Active noise-driven particles under space-dependent friction in one dimension
AActive noise-driven particles under space-dependent friction in one dimension
D. Breoni , H. L¨owen , and R. Blossey Institut f¨ur Theoretische Physik II: Weiche Materie,Heinrich Heine-Universit¨at D¨usseldorf, Universit¨asstraße 1, 40225 D¨usseldorf, Germany University of Lille, UGSF CNRS UMR8576, 59000 Lille, France
We study a Langevin equation describing the stochastic motion of a particle in one dimensionwith coordinate x , which is simultaneously exposed to a space-dependent friction coefficient γ ( x ), aconfining potential U ( x ) and non-equilibrium (i.e., active) noise. Specifically, we consider frictions γ ( x ) = γ + γ | x | p and potentials U ( x ) ∝ | x | n with exponents p = 1 , n = 0 , ,
2. Weprovide analytical and numerical results for the particle dynamics for short times and the stationaryprobability density functions (PDFs) for long times. The short-time behaviour displays diffusiveand ballistic regimes while the stationary PDFs display unique characteristic features dependingon the exponent values ( p, n ). The PDFs interpolate between Laplacian, Gaussian and bimodaldistributions, whereby a change between these different behaviours can be achieved by a tuningof the friction strengths ratio γ /γ . Our model is relevant for molecular motors moving on aone-dimensional track and can also be realized for confined self-propelled colloidal particles. I. INTRODUCTION
Particles moving under the influence of a stochasticdriving force in one dimension [1] are a fruitful labora-tory for the exploration of the statistical mechanics ofactive systems, since they allow, in suitably chosen cases,for an analytic treatment. Following the initial works onone-dimensional active particles [2, 3], the problem is cur-rently receiving increased attention, since the results canbe of relevance for various soft matter and biological sys-tems in a larger sense [4–8]. One-dimensional models foractive particles, in spite of their inherent simplicity, areindeed of relevance even for the description of collectiveeffects [9–12].A standard type of model under scrutiny is the persis-tent Brownian motion, the persistence being forced by ac-tivity. Maybe the simplest model for an active particle inone dimension is a discrete run-and-tumble process (RTP)where the direction of self-propulsion discretely flips, i.e.the driving is assured by a random directional velocity,see, e.g. [10, 13–18]. It is defined by the Langevin equa-tion ˙ x ( t ) = v σ ( t ) (1)where the stochastic term η ( t ) = v σ ( t ) is a telegraphicnoise with values ± v , with the sign flipped at a giventumbling rate. In particular, this model has been ex-plored for a single particle in the presence of externalpotentials [19–21] and random disorder [10, 16].On a second level of complexity, one can considera Brownian particle self-propelled along its orientationsuch that only the projection on the x -axis is contribut-ing to the actual particle propulsion but the orientationdiffuses on the unit circle or unit spheres [22]. Thesemodels of active Brownian particles (ABP) were exten-sively discussed in the literature [7] and can be realized byself-propelled Janus-colloids in channel-like confinement[23–25]. For low activity, the fluctuation-dissipation the-orem which couples the strength of the Brownian noise and the friction via the bath temperature should be ful-filled. Hence, in the limit of vanishing activity, the sta-tionary probability density function (PDF) is a Boltz-mann distribution. Also simpler variants of these modelswhere the drive just enters via colored noise, often calledactive Ornstein-Uhlenbeck particles have been exploredin one dimension [26–31].A third complementary approach starts from Langevinequations coupling an active white noise term to spa-tially dependent friction [32, 33]. The basic idea here isthe gradient in the friction induces a drift velocity whichdrives the particle at constant noise. In near-equilibriumsituations, a spatial dependence of the friction enforcesa spatial dependence of the noise strength according tothe fluctuation-dissipation theorem which guarantees arelaxation of the PDF to the stationary Boltzmann dis-tribution. Here we deliberately abandon the validity ofthe fluctuation-dissipation theorem and therefore postu-late a non-equilibrium noise in the presence of a frictiongradient to define a nonequilibrium model with inher-ent activity. We refer to this kind of noise as “active”noise in the sequel. The equilibrium limit of a stationaryBoltzmann distribution is reached if the friction gradientvanishes. Though these kind of non-equilibrium noisemodels were proposed more than a decade ago [32, 33]and bear interesting descriptions for the biologically mo-tivated case of molecular motors moving on a 1d-track[34–39] such as the action of chromatin remodeling mo-tors on nucleosomes [40], they have not yet been studiedsystematically.Here we propose a class of one-dimensional modelswith active noise in different friction gradients and exter-nal confining potentials which we solve analytically. Ourmotivation to do so is threefold: first, any exactly solublemodel in nonequilibrium is of fundamental importancefor a basic understanding of particle transport. Second,we obtain qualitatively different PDFs which can be cat-egorized such that we establish a “grammar” to classifyPDFs within these new active noise models. Third, ourresults are relevant for applications in the biological con- a r X i v : . [ c ond - m a t . s o f t ] F e b text and for artificial colloidal particles.The model we discuss is based on a Langevin equa-tion of a particle with nonequilibrium noise and space-dependent friction in one dimension with a spatial coor-dinate x . The particle is exposed to a space-dependentfriction coefficient γ ( x ) = γ + γ | x | p and an externalpotential U ( x ) ∝ | x | n with exponents p = 1 , n = 0 , ,
