17) Γ (cid:16) − iz (cid:17) . (61)1The survival probability, given by Eq. (46), also has a scaling form, S y ( t ; y ) = g (cid:16) y t (cid:17) , (62)where g ( z ) is given by, g ( z ) = Z ∞ dz [ f ( z − z ) − f ( z + z )] = 2 Z z dz f ( z ) . (63)Although this integral cannot be computed explicitly for arbitrary t, the large time behavior can be extracted easilyby taking z (cid:28) , g ( z ) = 2 z f (0) + O ( z ) . (64)Finally, we have, S y ( t ; y ) ≈ Γ(1 / √ π y Λ t . (65)Thus, going back to original notation t → v t/ (2 γ ), and Λ = 2 γD R /v , we have, S ( y , t ) ≈ Γ(1 / π / r γD R y v t for y v t (cid:28) s D R γ (cid:28) . (66)Using this result we conclude in the main text that the survival probability of a DRABP in the time regime (II) γ − (cid:28) t (cid:28) D − R has a power-law decay with persistence exponent α y = 1 . Note that, the exact first-passage distribution F y ( t ) = − ∂∂t S y ( t ; y ) can be easily computed from Eq. (62), F y ( t ) = y t f (cid:16) y t (cid:17) (67)which was obtained in Ref. [4] in the context of diffusing diffusivity.2 MOVIES
We show the time-evolution of a single particle trajectory and of many non-interacting particles in two separatemovies.
Supplementary Movie 1: DRABP-trajectory.gif
Time-evolution of a DRABP starting from the origin with θ = π/ σ = 1 . Here D R = 0 . γ = 0 . Supplementary Movie 2: DRABP-profile.gif
Time-evolution of N = 5000 non-interacting DRABP, each starting from the origin with θ = π/ σ = 1 . Here D R = 0 .
02 and γ = 0 . [1] We use the same notation P ( · ) to denote all probability density functions. The number of arguments as well as the actualfunctional form is different depending on the context.[2] R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals , McGraw-Hill, New York (1965).[3] NIST Digital Library of Mathematical Functions, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F.Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds[4] D. S. Grebenkov, V. Sposini, R. Metzler, G. Oshanin and F. Seno,
Exact first-passage time distributions for three randomdiffusivity models
J. Phys. A: Math. Theor.54