A First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
AA First-Order Dynamical Transition in thedisplacement distribution of a DrivenRun-and-Tumble Particle
Giacomo Gradenigo
Dipartimento di Fisica, Universit`a Sapienza, Piazzale Aldo Moro 5, I-00185,Rome, ItalyCNR-Nanotec, Institute of Nanotechnology, UOS-Roma, Rome, Italy
Satya N. Majumdar
LPTMS, CNRS, Univ. Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, France
Abstract.
We study the probability distribution P ( X N = X, N ) of the totaldisplacement X N of an N -step run and tumble particle on a line, in presence ofa constant nonzero drive E . While the central limit theorem predicts a standardGaussian form for P ( X, N ) near its peak, we show that for large positive andnegative X , the distribution exhibits anomalous large deviation forms. For largepositive X , the associated rate function is nonanalytic at a critical value of thescaled distance from the peak where its first derivative is discontinuous. Thissignals a first-order dynamical phase transition from a homogeneous ‘fluid’ phaseto a ‘condensed’ phase that is dominated by a single large run. A similar first-ordertransition occurs for negative large fluctuations as well. Numerical simulationsare in excellent agreement with our analytical predictions. a r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
1. Introduction
Recent years have seen immense theoretical and experimental interest in the studyof ‘active’ systems, consisting of self-propelled individual particles [1–4]. Theseactive particles exhibit novel collective nonequilibrium pheomena, such as the motilityinduced phase separation (MIPS) [5–11], clustering effect [12], spontaneous segregationof mixtures of active and passive particles [13] and many other interesting effects.These collective effects arise from a combination of self-propulsion and interactionbetween the active particles. However, even in the absence of interactions betweenparticles (noninteracting limit), the stochastic process associated with a single activeparticle is rather interesting due purely to the self-propulsion. This self-propulsioninduces a memory or ‘persistence’ in the effective noise felt by the particle, leadingoften to interesting non-Markovian effects. At the level of individual particles, thesimplest examples of such active particles are the so called active Brownian motion(ABM) or the ‘Run-and-Tumble’ particle (RTP) [for a recent pedagogical review,see [4]]. For a single ABM particle, both free as well as confined in a harmonictrap, there have been a number of recent theoretical and experimental studies intwo dimensions on the position distribution [14–20], as well as on its first-passageproperties [18]. In this paper, we will focus on the other well studied model of a singleactive particle, namely the RTP [5,8,10] in one dimension, but subjected to a constantexternal force
E > γ , i.e., the distribution of the durationof a single run between two successive tumblings is exponential with parameter γ . Wewill set γ = 1 for the rest of the paper. We will focus here in one dimension. At theend of each tumbling, the particle chooses a new velocity drawn independently (fromrun to run) from a probability distribution funtion (PDF) q ( v ), which is typicallysymmetric. In the standard RTP model known as the persistent random walk, q ( v ) ischosen to be bimodal: q ( v ) = 12 [ δ ( v − v ) + δ ( v + v )] . (1)In this model, the position x ( t ) of the RTP evolves in time via dxdt = v σ ( t ) , (2)where σ ( t ) = ± γ = 1. This persistent random walk model has been studiedextensively in the past and many properties are known, e.g. the propagator andthe mean exit time from a finite interval, amongst other observables [21, 22]. In onedimension, there have been a number of recent theoretical studies on the first-passageproperties of a free RTP [23–28] and, very recently, for an RTP subjected to an externalconfining potential [29].In this paper, we will study a variant of this standard RTP model in onedimension. In our model, while the duration of a run, say τ i for the i -th run, isstill exponentially distributed with rate γ = 1, the actual motion during a ‘run’ isdifferent. In our model there is an external force E >
First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
3a run. More precisely, at the begining of the i -th run, the particle again chooses anew velocity v i from a PDF q ( v ) (which is not necessarily bimodal). Then startingwith this initial velocity v i , the particle moves via Newton’s second law during therun duration 0 ≤ t ≤ τ i dxdt = v ( t ); m dvdt = E ; v (0) = v i . (3)Thus we assume that there is no friction due to the environment (the particle’s motionis thus not overdamped as in standard Brownian motion). We will also set the mass m = 1 for simplicity. Integrating Eq. (3) trivially, the displacement x i during the i -thrun is given by x i = v i τ i + E τ i (4)where both τ i and v i are independent random variables, drawn respectively from p ( τ ) = e − τ θ ( τ ) and q ( v ) which is arbitrary (albeit symmetric). For E = 0 and q ( v )bimodal as in Eq. (1), our model reduces to the standard RTP. We will focus hereon E > q ( v ) = e − v / / √ π , though our resultson condensation (see later) will hold for a large class of velocity distributions q ( v ),including the bimodel case discussed above. We consider N successive runs. We workhere in the ensemble where the total number N of runs (or tumbles) is fixed, ratherthan the total time elapsed t . However, our results can be easily extended to constanttime ensemble. In the presence of a constant force E , the total distance travelled bythe particle after N runs is X N = N (cid:88) i =1 x i = N (cid:88) i =1 (cid:20) v i τ i + E τ i (cid:21) (5)In this paper, we are interested in the PDF P ( X, N ) = Prob . [ X N = X ] of the totaldisplacement for large N . Our main new result is that for E > q ( v )’s including the Gaussian and the bimodal distributions, the PDF P ( X, N ), forlarge N , exhibits a non-analyticity as a function of X —signalling a condenation typefirst-order ‘phase transition’ in the system, as disussed below. We show that for thestandard RTP, i.e, for E = 0 and q ( v ) bimodal as in Eq. (1), this interesting phasetransition disappears. Our main results are summarized in the next section. Belowwe discuss some qualitative features of this PDF P ( X, N ) for large N and the physicsbehind the phase transition, before moving to a more quantitative detailed analysis inthe later sections.Due to the presence of a nonzero E >
0, the PDF P ( X, N ) is clearly asymmetricas a fuction of X (see Fig. 1), and it has three regimes which are denoted as I , II and III in Fig. 1. In the central regime II , P ( X, N ) describes the probability of typical fluctuations of X , while regimes I and III correspond to atypically large fluctuationsof X on the negative and the positive side respectively. A “kink”, which is shownschematically in Fig. 1 at X = X c , separates the typical fluctuations regime ( II )from positive large deviations (regime III ). Another similar kink at X = − X c (seeFig. (1)) separates regimes I and II on the side of negative fluctuations. The mainresult of this paper is to demonstrate that this change in the nature of the fluctuationsat X = X c corresponds to a dynamical first-order transition. A similar transitionseparates fluctuations in the center of the distribution from negative large deviations(at the second kink on the left at X = − X c , although in this case the transition is First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle -2000400 0 200 X ( t ) t -2000400 0 200 X ( t ) t -2000400 0 200 X ( t ) t10 -6 -3 -60 -40 -X c -
40 60
I II III P ( X , N ) X Figure 1:
Main : Schematic representation of P ( X, N ). Mean value is (cid:104) X (cid:105) = E N . Vertical lines separates three regions, I , II and III , correspondingrespectively to
X < − X c , − X c ≤ X < X c and X > X c . The two dynamicaltransitions are located at X c and − X c . Region II is the homogeneous phase.Regions I and III are the condensed phases. Dotted parabola is a guideto the eye. For X ∈ [ −(cid:104) X (cid:105) , (cid:104) X (cid:105) ] (inside the dotted vertical lines), thePDF P ( X, N ) is the result of a saddle-point approximation.
Insets : typicaltrajectories for each of the three regions: homogeneous trajectories for II ,dominated by one single run for I and III .“hidden” by an exponential prefactor e − E | X | . In the central fluid regime II , the totaldistance X is democratically distributed between N individual runs of average sizes,while in regime III (respectively in I ) there is a large positive (negative) “condensate”(i.e., a single run that is large) that coexists with ( N −
1) typical runs. By zoomingin close to the kinks or the critical points, we show that P ( X, N ) is described byan anomalous large deviation form near the kinks, with a local rate function that iscontinuous at the kink, but its first-derivative has a discontinuous jump (see the resultsof numerical simulations in Figs. 4)—thus signalling a first-order phase transition.Thus in our simple model, the PDF of the total displacement P ( X, N ) exhibitsa first-order phase transition, similar to the condensation transition that occurs invarious lattice models of mass-transport [30–37]. In these models, each site of a lattice(of N sites) has a certain mass m i ≥ M is conserved by the dynamics. The Zero Range Process (ZRP) is a special caseof these more general mass-transport models [30–33]. The dynamics drives the systeminto a nonequilibrium steady state and there is a whole class of models for which the First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
M < M c , to acondensed phase ( M > M c ) where one single site acquires a mass proportional to thetotal mass M [32, 33]. This single site is the so called condensate.The total distance X travelled by an RTP in N runs in our model is thecounterpart to the total mass M in mass-transport models on a lattice with N sites. Hence, the condensate (a single site carrying a mass proportional to N ) inthe mass transport model corresponds to a single extensive run in the RTP model.One difference is that in mass-transport models, the mass M is always positive, unlikein our case where X can be both positive and negative. Consequently, we have both“positive” and “negative” condensates, while in the standard mass-transport models,there is only a “positive” condensate. This explains why we have a pair of criticalpoints (see Figs. 1), as opposed to a single critical point in mass-transport models.Another difference lies in the observable of interest. In mass-transport models, thecentral object of interest is the mass distribution at a single lattice site and it showsdifferent behaviors across the condensation transition. In our case we focus on asimpler object, the distribution P ( X, N ) (playing the role of the partition function inmass-transport models with a factorised steady state), and we show how the signatureof the condensation transition is already manifest in P ( X, N ) itself. One of our mainresults is to show that near this critical point, P ( X, N ) exhibits an anomalous largedeviation form with an associated rate function that shows a discontinuity in its firstderivative.The rest of the paper is organized as follows. In Section 2, we define our modelprecisely and summarize the main results. Section 3 contains the most extensiveanalytical computation of the distribution P ( X, N ) of the total displacement for
X > (cid:104) X (cid:105) = EN (positive fluctuations). It also includes a discussion on the detailsof numerical simulations (Section 3.4). Section 4 contains analogus computation of P ( X, N ) for
X < −(cid:104) X (cid:105) = − EN , i.e, for negative fluctuations. Section V contains asummary and conclusions. Finally, some details of the computations are presented inthe Appendices.