2. For short times, we provide analytical re-sults for the mean displacement and the mean-squareddisplacement. Depending on the parameters, we find acrossover from an initial diffusive to a ballistic regimefor p = 1 , n (cid:54) = 0 as typical for any model of asingle free active particle. For long times and n > γ /γ . To test the robustness of our results,we evaluate the effect of additional thermal noise [32, 33].As already mentioned, our proposed model is relevantfor molecular motors moving on a one-dimensional trackand can also be realized for confined self-propelled col-loidal particles. In fact, colloids can be exposed to al-most any arbitrary external potential by using opticalfields [41–43] and almost any kind of noise can exter-nally be programmed by external fields [44, 45]. A space-dependent friction can be imposed be a viscosity gradientin the suspending medium on the particle scale, a situa-tion typically encountered for viscotaxis [46–49]. II. A PARTICLE UNDER NONEQUILIBRIUMNOISE: THE MODEL
Following [33], the model Langevin equation of a singleactive particle on a one-dimensional trajectory x ( t ) weuse in this work is given by the expression γ ( x ) ˙ x ( t ) = − U (cid:48) ( x ) + √ Aξ ( t ) (2)in which U ( x ) is the confining potential, and ξ ( t ) a Gaus-sian random noise with (cid:104) ξ ( t ) (cid:105) = 0 , (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) (3)and A > < ... > denote a noise-average. The Langevin equation(2) can be rewritten in the standard multiplicative noiseform as ˙ x ( t ) = − U (cid:48) ( x ) γ ( x ) + √ Aγ ( x ) ξ ( t ) , (4)which we will interpret in the Stratonovich sense.The factor γ ( x ) in Eqs.(2), (4) is a space–dependentfriction force. It has been introduced in models for molec-ular motors in [32] and been modeled by an expression ������� ���� (cid:1) (cid:2) � ���� [ � ] � (cid:3) � + ��� � ���� [ � ] (cid:4)(cid:5) � { �� - ���� ��� } � ����� → ����� ���������� → ���� (cid:7) ������� FIG. 1. Sketch of the confining potential U ( x ) = κ | x | , alinear friction gradient γ ( x ) = γ + γ | x | in arbitrary units.Theparticle, shown by a blue dot on the x -axis, is activated bynoise (indicated in red), under the influence of the potentialand the friction gradient. γ ( x ) = 1 + δ tanh( xβ ) with parameters δ , β (0 < δ < γ ( x ) = γ + γ | x | p (5)for the friction term with two parameters γ > γ ≥ p ≥
0, which, althoughunbounded, will allow us to uncover interesting proper-ties of the stationary probability density functions. Thesearise when we consider the particle in low-order poly-nomial confining potentials which we take to be of thegeneral form U ( x ) = κn | x | n (6)with κ ≥ n ≥ p = n = 1 corresponding to a wedge-like potential U ( x ) = κ | x | with a friction term γ ( x ) = γ + γ | x | , seeFigure 1. III. SHORT-TIME BEHAVIOUR
We start our discussion by determining the short-timebehaviour of the active-noise driven particle and computethe short-time mean displacement (MD) and the mean-square displacement (MSD) for the Langevin equation(2), as done previously [50]. Specifically, we address thecases of a freely moving particle, U (cid:48) ( x ) = 0 (i.e. n = 0)and a particle moving in the potential U ( x ) = ( κ/n ) | x | n for n = 1 ,
2, which respectively correspond to a particleon a (double) ramp (or, under gravity) and in a harmonicoscillator potential.
A. Constant friction gradient
Free particle.
First we consider the case of p = 1, i.e.a constant friction gradient acting on a free particle. Dueto the spatial dependence of the friction term, the choiceof initial position x = x ( t = 0) is important. In theimmediate vicinity of the origin, the initial motion willbe that of a free Brownian particle since γ (cid:29) γ | x | . Inorder to see an effect of the x -dependence of the frictionterm, we place the particle initially far away from theorigin with | x | (cid:29) x > x < γ ( x ) at the origin. We can then drop the modulus anduse separation of the variables in Eq.(2) to find γ ( x ( t ) − x ) + γ x ( t ) − x ) = √ A (cid:90) t dt (cid:48) ξ ( t (cid:48) ) (7)resulting in x ( t ) = γ − (cid:18) − γ + (cid:113) γ + c ( t ) (cid:19) (8)with c ( t ) ≡ γ (cid:18) γ x + γ x + √ A (cid:90) t dt (cid:48) ξ ( t (cid:48) ) (cid:19) . (9)The resulting mean displacement (MD) (cid:104) x ( t ) − x (cid:105) canthen be obtained by an expansion of the square root as (cid:104) x ( t ) − x (cid:105) = − sgn( x ) γ + γ x ξ ( t ) (10)+ ∞ (cid:88) m =2 ( − m (2 m − m − m !( m − γ m − ξ m ( t )( γ + γ x ) m − where ξ m ( t ) ≡ (cid:42)(cid:18) √ A (cid:90) t dt (cid:48) ξ ( t (cid:48) ) (cid:19) m (cid:43) = (11) (cid:40) m !2 m/ ( m/ ( At ) m/ m even0 m oddsuch that the final expression for the MD is: (cid:104) x ( t ) − x (cid:105) = (12) − sgn( x ) ∞ (cid:88) m =1 (4 m − m − m !(2 m − γ m − γ ( x ) m − ( At ) m . Let us now discuss this result for the MD in more de-tail: first of all, if the friction gradient vanishes (i.e.,in the case γ = 0), there is no drift at all as en-sured by left-right symmetry. Second, for positive fric-tion gradients γ the leading term for short times inthe MD is linear in time and in the friction gradient − sgn( x ) γ At/ γ ( x )+ O ( t ) resulting in a drift velocity of − sgn( x ) γ A/ γ ( x ). Interestingly the particle driftis along the negative gradient of the friction implyingthat the particle migrates on average to the place wherethe friction is small. This is plausible since at positionswith smaller friction there are stronger fluctuations whichpromote the particle to the position of even lower fric-tion on average. A similar qualitative argument was putforward for colloids moving under hydrodynamic interac-tions (see ref. [51], p.54), which represent another caseof multiplicative noise, see also [52]. Third, in a moremathematical sense, the series in Eq.(13) is an asymp-totic series which strictly speaking does not converge for m → ∞ but nevertheless gives a good approximation tothe MD to any finite order in time. This asymptotic ex-pansion even holds if the cusp in the friction at x = 0were to be included as any corrections do not contributeto the short-time expansion in powers of time.Similarly, one can calculate the mean-squared displace-ment (MSD), which we define as∆( t ) = (cid:104) ( x ( t ) − x ) (cid:105) . (13)One obtains a simple relation to the MD as follows∆( t ) = − sgn( x ) 2 γ ( x ) γ (cid:104) x ( t ) − x (cid:105) . (14)Taking the asymptotic series as an approximationfor finite times, we can now discuss the crossing times t m → m +1 between the m th regime and the next, definedas the ratio of the coefficients of t m and that of t m +1 [50].These crossing times are therefore given by t m → m +1 = 4( m + 1)(2 m − m + 1)(4 m − m − γ ( x ) Aγ . (15)The sequence of crossing times is monotonously decreas-ing, i.e. crossing times between larger regimes alwaysoccur before those of smaller ones. This in turn meansthat the only real regime for the free particle is the firstone, linear in time. The same reasoning applies to theMSD, as it is proportional to the MD.Generally, we characterize these regimes with time-dependent scaling exponents β ( t ) ≡ d (log( (cid:104) x ( t ) − x (cid:105) )) d (log( t )) (16)and α ( t ) ≡ d (log(∆( t ))) d (log( t )) . (17)If these exponents are constant over a certain regimeof time they indicate that the MD (or the MSD) are apower-law in time proportional to t β (or t α ).Finally, we define a typical passage time for the particleto reach the origin and cross the cusp in the friction at x = 0. Beyond such a passage time our theory shouldnot be applicable any longer, as we ignored the presenceof the cusp in the friction. Such a typical passage time t x is set by requiring (cid:104) x ( t x ) (cid:105) ≡ , (18)which means that on average the particle has reached theorigin. Of course this is only an estimate. The definitionof a passage time can be improved by requiring that theparticle is one standard deviation away from the originon average (cid:104) x ( t x ) (cid:105) + (cid:113) ∆( t x ) ≡ x >
0. This defines a second typical passage time t x which is in general smaller than t x . Taken together, thetwo passage times t x and t x provide a rough estimate forthe validity of our theory.Explicit data for the MD and MSD are shown in Fig-ure 2 a) and c), with the associated exponents β ( t ) and α ( t ) given in Figure 2 b) and d). The typical passagetimes t x (in purple) and t x (in orange) are also indicatedby vertical lines. In the figure we compare our analyticresults (taken by summing up the series up to a finite or-der of 5) with the full numerical solution of the Langevinequation, Eq.(4), in Stratonovich interpretation; detailsof the numerical method are discussed in the Appendix.First of all in the time regime t < t x the asymptotictheory is in good agreement with the simulation data.Both theory and simulations are dominated by the lineartime-dependence in the MD and MSD as indicated by theslope of the MD and MSD and likewise by the scalingexponents β ( t ) and α ( t ) which are both close to unity.In both theory and simulation the scaling exponents β ( t )and α ( t ) first show a trend to increase to transient valueslarger than unity, i.e. towards superdiffusive behaviour.Beyond t x this trend weakens in the simulations suchthat both exponents fall significantly below unity. Thisis due to the fact that the particle has arrived at theposition of minimal friction at the origin and thereforedecelerates. However, in the theory there is an artificialmonotonic increase in the slope due to the fact that thereis even unphysical negative frictions for position smallerthan γ /γ (for the case x > Linear confining potential.
Now we consider thecase n = 1 where U ( x ) = κ | x | , for p = 1. As before, weassume x (cid:29) U (cid:48) ( x ) = − κ and the equationof motion can be solved by separation of variables as inthe free case n = 0. The result for the mean displacementis (cid:104) x ( t ) − x (cid:105) = − sgn( x ) (cid:34) κtγ ( x ) + ∞ (cid:88) m =2 (2 m − m − ( m − × γ m − γ ( x ) m − (cid:98) m/ (cid:99) (cid:88) k =0 A k κ m − k ( m − k )!2 k k ! t m − k , (20)where the Gauss bracket (cid:98)·(cid:99) indicates the closest integer from below and the case x < (cid:104) x ( t ) − x (cid:105) = − sgn( x ) (cid:20)(cid:18) κγ ( x ) + γ A γ ( x ) (cid:19) t (21)+ (cid:18) γ κ γ ( x ) + 3 γ κA γ ( x ) + 15 γ A γ ( x ) (cid:19) t (cid:21) + O ( t )with an initial effective drift velocity − sgn( x ) (cid:18) κγ ( x ) + γ A γ ( x ) (cid:19) , (22)which is a superposition of two effects arising from: i) thedirect force − sgn( x ) κ already present in the equilibriumnoise case (where γ = 0), and ii) the linear friction gra-dient. As in the free particle case ( n = 0), the MD andthe MSD fulfil a linear relationship given by∆( t ) = − γ ( x ) γ (cid:20) κtγ ( x ) + sgn( x ) (cid:104) x ( t ) − x (cid:105) (cid:21) , (23)such that the short-time expansion for the MSD is givenby∆( t ) = Aγ ( x ) t + (cid:18) κ γ ( x ) + 3 γ κAγ ( x ) + 15 γ A γ ( x ) (cid:19) t + O ( t ) . (24)Clearly, for κ = 0, the free case is recovered.We see from the MSD that we have first a diffusive andlater a ballistic regime while for the MD the dominatingterm is the drift, as the particle feels the effects of theconstant force. In fact, the crossing time between thesetwo regimes in the MSD is t → = 4 Aγ ( x ) κ γ ( x ) + 12 γ γ ( x ) κA + 15 γ A (25)and can be made arbitrarily small by formally varying theparameters A and κ , meaning that one can in principlehave two wide regimes of initial diffusive and subsequentballistic dynamics. Two regimes with a crossover time t → already exist for equilibrium noise γ = 0 but theeffect is persistent and tunable via nonequilibrium noiseas documented by Eq.(25).Results for the MD and the MSD as well as the scal-ing exponents and passage times t x and t x are shownin Figure 3, obtained by both theory and simulation.The crossover between the initial diffusive and subse-quent ballistic behaviour in the MSD is clearly visible, inparticular in α ( t ), which shows a plateau around α ( t ) = 2for intermediate times. The simulation data even reveala transient subsequent superballistic behaviour, whichthen falls off once the particle arrives at the origin, whereit decelerates due to the opposed friction gradient. Again,for times smaller than the passage duration, theory andsimulation are in very good agreement. Harmonic potential.