2. The model and the summary of the main results
We consider a single RTP on a line, starting initially at X = 0. Each trajectory ismade of N independent runs. The i -th run starts with initial velocity v i and lasts arandom time τ i . The particle is also subjected to a constant force (field) E >
0. Thetotal displacement of the particle after N runs, using Newton’s law for each run, istherefore given by X N = N (cid:88) i =1 x i = N (cid:88) i =1 (cid:20) v i τ i + E τ i (cid:21) (6)where x i denotes the displacement during the i -th run. The velocity v i ’s and theduration τ i ’s for each run are i.i.d random variables drawn from the normalized PDF’s q ( v ) = 1 √ π exp (cid:2) − v / (cid:3) (7) p ( τ ) = Θ( τ ) exp( − τ ) (8) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle t ) is the Heaviside theta function: Θ( t ) = 1 for t ≥ t ) = 0 for t < q ( v ) in Eq. (7), our main conclusions concerning the first-order phase transition isvalid for a broad class of q ( v )’s, including the bimodal distribution in Eq. (1). Ourgoal is to compute the probability distribution P ( X, N ) = Prob . [ X N = X ] of the totaldisplacement X N = (cid:80) Ni =1 x i . Thus, X N is clearly a sum of N i.i.d. random variables.Each of the x i ’s has the normalized marginal PDF P ( x ) = ∞ (cid:90) −∞ dv ∞ (cid:90) dτ q ( v ) p ( τ ) δ ( x − vτ − Eτ / . (9)where q ( v ) and p ( τ ) are given in Eqs. (7) and (8) respectively. The mean and thevariance of the displacement during each run can be computed easily and one gets (cid:104) x (cid:105) = E (10) σ = (cid:104) x (cid:105) − (cid:104) x (cid:105) = 2 + 5 E (11)Computing explicitly P ( x ) from Eq. (9) is hard, however as we will see, what reallymatters for the large N behavior of P ( X, N ) is the asymptotic tail behavior of P ( x ).These tails can be explicitly obtained (see Appendix A). For large positive x we get P ( x → ∞ ) ≈ E e /E x − / e − √ x/E , (12)and for large negative x P ( x → −∞ ) = e − E | x | P ( | x | ) ≈ e − E | x | e /E E | x | − / e − √ | x | /E . (13)Thus the PDF P ( X N = X, N ) of the sum X N = (cid:80) Ni =1 x i reads P ( X, N ) = ∞ (cid:90) −∞ (cid:34) N (cid:89) i =1 dx i P ( x i ) (cid:35) δ (cid:32) X − N (cid:88) i =1 x i (cid:33) . (14)where P ( x ) is given in Eq. (9). Relation to mass transport models and a criterion for condensation transition.
Itis interesting to notice that P ( X, N ) is formally similar to the partition function oflattice models of mass-transport with a factorised steady state [30–35]. The latterreads as: Z ( M, N ) = ∞ (cid:90) (cid:34) N (cid:89) i =1 dm i f ( m i ) (cid:35) δ (cid:32) M − N (cid:88) i =1 m i (cid:33) . (15)where m i ≥ i , f ( m i ) the corresponding steady state weightand M being the total mass. Comparing Eqs. (14) and (15), and identifying the rundistance x i with the mass m i , X with M , and f ( m i ) with P ( x i ), we see that for-mally, our P ( X, N ) is exactly the counterpart of the partition function Z ( M, N ) inmass-transport models: the only difference is that, at variance with m i ’s which are First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle x i ’s can be both positive and negative, which give rise totwo condensed phases (respectively with a long positive and a long negative run).Before discussing our strategy for the computation of P ( X, N ) in Eq. (14) with P ( x ) given by (9), it is useful to recall, from the literature on the mass-transportmodels, which classes of P ( x ) may lead to the phenomenon of condensation. For themass-transport models with positive mass m distributed via the PDF f ( m ) in Eq. (15),it is known [33] that a condensation occurs when the tail of f ( m ) remains bounded inthe interval e − cm < f ( m ) < /m , as m → ∞ , where c > f ( m ) ∼ m − − µ with µ >
1, and hence exhibitscondensation [32, 33]. However, another class of f ( m )’s that satisfy these bounds forlarge m is the so called stretched exponential class: f ( m ) ∼ exp [ − a m α ] for large m ,with α >
0. Hence this class will also exhibit the condensation transition [34, 35]. Inour RTP model with p ( τ ) = e − τ and q ( v ) = e − v / / √ π , we see that for large x , P ( x )in Eq. (12) decays as a stretched exponential with the stretching exponent α = 1 / X . A similarargument on the negative side shows that we will have a condensation transition forlarge negative X as well. Hence, we expect that for any choice of p ( τ ) and q ( v )that leads via Eq. (9) to a marginal distribution P ( x ) which satisfies the bounds e − c x < P ( x ) < /x for large x , one will get a condensation. For example, for E > p ( τ ) = e − τ θ ( τ ) and with a bimodal velocity distribution q ( v ) as in Eq. (1), it is easyto show (see Appendix A) that as x → ∞ , P ( x ) ∼ √ E x e − √ x/E , (16)which again satisfies the criterion for condensation. However, for the standard RTPmodel, i.e., if E = 0, p ( τ ) = e − τ and q ( v ) is bimodal as in Eq. (1), one finds (seeAppendix A) P ( x ) = 12 v e −| x | /v , (17)which does not satisfy the condensation criterion above. Hence, for the standardRTP, this condensation transition is absent. Thus, we see that while we presentdetailed calaculations only for the Gaussian velocity distribution, the phenomenon ofcondensation that we have found for the RTP model is robust: it occurs for a broadclass of p ( τ ) and q ( v ) that lead to a marginal P ( x ) satisfying the asymptotic boundsmentioned above. Incidentally, to the best of our knowledge, our model provides thefirst physical realization of the condensation belonging to this stretched exponentialclass. Strategy for the large N analysis of P ( X, N ) . Let us now briefly outline our strategyto compute analytically the PDF P ( X, N ). By using the integral representation ofthe delta function: δ ( X ) = (cid:82) e s X ds/ (2 πi ), one can write P ( X, N ) as P ( X, N ) = 12 πi s + i ∞ (cid:90) s − i ∞ ds e Nh ( s ) h ( s ) = sx + log[ L ( s )] (18) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle x = X/N and L ( s ) = ∞ (cid:90) −∞ dx e − s x P ( x ) , = √ π e s ( E − s ) (cid:112) s ( E − s ) erfc (cid:34) (cid:112) s ( E − s ) (cid:35) . (19)where erfc( z ) = √ π (cid:82) ∞ z e − u du is the complementary error function.The integration contour in Eq. (18) is the Bromwich contour in the complex s plane. There are two possible situations: (A) the equation ∂h ( s ) /∂s = 0 has a solutionfor real s = s (see Fig. 3) and then the the integral in Eq. (18) can be computedfor large N using a saddle-point approximation; (B) there is no saddle point and onehas to carry out the integration along the complex Bromwich contour. While thebehaviour of P ( X, N ) in case (A) has been already considered in [38], the accuratestudy of the PDF in case (B) is the original result of this paper: it is in this regimethat the condensation takes place. Let us just mention that in regime (A), whichcorresponds to values X ∈ [ −(cid:104) X (cid:105) , (cid:104) X (cid:105) ] with (cid:104) X (cid:105) = EN (see inside the homogeneous regime ( II ) in Fig. 1), the PDF P ( X, N ) exhibits a large deviation form of the kind P ( X, N ) ∼ exp (cid:26) − N Φ (cid:18) x = XN (cid:19)(cid:27) , (20)where the rate function Φ( x ) was computed numerically in [38]. It is easy to see,by virtue of central limit theorem [39], that in the vicinity of X = (cid:104) X (cid:105) = EN andsimilarly around X = −(cid:104) X (cid:105) = − EN , the rate function simply readsΦ( x ) = (cid:40) ( x − E ) σ , for x (cid:46) E − Ex + ( x + E ) σ , for x (cid:38) − E, (21)where x = X/N and E = (cid:104) X (cid:105) /N .Consider now studying P ( X, N ) in Eq. (18) as a function of increasing X . Thereis a saddle point s on the real s axis as long as −(cid:104) X (cid:105) < X < (cid:104) X (cid:105) . As X → (cid:104) X (cid:105) frombelow, s →
0. Similarly, as X → −(cid:104) X (cid:105) from above, s → E (see Fig. 3). Our maininterest in this paper is to study what happens when X exceeds (cid:104) X (cid:105) on the positiveside (respectively when X goes below −(cid:104) X (cid:105) on the negative side), i.e., when there isno longer a saddle point on the real s axis in the complex s plane. A detailed studyof the inverse Laplace transform in Eq. (18), when there is no saddle point, reveals arich behavior of P ( X, N ) for
X > (cid:104) X (cid:105) (respectively for X < −(cid:104) X (cid:105) ). Summary of the main results.
Let us summarize our main results for
X > (cid:104) X (cid:105) (detailed calculations are provided in Section III). Similar computations for X < −(cid:104) X (cid:105) are done in Section IV. It turns out that when X exceeds (cid:104) X (cid:105) by O ( √ N ), the behaviorof P ( X, N ) still remains Gaussian (as expected from the central limit theorem).Actually this Gaussian form continues to hold all the way up to X − (cid:104) X (cid:105) ∼ N / .However, when X −(cid:104) X (cid:105) exceeds the critical value X c = z c N / (where z c is a constant First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle c z l Ψ ( z ) z Figure 2: Continuous (red) line: rate function of Eq. (22), analytical prediction. z c ≈ .
78 is the location of the first-order dynamical transition: Ψ (cid:48) ( z )is clearly discontinuous at z c . Dotted lines indicates χ ( z ) for z < z c and z / (2 σ ) for z > z c . z l is the lowest value of z such that χ ( z ) can becomputed via a saddle-point approximation.of order 1 that we compute explicitly), the Gaussian form ceases to hold. This is wherethe condensate starts to form. In this intermediate regime, where X − (cid:104) X (cid:105) = z N / (where z ∼ O (1)), P ( X, N ) exhibits an anomalous large deviation form. Finally, inthe extreme tail regime when X − (cid:104) X (cid:105) ∼ O ( N ), where the system is dominated byone single large condensate, P ( X, N ) has a stretched exponential form. These threebehaviors are summarized as follows: P ( X, N ) ≈ e − ( X − NE ) / (2 Nσ ) for ( X − EN ) ∼ N / e − N / Ψ( z ) for ( X − EN ) ∼ N / e − √ /E ( X − EN ) / for ( X − EN ) ∼ N where z = ( X − EN ) /N / . The rate function Ψ( z ) can be expressed asΨ( z ) = min (cid:20) z σ , χ ( z ) (cid:21) (22)where the function χ ( z ) can be computed exactly (see Section III) in the regime z ∈ [ z l , ∞ ] with z l = 32 (cid:18) σ E (cid:19) / . (23) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle z l < z < ∞ , the function χ ( z ) has the asymptotic behaviors χ ( z ) = (cid:0) σE (cid:1) / z → z l (cid:113) E √ z − σ E z + O (cid:0) z / (cid:1) , z (cid:29) z / σ and χ ( z ) in Eq. (22) are plotted in Fig. 2.Clearly, there exists a critical value z = z c where these two functions cross each other,such that one gets from Eq. (22)Ψ( z ) = (cid:40) z < z c = ⇒ z / (2 σ ) z > z c = ⇒ χ ( z ) . (25)where z c is given by the solution of the equation z σ = χ ( z ) (26)At z = z c , the two functions match continuously, but the derivative Ψ (cid:48) ( z ) isdiscontinuous at z = z c (see Fig. (2), signalling a first-order dynamical phasetransition. The two functions cross each other at z c , provided z c > z l . Indeed,by writing z in units of z l and solving the matching condition in Eq. (26), we findthat, independently of the value of E , z c = 2 / z l , (27)which shows that z c > z l for any choice of the field E . All the details on thecomputation of the function χ ( z ) and the determination of z c are given in Sec. 3and in Appendix B.Our analysis also clarifies that the mechanism of this dynamical transition is atypical one for a classic first-order phase transition: we show that the PDF P ( X, N ),for X − (cid:104) X (cid:105) = z N / where z ∼ O (1), can be written as a sum of two contributions, P ( X, N ) = P G ( z, N ) + P A ( z, N ) , (28)where P G ( z, N ) denotes Gaussian fluctuations, while P A ( z, N ) (where the subscript A is for anomalous ) denotes the rare fluctuations emerging from the formation of acondensate. These two terms compete with each other. In the vicinity of the transitionpoint z c both contributions can be written in a large deviation form: P G ( z, N ) ∼ e − N / z / (2 σ ) P A ( z, N ) ∼ e − N / χ ( z ) , (29)Since for z < z c one has z / (2 σ ) < χ ( z ), see Fig. 2, then lim N →∞ P A ( z, N ) /P G ( z, N ) =0: the Gaussian contribution dominates. On the contrary for z > z c onefinds z / (2 σ ) > χ ( z ) and the probability of the condensate takes over, i.e.lim N →∞ P G ( z, N ) /P A ( z, N ) = 0.The accurate description of the first-order dynamical phase transitioncharacterizing the tails of P ( X, N ) is the main theoretical prediction of this paper. Wehave verified it via direct numerical simulations (see Fig. 4) and have found excellentagreement between numerics and theory.
First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
3. First-order dynamical transition: calculation of the rate function
In this section, we compute the large N behaviour of P ( X, N ) for
X > (cid:104) X (cid:105) . Thestrategy consists in evaluating the leading contribution to the integral in Eq. (18)according to the scale of the deviation of X from the average (cid:104) X (cid:105) that one is interestedin. In particular we identify the three following regimes:(i) X − (cid:104) X (cid:105) ∼ N / : the Gaussian regime.We discuss it in Sec. 3.1.(ii) X − (cid:104) X (cid:105) ∼ N , the extreme large-deviation regime.This is discussed it in Sec. 3.2.(iii) X − (cid:104) X (cid:105) ∼ N / , the intermediate matching regime .We discuss it in Sec. 3.3.In the Gaussian regime, for completeness, we also repeat how to compute P ( X, N )when
X < (cid:104) X (cid:105) and | X − (cid:104) X (cid:105)| ∼ N / , just to show that the result is consistent withfluctuations above the mean.The three regimes listed above have one common feature: in order to compute P ( X, N ), the Bromwich contour appearing in its integral representation in Eq. (18)must be deformed in order to pass around the branch cut on the negative semiaxis,see Fig. 3. In Fig. 3 are represented the analytical properties in the complex s plane of L ( s ), the function defined in Eq. (18) and Eq. (19): it has two branch cuts on the realaxis. The branch cut on the semiaxis [ E, ∞ [ is related to the behaviour of P ( X, N )for
X < −(cid:104) X (cid:105) , the branch cut on ] − ∞ ,
0] to the behaviour for
X > (cid:104) X (cid:105) . In Fig. 3 areshown the examples of the two possible shapes of the Bromwich contour, dependingon whether X lies inside or outside the interval [ −(cid:104) X (cid:105) , (cid:104) X (cid:105) ]. For X ∈ [ −(cid:104) X (cid:105) , (cid:104) X (cid:105) ]the Bromwich contour is a straight vertical line crossing the real axis at s , where s is the saddle-point of the function h ( s ) = sx + log[ L ( s )], with x = X/N . For
X / ∈ [ −(cid:104) X (cid:105) , (cid:104) X (cid:105) ] the contour must be deformed in order to pass around the branchcut. In the following subsections we discuss the details of our calculations. Let us start with the calculation of the probability of O ( N / ) fluctuations around (cid:104) X (cid:105) = EN , considering separately the two cases X < EN and
X > EN . The resultin the second case is that the non-analiticity at the branch cut is negligible, and theprobability of fluctuations of order | X − EN | ∼ / √ N is Gaussian also for X > EN .The general strategy of all the following calculations is to first fix the scale of the fluc-tuations | X − EN | we are interested in, and then consider the corresponding ordersin the expansion of L ( s ) around s = 0.We start by computing P ( X, N ) for
X < EN and | X − EN | ∼ N / . Theexpansion of L ( s ) in Eq. (19) for small and positive s reads: L ( s ) = 1 − Es + (1 + 3 E )2 s + O ( s ) , (30)from which one then getslog[ L ( s )] = − Es + 12 σ s + O ( s ) , (31) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle σ = (2 + 5 E ) is the second cumulant of the distribution P ( x ) defined inEq. (9). Plugging the above expansions into integral of Eq. (18) one gets, for large N : P ( X, N ) ≈ s + i ∞ (cid:90) s − i ∞ ds πi e s ( X − EN )+ N σ s + O ( Ns ) . (32)Since we are interested in evaluating the contribution to P ( X, N ) at the scale | X − EN | ∼ N / , from X and s we change variables to z and ˜ s : X − EN = z N / s = ˜ s/N / , (33)and then take the limit N → ∞ . All the irrelevant contributions vanish and one isleft with a trivial Gaussian integral: P ( X, N ) = 1 √ N i ∞ (cid:90) − i ∞ d ˜ s πi e ˜ sz + σ ˜ s = e − ( X − EN ) / (2 σ N ) √ πσ N . (34)The same result can be obtained in a straightforward manner with the saddle-pointapproximation.More interesting is the calculation of P ( X, N ) for
X > EN . In this case theBromwich contour needs to be deformed as shown in Fig. 3. Due to the presence ofthe branch cut ] − ∞ , L ( s ) in Eq. (19) is non-analytic at s = 0for Re( s ) <
0, in particular it yields different results for the positive and the negativeimaginary semiplane: L ( s + i + ) = 1 − Es + (1 + 3 E ) s + . . . + (cid:114) πsE e sE + E L ( s + i − ) = 1 − Es + (1 + 3 E ) s + . . . (35)Accordingly, for the expansion of the logarithm one finds:log[ L ( s + i + )] = − Es + 12 σ s + . . . + (cid:114) πsE e sE + E log[ L ( s + i − )] = − Es + 12 σ s + . . . (36)Now, to compute P ( X, N ) at the scale X − EN ∼ √ N we consider separatelythe integration along the contour in the positive immaginary semiplane, denoted asΓ (+) in Fig. 3, and along the contour in the negative semiplane, denoted as Γ ( − ) , sothat P ( X, N ) = I ( − ) + I ( − ) , (37) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle • Es s Γ (+) Γ ( − ) Re( s )Im( s ) • •• Figure 3: Analyticity structure of L ( s ), see Eq. (19), in the complex s plane. Wiggledlines: the two branch cuts, respectively ] − ∞ ,