Finally, for the harmonic os-cillator: U ( x ) = κx , or n = 2, separation of variables log ( t / ) l o g (( x ( t / ) x ) / l ) = 1 (a) TheorySimulation log ( t / ) ( t / ) (b) t x t x log ( t / ) l o g ( ( t / ) / l ) = 1 (c) log ( t / ) ( t / ) (d) FIG. 2. Constant friction gradient and free particle ( p, n ) =(1 , β ( t ); (c) mean-squared displacement ∆( t ); (d) associated scal-ing exponent α ( t ). The length unit is l ≡ γ /γ , while thetime unit is τ ≡ l /A . The initial position is x = 5 l . Sim-ulation data are shown with error bars as red symbols. Thetheory is the solid line. The typical passage times t x and t x are indicated by purple and orange vertical lines. log ( t / ) l o g (( x ( t / ) x ) / l ) = 1 (a) log ( t / ) ( t / ) (b) log ( t / ) l o g ( ( t / ) / l ) = 2= 1 (c) log ( t / ) ( t / ) t (d) FIG. 3. Same as Figure 2, but now for n = 1. (a) meandisplacement; (b) scaling exponent β ( t ); (c) mean-squareddisplacement ∆( t ); (d) scaling exponent α ( t ). In (c) and (d)the crossing time t → is indicated by a vertical blue line.Parameter values are: κ = γ l /τ , x = 100 l . is no longer possible and we therefore resort to a short-time expansion gained by perturbation theory (see [50]).In doing so, first we take the solution of the ( p, n ) = (1 , − κx , and next we con-sider a harmonic oscillator potential centred in x as aperturbation. Following this procedure, the short-time log ( t / ) l o g (( x ( t / ) x ) / l ) = 1 (a) log ( t / ) ( t / ) (b) log ( t / ) l o g ( ( t / ) / l ) = 1 = 2 (c) log ( t / ) ( t / ) (d) t FIG. 4. Same as Figure 2, but now for n = 2: (a) mean dis-placement (cid:104) x ( t ) − x (cid:105) and (b) scaling exponent β ( t ); (c) mean-squared displacement ∆( t ) and (d) scaling exponent β ( t ) Theparameters are κ = γ /τ and x = 10 l . expansions of the MD and MSD are: (cid:104) x ( t ) − x (cid:105) = − sgn( x ) (cid:18)(cid:20) κ | x | γ ( x ) + γ A γ ( x ) (cid:21) t + (cid:20) − | x | κ γ ( x ) + γ κ x γ ( x ) − κAγ γ ( x ) (26)+ 32 | x | κγ Aγ ( x ) + 158 γ A γ ( x ) (cid:21) t (cid:19) + O ( t )and ∆( t ) = Aγ ( x ) t + (cid:20) x κ γ ( x ) − κAγ ( x ) (27)+ 3 γ κ | x | Aγ ( x ) + 154 γ A γ ( x ) (cid:21) t + O ( t ) . In this case, the MD only shows a linear behaviour, whilethe MSD displays two different regimes, diffusive and bal-listic, separated by the crossing time t → = (28)4 γ ( x ) A γ ( x ) x κ − γ ( x ) κA + 12 γ γ ( x ) κx A + 15 γ A . Figure 4 shows the comparison of the perturbation theorywith the full numerical simulations revealing very goodagreement for times smaller than a typical passage time.Clearly, for larger times, the particles becomes confinedby the harmonic potential around the origin as signaledby a plateau arising in the MD and MSD for times largerthan the typical passage time. Correspondingly, bothscaling exponents β ( t ) and α ( t ) drop to zero. B. Linear friction gradient
We now turn to a linear friction gradient, p = 2, wherethere is no nonanalyticity in the spatial dependence ofthe friction at the origin. Then Eq.(2) becomes( γ + γ x ) ˙ x ( t ) = − U (cid:48) ( x ) + √ Aξ ( t ) . (29)Bearing in mind that the free case is a simple special caseof the n = 1 one (for κ = 0), we directly show the resultsfor n = 0 , κ ≥
0. The MD is (cid:104) x ( t ) − x (cid:105) = ∞ (cid:88) m =1 a m (cid:104) ζ m ( t ) (cid:105) (30)= ∞ (cid:88) m =1 a m (cid:98) m/ (cid:99) (cid:88) k =0 m ! A k ( − sgn( x ) κ ) m − k ( m − k )!2 k k ! t m − k , where the factors a m are straightforwardly obtained byTaylor expanding the expression ( x ( t ) − x ), calculatedusing separation of variables, in powers of ζ ( t ) = − sgn( x ) κt + √ A (cid:90) t dt (cid:48) ξ ( t (cid:48) ) . (31)Here a = γ ( x ) − , but the expressions for the coeffi-cients a m for m ≥ t ) = ∞ (cid:88) m =2 b m (cid:104) ζ m ( t ) (cid:105) (32)= ∞ (cid:88) m =2 b m (cid:98) m/ (cid:99) (cid:88) k =0 m ! A k ( − sgn( x ) κ ) m − k ( m − k )!2 k k ! t m − k , where b = γ ( x ) − and the coefficients b m for m ≥ p = 1case, with a simple diffusive behaviour if κ = 0 and both adiffusive and ballistic behaviour otherwise. A comparisonbetween theory and simulations is shown in Figure 5 forthe free case and in Figure 6 for n = 1.For the case n = 2 we used perturbation theory tocalculate up to the first order in time for the MD and upto the second order in time for the MSD: (cid:104) x ( t ) − x (cid:105) = (cid:18) − κx γ ( x ) + a A (cid:19) t + O ( t ) , (33)∆( t ) = Atγ ( x ) + (34) (cid:20) κ x γ ( x ) − κAγ ( x ) − b Aκx + 3 b A (cid:21) t + O ( t ) , where the a i and b i are the coefficients already used inEqs. (30) and (32). We see again a linear behaviourfor the MD while the MSD goes from diffusive to ballis-tic. In Figure 7 we compare these results with numericalsimulations. log ( t / ) l o g (( x ( t / ) x ) / l ) = 1 (a) log ( t / ) ( t / ) (b) log ( t / ) l o g ( ( t / ) / l ) = 1 (c) log ( t / ) ( t / ) (d) FIG. 5. Linear friction gradient p = 2 for a free particle( n = 0): (a) mean displacement (cid:104) x ( t ) − x (cid:105) and (b) scalingexponent β ( t ); (c) mean-squared displacement ∆( t ), (d) scal-ing exponent α ( t ). The length units used is l = (cid:112) γ /γ and the time unit is τ = l /A . The chosen initial position is x = 3 l . log ( t / ) l o g (( x ( t / ) x ) / l ) = 1 (a) log ( t / ) ( t / ) (b) log ( t / ) l o g ( ( t / ) / l ) = 2= 1 (c) log ( t / ) ( t / ) t (d) FIG. 6. Same as Figure 5, but now for n = 1: (a) meandisplacement (cid:104) x ( t ) − x (cid:105) and (b) scaling exponent β ( t ); (c)mean-squared displacement ∆( t ), (d) scaling exponent α ( t )and indicated crossing time t → . Parameter values: κ = γ l /τ , x = 10 l . IV. LONG-TIME BEHAVIOUR
We now consider the stationary long-time behaviour.In order to keep a normalized probability distributionfunction, we confine the system in a potential ( n = 1 , x -coordinate which can be com-puted from the Fokker-Planck equation corresponding tothe process Eq.(2). We rewrite, analogous to Eq.(4),˙ x ( t ) = a ( x ) + b ( x ) ξ ( t ) (35) log ( t / ) l o g (( x ( t / ) x ) / l ) = 1 (a) log ( t / ) ( t / ) (b) log ( t / ) l o g (( t / ) / l ) = 1 = 2 (c) log ( t / ) ( t / ) (d) t FIG. 7. Same as Figure 5, but now for n = 2: (a) meandisplacement (cid:104) x ( t ) − x (cid:105) and (b) scaling exponent β ( t ); (c)mean-squared displacement ∆( t ), (d) scaling exponent α ( t )and indicated crossing time t → . Parameter values: κ = γ /τ , x = 10 l . with a ( x ) ≡ − U (cid:48) ( x ) γ ( x ) , b ( x ) ≡ √ Aγ ( x ) . (36)The Fokker-Planck equation for this case has been de-rived in [33, 53] and reads as ∂ t p ( x, t ) = − ∂ x [ a ( x ) p ( x, t )] + 12 ∂ x [ b ( x )[ ∂ x [ b ( x ) p ( x, t )]]] , (37)admitting a stationary solution at zero flux which is givenby p ( x ) = Nb ( x ) exp (cid:20)(cid:90) x dy a ( y ) b ( y ) (cid:21) , (38)where N is a normalization factor. The integrand in theexponential of Eq.(38), denoted by I ( y ), can be expressedin terms of the confining potential and the friction termas I ( y ) = − A U (cid:48) ( y ) γ ( y ) (39)which shows that it is given by polynomial expressionsfor the cases we address now.Taking γ ( x ) = γ + γ | x | p and U ( x ) = ( κ/n ) | x | n , whichcovers both our cases of interest for p = 1 , n = 1 , p ( x ) = N √ A ( γ + γ | x | p ) × (40)exp (cid:20) − κA (cid:18) γ n | x | n + γ n + p | x | n + p (cid:19)(cid:21) . We can now discuss the different cases as a functionof the exponent pairs ( p, n ). For the lowest-order case
1: Gauss-like-distribution; γ = 1: flat-topdistribution; γ = 2; bimodal Gaussian-like distribution. ( p, n ) = (1 ,
1) one has the superposition of the exponen-tials of a Laplace- and a Gaussian distribution, as shownin Figure 8. The resulting PDF therefore interpolatesbetween a Laplace-like distribution in the limit γ (cid:29) γ and a Gaussian-like distribution up to γ = 2 γ a , wherethe coefficient a ≡ κ/A takes care of the different physi-cal dimensions of γ and γ ; we set a ≡
1. For still largervalues of γ (cid:29) γ , the monomodal Gaussian distributionsplits in what we call a bimodal “mirrored” Gaussiandistribution. This name reflects the observation that theresulting distribution looks like a Gaussian placed closeto a mirror, with the parts of the image behind the mirrorcut out. It is important to note that for the presence ofthese different distribution forms the friction-dependentprefactor is important; at x = 0 it is a constant, butwithin a range of x -values around zero it reweights thedistribution away from that constant, before for large val-ues of x the exponential contribution becomes dominant.The PDF in the case ( p, n ) = (2 ,
1) shows the same be-haviour, which can be read off from the exponents. Theleading Laplacian terms in unaltered since n = 1, whilethe subsequent term now acquires a cubic nonlinearity.In the case ( p, n ) = (1 ,
2) the leading order term is now aGaussian term, which therefore dominates at small val-ues of γ . As in the previous cases, for increasing valuesof γ , the distribution immediately turns into a mirroredGaussian-distribution, i.e. the maximum of the distribu-tion splits into two maxima.Finally, ( p, n ) = (2 ,
2) the polynomial in the exponentis even and of fourth-order, with Gaussian behaviourdominating at low values of γ . Going from small tolarge γ , one now crosses over from a Gaussian-like to abimodal Gaussian-like-distribution, which now is smoothat x = 0 due to the absence of modulus terms. This formis shown in Figure 9. All behaviours found are summa-rized in Table I. TABLE I. Graphic summary of the PDFs for the cases ( p, n )for p = 1 , n = 1 ,
2, varying only the friction strengths γ and γ . For γ = 0, the distributions are either Laplacian(L) or Gaussian (G); left-most points. Increasing γ leadsto mirrored Gaussian behaviour (MG), passing via Gaussianbehaviour at γ = 2 γ a , a = ( κ/A ) ≡