0] and [ E, ∞ [. Straight (blue)line: Bromwich contour for the calculation of P ( X, N ) when −(cid:104) X (cid:105) < X < (cid:104) X (cid:105) , with s indicating the location of the saddle-point. Deformed (red)line: Bromwich contour to compute P ( X, N ) when
X > (cid:104) X (cid:105) , s indicatesthe new saddle point. Γ (+) and Γ ( − ) are labels for contour pieces in thepositive and negative imaginary semiplanes.where the symbols I ( − ) and I (+) denote respectively the contour integrations alongΓ ( − ) and Γ (+) . By plugging the expansions of Eq. (36) in the two integrals and changingof variables to z = ( X − EN ) /N / and s = ˜ s/N / one gets respectively: I ( − ) = 1 √ N (cid:90) Γ ( − ) d ˜ s πi e ˜ sz + σ ˜ s + O ( N − ) , (38)and I (+) = 1 √ N (cid:90) Γ (+) d ˜ s πi e ˜ sz + σ ˜ s + O ( N − )+ N / √ π ˜ sE e √ N sE + 12 E , (39) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle s ) < L ( s )] in Eq. (39) is exponentially small in √ N and can be neglected, so that tothe leading order the integrands of I ( − ) and of I (+) are identical. By dropping alsothe terms O ( N − ) in the argument of exponential one ends up with the formula: P ( X, N ) = 1 √ N (cid:90) Γ ( − ) +Γ (+) d ˜ s πi e ˜ sz + σ ˜ s = 1 √ N i ∞ (cid:90) − i ∞ d ˜ s πi e ˜ sz + σ ˜ s = e − ( X − EN ) / (2 σ N ) √ πσ N . (40)The last equation completes the demonstration that at the scale | X − EN | ∼ N / the distribution P ( X, N ) is a Gaussian centered at (cid:104) X (cid:105) = EN . This is, in fact, justa consequence of the validity of the central limit theorem. We now focus on the extreme right tail of P ( X, N ), where X − N E ∼ O ( N ). Tocompute the leading contributions to P ( X, N ) on this scale, we change variables fromfrom X and s to z and ˜ s as follows: X − EN = z Ns = ˜ s/N. (41)Also in this case, see for comparison Sec. 3.1, it is then convenient to split theintegral expression of P ( X, N ) in the positive and negative immaginary semiplanecontributions, denoted respectively as I (+) and I ( − ) . The function log[ L ( s )] is notanalytic at s = 0 and for Re( s ) < I ( − ) and I (+) the expressions of Eq. (36) and the change of variables of Eq. (41)one finds respectively I ( − ) = 1 N (cid:90) Γ ( − ) d ˜ s πi e ˜ sz + σ N ˜ s + O ( N − ) = 1 N (cid:90) Γ ( − ) d ˜ s πi e ˜ sz (cid:20) σ N ˜ s + O ( N − ) (cid:21) , (42)and I (+) = 1 N (cid:90) Γ (+) d ˜ s πi e ˜ sz + σ N ˜ s + O ( N − )+ N / √ π ˜ sE e N sE + 12 E . (43)Note that all terms except ˜ sz inside the exponential are small for large N (includingthe term containing e N/ (2˜ sE ) , since the real part of s is negative along the contourΓ (+) ). Hence, we can expand the exponential for large N . Keeping only leading orderterms, we get I (+) ≈ N (cid:90) Γ (+) d ˜ s πi e ˜ sz (cid:34) σ N ˜ s + O ( N − ) + N / (cid:114) π ˜ sE e N sE + E (cid:35) (44) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle I (+) + I ( − ) ≈ N i ∞ (cid:90) − i ∞ ds πi e sz (cid:20) σ N s + O ( N − ) (cid:21) ++ N / (cid:90) Γ (+) ds πi e sz (cid:114) πsE e N sE + E . (45)One can easily show that the integrals in the first line of Eq. (45) (coming from theanalytic terms) all vanish. For example, the first term just gives a delta function δ ( z ) /N that vanishes for any z >
0. The other analytic terms similarly can beevaluated using i ∞ (cid:90) − i ∞ ds πi e sz s n = δ ( n ) ( z ) (46)and thus contribute Dirac delta’s derivatives of increasing order which all vanish for z > N comes from the integral inthe second line of of Eq. (45). To evaluate this integral, it is convenient to first rescale s → √ N s and rewrite it as I (+) + I ( − ) ≈ N / (cid:114) πE e E (cid:90) Γ (+) ds πi √ s e √ N ( sz + sE ) . To evaluate this integral, it is first convenient to rotate the contour Γ (+) anticlockwiseby angle π/
2. We are allowed to do this since the function is analytic in the left upperquadrant in the complex s plane. So, the deformed (rotated) contour now runs alongthe real axis from 0 to −∞ . This amounts to setting s = − x with x running from0 to ∞ , and the integral in Eq. (47) reduces to an integral on the real positive axis x ∈ [0 , ∞ ] I (+) + I ( − ) ≈ N / (cid:114) π E e E ∞ (cid:90) dx √ x e −√ N ( zx + xE ) . (47)This integral can now be evaluated using the saddle point method. Defining, u ( x ) = xz + 12 xE . (48)it is easy to check that u ( x ) has a unique minimum at x ∗ = 1 / √ zE (where u (cid:48)(cid:48) ( x ∗ ) > x ∗ = − / √ zE into the integral of Eq. (47) andevaluating carefully the integral (including the Gaussian fluctuations around the saddlepoint) [40, 41], we get, for large N and with z = ( X − EN ) /N , P ( X, N ) ≈ N e E exp (cid:104) − (2 /E ) / (cid:112) ( X − EN ) (cid:105)(cid:112) E ( X − EN ) , (49) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle N times the asymptotic behaviour of the marginal probabilitydistribution of the displacement in a single run, given in Eq. (12). This fact is perfectlyconsistent with the existence of a single positive condensate for X − N E ∼ O ( N ), thatdominates the sum of N i.i.d random variables each distributed via the marginaldistribution; the combinatorial factor N in front indicates that any one of the N variables can be the condensate. intermediate matching regime The main new result of this paper is the detailed study of the intermediate regimewhere Gaussian fluctuation and extreme large positive fluctuation both are of thesame order: their competition is at the heart of the first-order nature of the dynamicaltransition that we find. This conditionexp (cid:20) − ( X − EN ) σ N (cid:21) ∼ exp (cid:34) − (cid:114) E (cid:112) ( X − EN ) (cid:35) , (50)simply sets the scale of the matching to be X − EN ∼ N / . (51)As a consequence, in order to single out the leading contributions to P ( X, N ) at thisscale we must change variable from X to z in the integral of Eq. (18), with z ∼ O (1)such that: X − EN = zN / . (52)The trick is then to chose the proper rescaling of s so that the analytic terms of thelog[ L ( s )] expansions (responsible for the Gaussian fluctuations), and the non-analyticones (responsible for the anomalous fluctuations coming from the formation of thecondensate), are of the same order. As is shown below, this is achieved by rescaling s as s = ˜ s/N / . (53)Once again it is useful to evaluate separately the two contributions I (+) and I ( − ) afterthe change of variables. Taking into account the expansion of log[ L ( s )] in Eq. (36)one gets respectively: I ( − ) = (cid:90) Γ ( − ) ds πi e sN / z + N σ s + O ( Ns ) = 1 N / (cid:90) Γ ( − ) d ˜ s πi e N / (cid:16) ˜ sz + σ ˜ s (cid:17) + O (1) , (54) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle I (+) = i ∞ (cid:90) ds πi e s ( X − NE )+ N σ s + O ( Ns )+ N √ πsE e sE + 12 E = i ∞ (cid:90) d ˜ s πiN e N / ˜ sz + N / σ ˜ s + O (1)+ N (cid:113) πN / sE e N / sE + 12 E = i ∞ (cid:90) d ˜ s πiN e N / ˜ sz + N / σ ˜ s (cid:34) N (cid:114) πN / ˜ sE e N / sE + E (cid:35) . (55)By summing the expression of the two integrals in Eq. (54) and Eq. (55) it is theneasy to write P ( X, N ), with X − E N = z N / , explicitly as the sum of a Gaussianand an anomalous contribution: P ( X, N ) = P G ( z, N ) + P A ( z, N ) P G ( z, N ) = 1 N / i ∞ (cid:90) − i ∞ ds πi e N / (cid:16) sz + σ s (cid:17) P A ( z, N ) = N / e / (2 E ) i √ πE (cid:90) Γ (+) ds √ s e N / F z ( s ) (56)where the function F z ( s ) reads: F z ( s ) = sz + 12 σ s + 12 sE . (57)The integral in the second line of Eq. (56) can be easily performed using thesaddle point method, and gives a Gaussian contribution, justifying its name P G ( z, N ) ≈ √ πσ N exp (cid:20) − N / z σ (cid:21) . (58)The integral in the second line of Eq. (56), giving rise to the anomalous part, canbe also be computed with a saddle point approximation only when the saddle-pointequation ∂F z ( s ) ∂s = 0 (59)has a real root s ∗ . The real roots of Eq. (59) and their properties are studied in fulldetail in Appendix B. In the same appendix, we also discuss the domain of existenceof the saddle point solution and give the explicit solution.Skipping further details, we find that the saddle-point equation F (cid:48) z ( s ) = 0 has areal solution s ∗ only for z ∈ [ z l , ∞ ], with z l = 3 σ / / E / (see Appendix B): in thisrange the integral P A ( z, N ) can be explicitly evaluated. For z < z l , computing the First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle -4-3-2-1 0 0 5 10 15 20 25 - Ψ ( z ) = N - / l og P ( z ) z=( X - N
Top : numerical data for the rate function Ψ( z ). Different curves correspondto different number of runs N in the trajectory: N = 10 , , .Acceleration is set to E = 2. Dotted (black) line: guide to the eye, Gaussian(inverted) parabola. Bottom : numerical data for the rate function derivativeΨ (cid:48) ( z ). Continuous black line is the analytical prediction in the limit N → ∞ ,the coordinate of the transition point is z c ≈ .
78 (for E = 2). First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle s axis. However, as wewill see, we do not need the information on P A ( z, N ) for z = ( X − EN ) /N / < z l .We will see that the transition occurs at z = z c > z l , so it is enough to compute P A ( z, N ) for z > z l . Hence, for our purpose, evaluating P A ( z, N ) by saddle point issufficient. Assuming the existence of a saddle point at s = s ∗ and plugging the explicitexpression of s ∗ as a function of z (see Appendix B) into Eq. (56) one gets: P A ( z, N ) ∼ e − N / χ ( z ) . (60)The shape of the function χ ( z ) is shown in Fig. 2, where its behavior is compared withthe parabola z / (2 σ ) of the Gaussian term. All the details on the derivation of χ ( z )are in Appendix B. The asymptotics are: χ ( z ) = (cid:0) σE (cid:1) / z → z l (cid:113) E √ z − σ E z + O (cid:0) z / (cid:1) z (cid:29) P ( X, N ) ≈ exp (cid:26) − N / z σ (cid:27) + exp (cid:110) − N / χ ( z ) (cid:111) . (62)The equation z / (2 σ ) = χ ( z ) can be solved exactly with a simple argument(see Appendix B.3), yielding the following value of z c in units of z l : z c z l = 2 / (63)In the numerical simulations we set the acceleration to E = 2. By plugging this valuein the expression of z l given in Appendix B we finally get: z c ≈ . , (64)the value indicated by the dotted vertical line in Fig. 4.The mechanism of the first-order transition is now transparent. Recall that X − EN = z N / . When z < z c , the probability distribution P ( X, N ) is dominatedby the Gaussian contribution P G ( z, N ), since z / (2 σ ) < χ ( z ). On the contrary for z > z c the distribution is dominated by the anomalous contribution P A ( z, N ), since z / (2 σ ) > χ ( z ). The result can be summarized as follows: z < z c = ⇒ z σ < χ ( z ) = ⇒ P ( X, N ) ≈ e − N / z / (2 σ ) z > z c = ⇒ z σ > χ ( z ) = ⇒ P ( X, N ) ≈ e − N / χ ( z ) (65)Thus the mechanism behind the first order transition corresponds to a classic first-order phase transition scenario in standard thermodynamics: in the vicinity of thetransition point there is a competition between two phases characterized by a differ-ent free-energy (here the value of the rate function) and the transition point itselfis defined as the value of the control parameter (here the value of the displacement)where the free-energy difference between the two phases changes sign. First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle The direct consequence of the description in Eq. (65) is that the rate function Ψ( z ),which is Ψ( z ) = z / (2 σ ) for z < z c and Ψ( z ) = χ ( z ) for z > z c , has a discontinuity inits first-order derivative Ψ (cid:48) ( z ) at z c : this happens because the two functions z / (2 σ )and χ ( z ) match continuously at z c , but with a different slope.We show in this section that the discontinuity of the rate function Ψ( z ) appearsfor large enough N when one tries to sample numerically the tails of P ( X, N ). We havestudied the behaviour of P ( X, N ) for N = 10 , , runs in the trajectory. The be-haviour of the rate function Ψ( z ) and of its derivative Ψ (cid:48) ( z ) are shown in Fig. 4. Whileat N = 10 the transition from the Gaussian to the large deviations regime is stilla smooth crossover, the trend for increasing N goes clearly towards a discontinuousjump of Ψ (cid:48) ( x ). The location of the discontinuity revealed by the numerical simula-tions is in agreement with the analytic prediction z c ≈ .
78 given for the value E = 2.Simulations are straightforward but one has to chose a clever strategy: just look-ing at the probability distribution of independent identically distributed random vari-ables is not sufficient, since doing like that one can only probe the typical fluctuationsregime, | X − (cid:104) X (cid:105)| ∼ N / , but not the large deviations. In order to sample P ( X, N )in the whole regime of interest a set of many simulations is needed, each probing thebehaviour of the PDF in a narrow interval [ X ∗ , X ∗ + ∆]. We provide in what followsa detailed description of the numerical protocol.In order to achieve an efficient sampling of P ( X, N ) also in the matching and inthe large deviations regime we follow here the strategy to compute the tails of randommatrices eigenvalues distribution used in [45,46]. The basic idea is to sample P ( X, N )in many small intervals [ X ∗ , X ∗ + ∆] varying X ∗ , in order to recover finally the wholedistribution. The sampling of P ( X, N ) in each interval [ X ∗ , X ∗ + ∆] correspondsto an independent Monte Carlo simulation. Since the total number of runs in thetrajectory is fixed to N , for each value of X ∗ we fix the initial condition chosing a set( τ in1 , . . . , τ in N ) of runs durations and a set ( v in1 , . . . , v in N ) of initial velocities for each runsuch that: N (cid:88) i =1 (cid:20) v in i τ in i + E τ in i ) (cid:21) > X ∗ . (66)In the intial condition all the local variables are of order unit: v in i , τ in i = O (1).The stochastic dynamics to sample P ( X, N ) in the vicinity of X ∗ then goes onas any standard Metropolis algorithm: attempted updates are accepted or rejectedwith probability p = min[1 , p ( C old ) /p ( C new )] where the stationary probability of aconfiguration p ( C ) reads as: p ( τ , v , . . . , τ N , v N ) ≈ exp (cid:32) − N (cid:88) i =1 (cid:2) τ i + v i / (cid:3)(cid:33) . (67)The only additional ingredient with respect to a standard Metropolis algorithm isthat all attempts which brings X N below X ∗ are rejected. The precise form of theprobability distribution sampled for each value of X ∗ in the Monte Carlo dynamics is First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle P ( X, N | X > X ∗ ) = e − (cid:80) Ni =1 [ τ i + v i / Θ (cid:32) N (cid:88) i =1 (cid:20) v i τ i + E τ i (cid:21) − X ∗ (cid:33) , (68)where Θ( x ) is the Heavyside step function.We define as a Monte Carlo sweep the sequence of N local attempts of the kind( v i , τ i ) ⇒ ( v new i , τ new i ) where v new i = v i + δv,τ new i = τ i + δτ. (69)The shifts δv , δτ are random variables drawn from uniform distributions. Then, ifand only if the constraint implemented by the Heavyside step function in Eq. (68) issatisfied, otherwise the attempt is rejected immediately, the new values ( v new i , τ new i )are accepted with probability: p acc = min (cid:104) , e − [ ( v new i ) / τ new i − ( v i / τ i ) ] (cid:105) . (70)In order to recover the full probability distribution P ( X, N ) within the interval X ∈ I = [ (cid:104) X (cid:105) , (cid:104) X (cid:105) + 2 z c (cid:104) X (cid:105) / ] we have divided I in a grid of M = 100 elements.More precisely, we have run M simulations for a set of equally-spaced values X ∗ ∈ I .Each simulation allows one to sample P ( X, N | X > X ∗ ) only in a small interval onthe right of X ∗ . The value z c is the critical one predicted by the theory. The choiceof the interval I has been somehow arbitrary, we just took care that it was centeredaround the expected critical value X c for the condensation transition. We have takena number of sweeps N sweeps ≈ , enough to forget the initial conditions.The relation between the PDF P ( X, N | X > X ∗ ) sampled in the MC numericalsimulations and the PDF we want to investigate, P ( X, N ), is as follows: P ( X, N ) = P ( X, N | X > X ∗ ) P ( X > X ∗ , N ) (71)In particular, what we are interested in is the rate function Ψ( z ) defined as:Ψ (cid:18) z = X − (cid:104) X (cid:105) N / (cid:19) = − N / log [ P ( X, N )] . (72)The rate function Ψ z ∗ ( z ) that we measure in the vicinity of X ∗ by sampling theprobability distribution in Eq. (68) differs from the original one, due to Eq. (71), byan additive constant:Ψ( z ) = − N / log [ P ( X, N | X > X ∗ )] − N / log [ P ( X > X ∗ , N )]= Ψ z ∗ ( z ) + f ( z ∗ ) , (73)where f ( z ∗ ) is a function that depends only on z ∗ . By taking the derivative withrespect to z (and taking into account that dz = dX/N / ) one gets rid of the additiveconstant and obtain the following expression: d Ψ( z ) dz = d Ψ z ∗ ( z ) dz = − N / P ( X, N | X > X ∗ ) dP ( X, N | X > X ∗ ) dX (74) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle d Ψ z ∗ ( z ) /dz in the vicin-ity of many values z ∗ by means of the biased Monte Carlo dynamics. The functionΨ( z ) has been then obtained from the numerical integration of the first-order deriva-tive. Both the rate function Ψ( z ) and its first derivative Ψ (cid:48) ( z ) are shown in Fig. 4.