1. This applies toboth ( p, n ) = (1 ,
1) and ( p, n ) = (2 , p, n ) = (1 , γ = 0, the PDF changes intoa mirrored Gaussian shape for finite positive γ . Finally,for ( p, n ) = (2 , γ = γ a , with again a = 1.( p, n ) • γ −−−−−−−−−−−−−−−−−−−−−−→ (1 , • L G −−−−−−−−−−→ • MG −−−−−−−−−−−→ (2 , γ = 2 γ (1 , • G MG −−−−−−−−−−−−−−−−−−−−−−−→ (2 , • G FT −−−−−→ • BG −−−−−−−−−−−−−−−−→ γ = γ We end by considering the robustness of our resultswith respect to thermal fluctuations. Following Baule etal. [33], we consider the Langevin equation γ ( x ) ˙ x ( t ) = − U (cid:48) ( x ) + (cid:112) γ ( x ) k B T η ( t ) + √ Aξ ( t ) (41)where η ( t ) is a Gaussian white noise. The thermal andactive processes η ( t ) and ξ ( t ) being uncorrelated, theycan be superimposed to ξ T ( t ) = η ( t ) + ξ ( t ), leading to˙ x ( t ) = a T ( x ) + b T ( x ) ξ T ( t ) , (42) with a T ( x ) = − γ ( x ) (cid:18) U (cid:48) ( x ) + k B T γ (cid:48) ( x ) γ ( x ) (cid:19) (43)and b T ( x ) = 1 γ ( x ) (cid:18) k B T + Aγ ( x ) (cid:19) . (44)The integrand I ( y ) in the exponential of the PDF readsas I ( y ) = − (cid:20) U (cid:48) ( y ) γ ( y ) + k B T γ (cid:48) ( x ) A + 2 k B T γ ( x ) (cid:21) , (45)which can be compared with Eq.(39) in the purely activecase. As a robustness check it suffices to examine thebehaviour of the integrand I ( y ) near the origin for smallvalues of y and for y → ±∞ , for our four cases ( p, n ), n = 1 , p = 1 ,
2. For the behaviour near the originone finds that the dominator behaves in a similar fashionas I ( y ) of Eq.(39), generating a polynomial with identi-cal powers, since the temperature-dependent term eithercontributes a sgn( x ) for p = 1 or a linear term for p = 2.The qualitative behaviour of the PDFs remains thus un-altered. For large arguments, one sees that generally I ( y )behaves as I ( y ) ∝ − U (cid:48) ( y ) k B T , (46)such that the tails of the distributions are determinedby thermal fluctuations and decay exponentially, i.e.Laplace-like for n = 1 or Gaussian-like for n = 2; theactive noise and the friction term then only play a rolein the prefactor of the PDF. V. DISCUSSION AND CONCLUSIONS
In this work we have studied the stochastic dynamicsof an active-noise driven particle under the influence ofa space-dependent friction and confinement. In order toelucidate the effect of the space-dependence of the fric-tion term, we start the dynamics for large initial values,so that the friction term dominates the dynamics. Forthe case of a free particle, a particle running down aramp and a harmonic potential we have determined themean displacement and mean-squared displacement andthe corresponding scaling exponents β ( t ) and α ( t ) in ashort-time expansion. The mean displacements generallyshow diffusive behaviours, while a crossover to a ballisticregime is observed for the mean-squared displacement,except for the free particle case.Further, we have determined the effect of the frictionterm in the presence of a confining potential U ( x ) ∝ | x | n for n = 1 , p, n ) and the rela-tive magnitude of the friction coefficients γ and γ . Oneobserves that the friction law and the confinement po-tential conspire to generate a set of generic behaviours:Laplace-like and Gaussian-like distributions for n = 1and n = 2, respectively, if the spatially-dependent fric-tion term is small ( γ (cid:28) γ ); this behaviour crosses overfor γ = γ to Gaussian behaviour for both p = 1 , n = 2, Laplace-like behaviour is absent.For all cases of ( p, n ) with n = 1 , p = 1 ,
2, one observesthat for γ (cid:29) γ , the stationary PDF displays a mirroredor bimodal Gaussian-like behaviour. Therefore, generallyfor all combinations of ( p, n ), at sufficiently strong space-dependent friction, the PDF becomes a bimodal distri-bution with a symmetrically increased weight off-centerof the potential minimum.To conclude, our study extends current studies on ac-tive particles in 1d by the inclusion of a space-dependentfriction and therefore links the problem to earlier stud-ies of molecular motors on linear tracks. Investigationsof the stationary probability density functions for therun-and-tumble process have already generated an ex-tended catalogue of distributions, see, e.g. [15], in whichalso bimodal-type PDFs appear (see their Fig.7), or [54].Placed in this context, the present study reveals a basicgrammar in which such complex distributions are gen-erated for the case of a space-dependent friction. Ourmodel system allows to extract the mechanism of shapechange of the PDFs in a particularly clear manner.For the future, our model and analysis should beextended to two spatial dimensions to take into accountfull viscosity landscapes [55–58]. Moreover inertialeffects can be included in the particle dynamics [59–62].Finally collective effects for many active-noise drivenparticles such as motility-induced phase separationshould be explored [63, 64]. Acknowledgement.