4. First-Order transition for negative fluctuations.
So far we focused only on the right tail of P ( X, N ), but the probability distributionis not symmetric due to E , as is shown in the pictorial representation of Fig. 1. Weneed to comment also on the behaviour of the left tail of P ( X, N ). In this section wedemonstrate that even for negative fluctuations a first-order dynamical transition takesplace, and that, following the same arguments of Sec. 3, it is located at X ( − ) c = − X c ,where X c = z c N / + N E , with z c given in Eqs. (63) and (64). The location ofthe transition for negative fluctuations is symmetric to that for positive ones. Theonly difference with the case of positive fluctuations is that for X < E . Here is the summary ofthe behavior of P ( X, N ) in the three regimes, i.e. typical fluctuations, extreme largenegative deviations and the intermediate matching regime, for
X < P ( X, N ) ≈ e − E | X | e − ( X + NE ) / (2 Nσ ) for ( X + EN ) ∼ N / e − E | X | e − N / Ψ( z ) for ( X + EN ) ∼ N / e − E | X | e − √ /E | X + EN | / for ( X + EN ) ∼ N , (75)where z = − ( X + N E ) /N / and the rate function Ψ( z ) is the same as te one we havecomputed for positive fluctuations.The calculations to obtain the behaviours in Eq. (75) are identical to those for X >
0, which are described in full detail in Sec. (3) and in Appendix B. We arenot going to repeat all of them here. We will sketch the derivation of the results inEq. (75) only for the matching regime , which is the most interesting among the three.This discussion has also the purpose to highlight the (small) differences with the calcu-lations in the case
X >
0, in particular to show where the prefactor e − E | X | comes from.First, as can be easily noticed looking at Eq. (19), the function L ( s ) has thefollowing symmetry L ( s ) = L ( E − s ) , (76)which is important for the following reason. To compute P ( X, N ) for values of thetotal displacemente
X > EN we needed to wrap the Bromwich contour around thebranch cut at ] − ∞ ,
0] (see Fig. 3). This was done by taking the analytic continuationof log[ L ( s )] in the complex plane in the neighbourhood of s = 0. In the same manner,in order to compute P ( X, N ) for
X < − N E , we must wrap the Bromwich contouraround the branch cut at [ E, ∞ [. To do this we need the analytic continuation oflog[ L ( s )] in the neighbourhood of s = E . Due to the symmetry in Eq. (76) theexpansion of log[ L ( s )] in the neighbourhood of s = E is identical to the one in the First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle s = 0, including the non-analyticities due to the branch cut. Inparticular we have that:log[ L ( E − s + i + )] = − E ( E − s ) + 12 σ ( E − s ) + . . . + (cid:114) π E ( E − s ) e s ( E − s ) + E log[ L ( E − s + i − )] = − E ( E − s ) + 12 σ ( E − s ) + . . . (77)As done in the case of positive fluctuations, also for X < − N E is convenient tosplit the expression of the inverse Laplace transform of P ( X, N ), see Eq. (18), in twocontributions: the contour integral in the negative semiplane, I ( − ) , and the contourintegral in the positive semiplane, I (+) . Let us consider first the integral I ( − ) : I ( − ) = (cid:90) Γ ( − ) ds πi e sX + N [ − E ( E − s )+ σ ( E − s ) + ... ] . (78)In this case ( X <
0) is convenient to change variable from s to y = E − s : I ( − ) = (cid:90) − Γ ( − ) dy πi e ( E − y ) X + N [ − Ey + σ y + ... ]= e XE (cid:90) Γ ( − ) dy πi e − y ( X + EN )+ N σ y + ... . (79)Then, in order to have a variable which is positive and is of order O (1) when X + EN ∼ N / we introduce: z = − X + ENN / . (80)By rescaling the integration variable y = ˜ y/N , appropriate for the matchingintermediate regime, we can rewrite I ( − ) = e EX N / (cid:90) Γ ( − ) d ˜ y πi e N / [ ˜ yz + σ ˜ y ] + O (1) . (81)The expression of I ( − ) in Eq. (81), is, apart from the prefactor e EX , identical to theanalogous one evaluted for X >
0, see Eq. (54). The only difference is that now thescaling variable z is defined as z = − ( X + EN ) /N / rather than z = ( X − EN ) /N / .In the same way for the integral in the positive complex semiplane we find: I (+) = e EX N / (cid:90) Γ (+) d ˜ y πi e N / [ ˜ yz + σ ˜ y ] + ... + N (cid:113) πN / E ˜ y e N / E ˜ y + 12 E (82)Recalling that we are expanding for Re( y ) = Re( E − s ) < First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle y <
0, we can further expand: I (+) = e EX N / (cid:90) Γ (+) d ˜ y πi e N / [ ˜ yz + σ ˜ y ] N (cid:115) πN / E ˜ y e N / E ˜ y + E = e EX N / (cid:90) Γ (+) d ˜ y πi e N / [ ˜ yz + σ ˜ y ] + N / e EX e / (2 E ) i √ πE (cid:90) Γ (+) d ˜ y √ ˜ y e N / F z (˜ y ) (83)so that I (+) + I ( − ) = e EX N / i ∞ (cid:90) − i ∞ d ˜ y πi e N / [ ˜ yz + σ ˜ y ] + N / e EX e / (2 E ) i √ πE (cid:90) Γ (+) d ˜ y √ ˜ y e N / F z (˜ y ) F z (˜ y ) = ˜ yz + 12 σ ˜ y + 12˜ yE , (84)where the function F z (˜ y ) is identical to that of Eq. (57), hence leading to the sameconclusions. For negative fluctuations as well, it is then straightforward to see thatin the intermediate matching regime we have two competing contributions, i.e. theGaussian, P G ( z, N ), and anomalous one, P A ( z, N ): P G ( z, N ) = 1 N / i ∞ (cid:90) − i ∞ d ˜ y πi e N / [ ˜ yz + σ ˜ y ] P A ( z, N ) = N / e EX e / (2 E ) i √ πE i ∞ (cid:90) − i ∞ d ˜ y √ ˜ y e N / F z (˜ y ) , (85)where z = − ( X + N E ) /N / . The probability distribution for negative fluctuationsin the matching regime reads therefore as: P ( X, N ) = e EX [ P G ( z, N ) + P A ( z, N )] , (86)Apart from the prefactor e EX the expression of P ( X, N ), for negative fluctuations inthe matching regime, is the same as that for positive fluctuations: the condensationtransition at z c is driven by the same mechanism, the competition between theGaussian fluctuations of P G ( z, N ) and the anomalous one of P A ( z, N ). The calculationof the probability of typical fluctuations X + N E ∼ N / and of large deviations X + N E ∼ N can be very easily done following the same steps of Sec. 3, which we donot repeat here.