DB is supported by the EUMSCA-ITN ActiveMatter, (proposal No. 812780). RB isgrateful to HL for the invitation to a stay at the Heinrich-Heine University in D¨usseldorf where this work was per- formed.
AppendixNumerical treatment of the Langevin equation
The stochastic equation˙ x ( t ) = a ( x ) + b ( x ) ξ ( t ) (47)is of the standard form dx t = a ( x t ) dt + b ( x t ) dW t , (48)where W t represents a Wiener process. In order to solvethis equation numerically in the Stratonovich paradigm,we implement a predictor-corrector scheme. In such ascheme, one first performs a full time step evolution ofthe position of the particle x ( t i ) using the same timecoefficients a ( x ( t i )) and b ( x ( t i )). This predicted position x p is used to calculate a ( x p ) and b ( x p ) and proceed tofinally calculate the position at time step t i +1 using theaverages of the coefficients calculated for x ( t i ) and x p . Toimplement the Stratonovich paradigm, using this kind ofaverage only for the stochastic part (and hence the b ( x ))is necessary, but we preferred to apply this procedure aswell to the deterministic part in order to improve stabilityof the result. The method we decided to use for the timeevolution is thus a Milstein scheme, of order O (∆ t ) [65].The Milstein evolution of Eq.48 can be written as: x ( t i +1 ) = x ( t i ) + a ( x ( t i ))∆ t + (49) b ( x ( t i ))∆ W ( t i ) + 12 b ( x ( t i )) db ( x ( t i )) dx ((∆ W ( t i )) − ∆ t ) , where ∆ W ( t i ) = W ( t i +1 − W ( t i ) is a normal-distributedrandom variable.It should be noted that the fact that the Milsteinscheme uses the derivative of the function b ( x ), which forour model is discontinuous at x = 0 for the case p = 1.This can be treated by adopting an algorithm devel-oped in [66], employing colored noise from the Ornstein-Uhlenbeck process. [1] J. P. Bouchaud, A. Comtet, A. Georges, and P. Le Dous-sal, Annals of Physics , 285 (1990).[2] J. Tailleur and M. E. Cates, Physical Review Letters ,218103 (2008).[3] B. Lindner and E. M. Nicola, The European PhysicalJournal Special Topics , 43 (2008).[4] J. Toner, Y. Tu, and S. Ramaswamy, Annals of PhysicsSpecial Issue, , 170 (2005).[5] S. Ramaswamy, Annual Review of Condensed MatterPhysics , 323 (2010).[6] M. E. Cates and J. Tailleur, Annual Review of CondensedMatter Physics , 219 (2015).[7] C. Bechinger, R. Di Leonardo, H. L¨owen, C. Reichhardt,G. Volpe, and G. Volpe, Reviews of Modern Physics , 045006 (2016).[8] J. Elgeti, R. G. Winkler, and G. Gompper, Reports onProgress in Physics , 056601 (2015).[9] P. Romanczuk and U. Erdmann, The European PhysicalJournal Special Topics , 127 (2010).[10] Y. Ben Dor, E. Woillez, Y. Kafri, M. Kardar, and A. P.Solon, Physical Review E , 052610 (2019).[11] P. Illien, C. de Blois, Y. Liu, M. N. van der Linden, andO. Dauchot, Physical Review E , 040602 (2020).[12] E. F. Teixeira, H. C. M. Fernandes, and L. G. Brunnet,“Single active ring model,” (2021), arXiv:2102.03439[cond-mat.soft].[13] T. Demaerel and C. Maes, Physical Review E , 032604(2018). [14] K. Malakar, V. Jemseena, A. Kundu, K. V. Kumar,S. Sabhapandit, S. N. Majumdar, S. Redner, andA. Dhar, Journal of Statistical Mechanics: Theory andExperiment , 043215 (2018).[15] A. Dhar, A. Kundu, S. N. Majumdar, S. Sabhapandit,and G. Schehr, Physical Review E , 032132 (2019).[16] P. Le Doussal, S. N. Majumdar, and G. Schehr, EPL(Europhysics Letters) , 40002 (2020).[17] K. Bia(cid:32)las, J. (cid:32)Luczka, P. H¨anggi, and J. Spiechowicz,Physical Review E , 042121 (2020).[18] D. S. Dean, S. N. Majumdar, and H. Schawe, Phys. Rev.E , 012130 (2021).[19] L. Angelani, Journal of Physics A: Mathematical andTheoretical , 325601 (2017).[20] N. Razin, R. Voituriez, J. Elgeti, and N. S. Gov, PhysicalReview E , 032606 (2017).[21] N. Razin, Physical Review E , 030103 (2020).[22] B. ten Hagen, S. van Teeffelen, and H. L¨owen, Journalof Physics: Condensed Matter , 194119 (2011).[23] Q.-H. Wei, C. Bechinger, and P. Leiderer, Science ,625 (2000).[24] C. Lutz, M. Kollmann, and C. Bechinger, Physical Re-view Letters , 026001 (2004).[25] S. Herrera-Velarde, A. Zamudio-Ojeda, andR. Casta˜neda-Priego, The Journal of Chemical Physics , 114902 (2010).[26] G. Szamel, Physical Review E , 012111 (2014).[27] R. Wittmann, C. Maggi, A. Sharma, A. Scacchi, J. M.Brader, and U. M. B. Marconi, Journal of Statistical Me-chanics: Theory and Experiment , 113207 (2017).[28] S. Das, G. Gompper, and R. G. Winkler, New Journalof Physics , 015001 (2018).[29] L. Caprini and U. Marini Bettolo Marconi, Soft Matter , 9044 (2018).[30] L. Caprini, U. M. B. Marconi, and A. Vulpiani, Journalof Statistical Mechanics: Theory and Experiment ,033203 (2018).[31] F. J. Sevilla, R. F. Rodr´ıguez, and J. R. Gomez-Solano,Physical Review E , 032123 (2019).[32] K. V. Kumar, S. Ramaswamy, and M. Rao, PhysicalReview E , 020102 (2008).[33] A. Baule, K. V. Kumar, and S. Ramaswamy, Journalof Statistical Mechanics: Theory and Experiment ,P11008 (2008).[34] A. Mogilner, M. Mangel, and R. J. Baskin, Physics Let-ters A , 297 (1998).[35] H. C. Fogedby, R. Metzler, and A. Svane, Physical Re-view E , 021905 (2004).[36] A. B. Kolomeisky and H. Phillips, Journal of Physics:Condensed Matter , S3887 (2005).[37] Y. A. Makhnovskii, V. M. Rozenbaum, D.-Y. Yang, S. H.Lin, and T. Y. Tsong, The European Physical JournalB - Condensed Matter and Complex Systems , 501(2006).[38] V. M. Rozenbaum, Y. A. Makhnovskii, D.-Y. Yang, S.-Y.Sheu, and S. H. Lin, The Journal of Physical ChemistryB , 1959 (2010).[39] Y. A. Makhnovskii, V. M. Rozenbaum, S.-Y. Sheu, D.-Y.Yang, L. I. Trakhtenberg, and S. H. Lin, The Journal ofChemical Physics , 214108 (2014).[40] R. Blossey and H. Schiessel, Journal of Physics A: Math-ematical and Theoretical , 085601 (2019). [41] F. Evers, R. D. L. Hanes, C. Zunke, R. F. Capell-mann, J. Bewerunge, C. Dalle-Ferrier, M. C. Jenkins,I. Ladadwa, A. Heuer, R. Casta˜neda-Priego, and S. U.Egelhaaf, The European Physical Journal Special Topics , 2995 (2013).[42] C. Lozano, B. ten Hagen, H. L¨owen, and C. Bechinger,Nature Communications , 12828 (2016).[43] S. Jahanshahi, C. Lozano, B. Liebchen, H. L¨owen, andC. Bechinger, Communications Physics , 1 (2020).[44] M. A. Fernandez-Rodriguez, F. Grillo, L. Alvarez,M. Rathlef, I. Buttinoni, G. Volpe, and L. Isa, NatureCommunications , 4223 (2020).[45] A. R. Sprenger, M. A. Fernandez-Rodriguez, L. Alvarez,L. Isa, R. Wittkowski, and H. L¨owen, Langmuir , 7066(2020).[46] B. Liebchen, P. Monderkamp, B. ten Hagen, andH. L¨owen, Physical Review Letters , 208002 (2018).[47] M. R. Stehnach, N. Waisbord, D. M. Walkama, and J. S.Guasto, bioRxiv , 2020.11.05.369801 (2020).[48] P. U. Shirke, H. Goswami, V. Kumar, D. Shah, S. Das,J. Bellare, K. V. Venkatesh, J. R. Seth, and A. Ma-jumder, bioRxiv , 804492 (2019).[49] C. E. Lopez, J. Gonzalez-Gutierrez, F. Solorio-Ordaz,E. Lauga, and R. Zenit, arXiv:2012.04788 [physics](2020).[50] D. Breoni, M. Schmiedeberg, and H. L¨owen, Phys. Rev.E , 062604 (2020).[51] M. Doi and S. Edwards, The Theory of Polymer Dynam-ics, Clarendon Press, Oxford University Press, New York(1986).[52] A. W. C. Lau and T. C. Lubensky, Phys. Rev. E ,011123 (2007).[53] D. Ryter, Zeitschrift f¨ur Physik B Condensed Matter ,39 (1981).[54] F. J. Sevilla, A. V. Arzola, and E. P. Cital, Phys. Rev.E , 012145 (2019).[55] S. Coppola and V. Kantsler, Scientific Reports , 399(2021).[56] C. Datt and G. J. Elfring, Phys. Rev. Lett. , 158006(2019).[57] R. Dandekar and A. M. Ardekani, Journal of Fluid Me-chanics , R2 (2020).[58] B. Liebchen and H. L¨owen, Europhysics Letters ,34003 (2019).[59] C. Scholz, S. Jahanshahi, A. Ldov, and H. L¨owen, NatureCommunications , 5156 (2018).[60] H. L¨owen, The Journal of Chemical Physics , 040901(2020).[61] A. R. Sprenger, S. Jahanshahi, A. V. Ivlev, andH. L¨owen, arXiv:2101.01608 (2021).[62] L. Caprini and U. Marini Bettolo Marconi, The Journalof Chemical Physics , 024902 (2021).[63] D. Marenduzzo, The European Physical Journal SpecialTopics , 2065 (2016).[64] Z. Ma, M. Yang, and R. Ni, Advanced Theory and Sim-ulations , 2000021 (2020).[65] G. N. Mil’shtejn, Theory of Probability & Its Applica-tions , 557 (1975).[66] R. Perez-Carrasco and J. M. Sancho, Phys. Rev. E81