5. Conclusions
We have studied the probability distribution P ( X, N ) of the total displacement X N = (cid:80) Ni =1 x i for a Run-and-Tumble (RTP) particle on a line, subject to a constantforce E >
0. The PDF p ( τ ) for the distribution of duration of a run and the PDF q ( v ) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
E > E = 0, p ( τ ) = e − τ θ ( τ ) (Poisson tumblingwith rate 1) and a bimodal q ( v ) = [ δ ( v − v ) + δ ( v + v )]. The main conclusion ofthis paper is that a broad class of p ( τ ) and q ( v ), for E >
0, leads to a condensationtransition. This is manifest as a singularity in the displacement PDF P ( X, N ) forlarge N and the transition is first-order. A criterion for the condensation is providedfor different choices of p ( τ ) and q ( v ). As a representative case, we have provideddetailed analysis and results for the specific choice: arbitrary E > p ( τ ) = e − τ θ ( τ )and q ( v ) = e − v / / √ π . We have also argued that the standard RTP does not havethis interesting phase transition.By a detailed computation of the PDF P ( X, N ) of the total displacement after N runs, we have shown that while the central part of the PDF P ( X, N ) is characterizedby a Gaussian form (as dictated by the central limit theorem), both the right and lefttails of P ( X, N ) have anomalous large deviations forms. On the positive side, as thecontrol parameter X − EN exceeds a critical value z c N / , a condensate forms, i.e,the sum starts getting dominated by a single long run. This signals a phase transition,as a function of X , from the central regime dominated by Gaussian fluctuations tothe condensate regime dominated by a single long run. A similar transition occurs forlarge negative X where a negative long run dominates the sum. The phase transitionis qualitatively similar to condensation phenomenon in mass transport models, wherethe role of the large condensate mass is played here by the macroscopic extent of thedisplacement travelled without tumbles in one single run.The main new result of our study is the uncovering of an intermediate matchingregime where the PDF P ( X, N ) of the total displacement exhibits an anomalous largedeviation form, P ( X, N ) ∼ e − N / Ψ( z ) with z = ( X − (cid:104) X (cid:105) ) /N / . Quite remarkableis the non-analytic behaviour of the associated rate function Ψ( z ) at the critical point z = z c , here the function is continuous but its first derivative jumps: we are in pres-ence of a first-order phase transition. The two phases on either side of the criticalpoint z c corresponds respectively to a fluid phase ( z < z c ) and a phase with a sin-gle large condensate ( z > z c ). The mechanism behind this transition is typical of athermodynamic first-order phase transition, where there is an energy jump (first orderderivative of the free energy with respect to the inverse temperature β ) emerging fromthe competition between two phases. Here we have homogeneous trajectories withGaussian probability P G ( X, N ) competing with trajectories dominated by one singlerun characterized by the anomalous part of the distribution P A ( X, N ). The transitiontakes place when the two competing terms are of the same order. An interesting fea-ture of the analysis presented here is that the first-order dynamical transition studiedtakes place in a regime where the natural scale (speed) of large deviations is N / andnot N , as is typical in extensive thermodynamic systems.In this paper, we have shown that the problem of computing the total displace-ment of the RTP reduces to the computation of the distribution of the linear statistics(in this case just the sum) of a set of i.i.d random variables, each drawn from a marginaldistribution that has a stretched exponential tail. Our study shows that even for sucha simple system, the distribution P ( X, N ) has an anomalous large deviation regimethat exhibits a discontinuity in the first-derivative of the rate function. It is worthpointing out that in a certain class of strongly correlated random variables (typicallyarising in problems involving the eigenvalues of a random matrix), the distribution of
First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
E > E = 0. We already know that thelimit E → P ( X, N )in the case of zero external field are different from the finite field case. All the detailson P ( X, N )’s large deviation form in the case of E = 0 are going to be presentedelsewhere [54]. The results of the present paper also have important implications foran equilibrium thermodynamics study of wave-function localization in the nonlinearSchr¨odinger equation: this is the subject of another forthcoming work [55]. Acknowledgments
We thank E. Bertin, F. Corberi, A. Puglisi and G. Schehr for useful discussions. Wealso warmly thank N. Smith for pointing out an algebraic error in Appendix B in theprevious version of the manuscript and for suggesting an argument for computing r c ,which is now reported in Appendix B.3. G.G. acknowledges Financial support fromthe Simons Foundation grant No. 454949 (Giorgio Parisi). G.G. aknowledges LIPhy,Universit`e Grenoble-Alpes, for kind hospitality during the first stages of this work(support from ERC Grant No. ADG20110209, Jean-Louis Barrat). First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle Appendix A. Asymptotic tails of P ( x )In this Appendix, we present the asymptotic bahaviors of the distribution of thedisplacement in a single run, namely the marginal distribution P ( x ) written in Eq. (9)of Sec.2. We first consider p ( τ ) = e − τ θ ( τ ) and q ( v ) = e − v / / √ π . Let us first definethe mean and the variance of P ( x ), which can be easily computed. The mean is givenby (cid:104) x (cid:105) = (cid:104) v (cid:105)(cid:104) τ (cid:105) + 12 E (cid:104) τ (cid:105) = E . (A.1)Similarly, the second moment is simply, (cid:104) x (cid:105) = (cid:104) v (cid:105)(cid:104) τ (cid:105) + E (cid:104) v (cid:105)(cid:104) τ (cid:105) + E (cid:104) τ (cid:105) = 2 + 6 E (A.2)and hence the variance is given by σ = (cid:104) x (cid:105) − (cid:104) x (cid:105) = 2 + 5 E . (A.3)To compute the full marginal distribution P ( x ), we perform the Gaussian integralover v to get P ( x ) = 1 √ π ∞ (cid:90) dττ exp (cid:20) − τ − ( x − Eτ / τ (cid:21) . (A.4)This integral is hard to compute exactly. However, we are only interested in the large | x | asymptotic tails of P ( x ).To derive the asymptotics of P ( x ) in Eq. (A.4), it is first convenient to rewrite itas P ( x ) = 1 √ π e x E/ ∞ (cid:90) dττ exp (cid:20) − τ − x τ − E τ (cid:21) . (A.5)Since P ( x ) is manifestly asymmetric, let us consider the two limits x → ∞ and x → −∞ separately. Consider first the positive side x ≥
0. Let us first rescale τ = √ x y in Eq. (A.5), and rewrite the integral for any x ≥ P ( x ) = 1 √ π ∞ (cid:90) dyy exp (cid:34) −√ x y − x (cid:18) Ey − y (cid:19) (cid:35) . (A.6)This is a convenient starting point for analysing the asymptotic tail x → ∞ . Thedominant contribution to this integral for large x comes from the vicinity of y = y ∗ = (cid:112) /E that minimizes the square inside the exponential. Setting y = (cid:112) /E + z ,expanding around z = 0 (keeping terms up to O ( z )) and performing the resultingGaussian integration gives, to leading order for large positive x P ( x ) ≈ E e /E x − / e − √ x/E . (A.7)Turning now to the large negative x , we set x = −| x | in Eq. (A.5) and rewrite it,for x < P ( x <
0) = e − E | x | / ∞ (cid:90) dττ exp (cid:20) − τ − | x | τ − E τ (cid:21) = e − E | x | P ( | x | ) (A.8) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle P ( x ) in Eq. (A.5) with argument | x | >
0. Hence, for large x → −∞ , we can use the already derived asymptotics of P ( x ) for large positive x in Eq. (A.7). This then gives, to leading order as x → −∞ , P ( x ) ≈ E e /E e − E | x | | x | − / e − √ | x | /E . (A.9)The results in Eqs. (A.7) and (A.9) can then be combined into the single expression P ( x ) ≈ (cid:40) c | x | − / e − (2 /E ) / | x | / , x → ∞ c | x | − / e − E | x | − (2 /E ) / | x | / , x → −∞ (A.10)where c = e /E /E is a constant. Thus the marginal PDF of x has stretchedexponential tails on both sides with stretching exponent α = 1 /
2, but in addition onthe negative side it has an overall multiplicative exponential factor e − E | x | . We notethat this model with a field E >
E > p ( τ ) = e − τ θ ( τ ), but the velocity distribution q ( v ) is bimodal as in Eq. (1). Substituting q ( v ) in Eq. (9) and carrying out the v integration gives P ( x ) = 12 ∞ (cid:90) dτ e − τ (cid:20) δ (cid:18) x − v τ − E τ (cid:19) + δ (cid:18) x + v τ − E τ (cid:19)(cid:21) . (A.11)Now, for large x >
0, the leading contribution comes from large τ , hence one canneglect v τ terms leading to P ( x ) ≈ ∞ (cid:90) dτ e − τ δ (cid:18) x − E τ (cid:19) = 1 √ E x e − √ x/E . (A.12)Hence, for large x >
0, the marginal distribution P ( x ) has a stretched exponentialdecay and it satisfies the criterion for positive condensation. In contrast, for negative x and E >
0, it is easy to see that P ( x ) strictly vanishes for x < − v / E . Thusthe marginal distribution is bounded on the negative side. Consequently it does notsatisfy the condensation criterion for negtaive x . Thus, in this example, we only haveone sided condensation in the displacement PDF P ( X, N ).We next consider the standard RTP case: E = 0, p ( τ ) = e − τ θ ( τ ) and q ( v )bimodal as in Eq. (1). Substituting q ( v ) in Eq. (9) and carrying out the v integralgives P ( x ) = 12 ∞ (cid:90) dτ e − τ [ δ ( x − v τ ) + δ ( x + v τ )]= 12 v e −| x | /v . (A.13)Thus, in this case, the marginal P ( x ) decays exponentially on both sides and hencedoes not satisfy the condensation criterion. Consequently, for the standard RTP, wedo not have condensation on either side. However, for E = 0, and arbitrary velocitydistribution q ( v ) with a finite width, the condensation transition is restored [54]. First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle Appendix B. Derivation of the rate function χ ( z ) in the intermediatematching regime In this Appendix we study the leading large N behavior of the integral that appearsin the expression for P A ( z, N ) in Eq. (56): I N ( z ) = (cid:90) Γ (+) ds √ s e N / F z ( s ) (B.1)where z ≥ F z ( s ) = sz + 12 σ s + 12 sE , (B.2)with σ = 2 + 5 E . It is important to recall that the contour Γ (+) is along a verticalaxis in the complex s -plane with its real part negative, i.e. Re( s ) <
0. Thus, we candeform this contour only in the upper left quadrant in the complex s plane (Re( s ) < s ) > s -plane where Re( s ) >
0. A convenient choice of the deformedcontour, as we will see shortly, is the Γ (+) rotated anticlockwise by an angle π/
2, sothat the contour now goes along the real negative s from 0 to −∞ .To evaluate the integral in Eq. (B.1), it is natural to look for a saddle point of theintegrand in the complex s plane in the left upper quadrant, with fixed z . Hence, welook for solutions for the stationary points of the function F z ( s ) in Eq. (B.2). Theyare given by the zeros of the cubic equation F (cid:48) z ( s ) = dF z ( s ) ds = z + σ s − Es ≡ z ≥ s plane. It turns out thatfor z < z l (where z l is to be determined), there is one positive real root and twocomplex conjugate roots. For example, when z = 0, the three roots of Eq. (B.3) arerespectively at s = (2 Eσ ) − / e iφ with φ = 0, φ = 2 π/ φ = 4 π/
3. However, for z > z l , all the three roots collapse on the real s axis, with s < s < s . The roots s < s < s > F (cid:48) z ( s ) in Eq. (B.3) as a function of real s , for z = 12 and E = 2(so σ = 2 + 5 E = 22). One finds, using Mathematica, three roots at s = − / s = − . . . . and s = 0 . . . . . Wecan now determine z l very easily. As z decreases, the two negative roots s and s approach each other and become coincident at z = z l and for z < z l , they split apartin the complex s plane and become complex conjugate of each other, with their realparts identical and negative. When s < s , the function F (cid:48) z ( s ) has a maximum at s m with s < s m < s (see Fig. B1). As z approaches z l , s and s approach each other,and consequently the maximum of F (cid:48) z ( s ) between s and s approach the height 0.Now, the height of the maximum of F (cid:48) z ( s ) between s and s can be easily evaluated.The maximum occurs at s = s m where F (cid:48)(cid:48) z ( s ) = 0, i.e, at s m = − ( Eσ ) − / . Hencethe height of the maximum is given by F (cid:48) z ( s = s m ) = z + σ s m + 12 s m E = z − (cid:18) σ E (cid:19) / . (B.4) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle -30-20-10 0 10 20 30-1 -0.5 0 0.5 1 s s s F z ' ( s ) s Figure B1: A plot of F (cid:48) z ( s ) = z + σ s − Es as a function of s ( s real) for z = 12, E = 2and σ = 2 + 5 E = 22. There are three zeros on the real s axis (obtainedby Mathematica ) at s = − . s = − . . . . and s = 0 . . . . .Hence, the height of the maximum becomes exactly zero when z = z l = 32 (cid:18) σ E (cid:19) / . (B.5)Thus we conclude that for z > z l , with z l given exactly in Eq. (B.5), the function F (cid:48) z ( s ) has three real roots at s = s < s < s >
0, with s being the smallestnegative root on the real axis. For z < z l , the pair of roots are complex (conjugates).However, it turns out (as will be shown below) that for our purpose, it is sufficientto consider evaluating the integral in Eq. (B.1) only in the range z > z l where theroots are real and evaluating the saddle point equations are considerbaly simpler. So,focusing on z > z l , out of these 3 roots as possible saddle points of the integrand inEq. (B.1), we have to discard s > s plane. This leaves us with s < s < (+) by rotating it anticlockwise by π/ s and s , itis easy to see (see Fig. (B1)) that F (cid:48)(cid:48) z ( s ) > s axis) and F (cid:48)(cid:48) z ( s ) < s for large N , we should choose s to be the correct root, i.e., the largest among the negativeroots of the cubic equation z + σ s − / (2 Es ) = 0.Thus, evaluating the integral at s ∗ = s (and discarding pre-exponential terms) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
N I N ( z ) ≈ exp[ − N / χ ( z )] (B.6)where the rate function χ ( z ) is given by χ ( z ) = − F z ( s = s ) = − s z − σ s − s E (B.7)The right hand side can be further simplified by using the saddle point equation (B.3),i.e., z + σ s − / Es = 0. This gives χ ( z ) = − zs − Es . (B.8) Appendix B.1. Asymptotic behavior of χ ( z )We now determine the asymptotic behavior of the rate function χ ( z ) in the range z l < z < ∞ , where z l is given in Eq. (B.5). Essentially, we need to determine s (the largest among the negative roots) as a function of z by solving Eq. (B.3), andsubstitute it in Eq. (B.8) to determine χ ( z ).We first consider the limit z → z l from above, where z l is given in Eq. (B.5). As z → z l from above, we have already mentioned that the two negative roots s and s approach each other. Finally at z = z l , we have s = s = s m where s m = − ( Eσ ) − / is the location of the maximum between s and s . Hence as z → z l from above, s → s m = − ( Eσ ) − / . Substituting this value of s in Eq. (B.8) gives the limitingbehavior χ ( z ) → (cid:16) σE (cid:17) / as z → z l (B.9)as announced in the first line of Eq. (24).To derive the large z → ∞ behavior of χ ( z ) as announced in the second line ofEq. (24), it is first convenient to re-parametrize s and define s = − √ Ez θ z . (B.10)Substituting this in Eq. (B.3), it is easy to see that θ z satisfies the cubic equation − b ( z ) θ z + θ z − , (B.11)where b ( z ) = σ √ E z / . (B.12)Note that due to the change of sign in going from s to θ z , we now need to determinethe smallest positive root of θ z in Eq. (B.11). In terms of θ z , χ ( z ) in Eq. (B.8) reads χ ( z ) = √ z √ E θ z + 3 θ z . (B.13)The formulae in Eqs. (B.11), (B.12) and (B.13) are now particularly suited for thelarge z analysis of χ ( z ). From Eqs. (B.11),(B.12) it follows that in the limit z → ∞ we have that b ( z ) →
0, so that θ z →
1. Hence, for large z or equivalently small b ( z ), First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle θ z = 1 + b ( z )2 + O (cid:16) b ( z ) (cid:17) . (B.14)with b ( z ) given in Eq. (B.12). Substituting this in Eq. (B.13) gives the large z behaviorof χ ( z ) χ ( z ) = (cid:114) E √ z − σ E z + O (cid:18) z / (cid:19) . (B.15)as announced in the second line of Eq. (24). Appendix B.2. Explicit expression of χ ( z )While the excercises in the previous subsections were instructive, it is also possibleto obtain an explicit expression for χ ( z ) by solving the cubic equation (B.11) with Mathematica . The smallest positive root of Eq. (B.11), using
Mathematica , reads θ z = 13 b z + 13 · / b z (1 − i √ (cid:16) − b z + 3 (cid:112) −
12 + 81 b z (cid:17) / + 13 · / b z (1+ i √ (cid:16) − b z + 3 (cid:112) −
12 + 81 b z (cid:17) / (B.16)where b z , used as an abbreviation for b ( z ), is given in Eq. (B.12). Using the expressionof z l in Eq. (B.5), we can re-express b z conveniently in a dimensionless form b z = 12 (cid:18) z l z (cid:19) . (B.17)Consequently, the solution θ z in Eq. (B.16) in terms of the adimensional parameter r = z/z l ≥ θ z ≡ θ ( r ) = √ r / (cid:34) − i √ g ( r ) + (1 + i √ g ( r ) (cid:35) (B.18)where g ( r ) = 1 r (cid:16) i (cid:112) r − (cid:17) / . (B.19)By multiplying both numerator and denominator of θ ( r ) by (1 − i √ r − / oneends up, after a little algebra, with the following expression θ ( r ) = √ r / (cid:20) r (cid:16) ξ ζ / r + ξ ζ / r (cid:17)(cid:21) , (B.20)where ξ and ζ r denotes, respectively, a complex number and a complex function ofthe real variable r : ξ = 1 + i √ ζ r = 1 + i (cid:112) r − , (B.21)and we have also introduced the related complex conjugated quantities: ξ = 1 − i √ ζ r = 1 − i (cid:112) r − , (B.22) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle ζ r = ρ r e iφ r and ξ = ρe iφ , with, respectively: ρ r = r / φ r = arctan( (cid:112) r −
1) (B.23)and ρ = 2 φ = arctan( √
3) = π . (B.24)Finally, by writing ξ and ζ r inside Eq. (B.20) in their polar form and taking advantageof the expressions in Eqns. (B.23),(B.24) we get: θ ( r ) = √ r / (cid:20) r ρ ρ / r (cid:16) e i ( φ + φ r ) + e − i ( φ + φ r ) (cid:17)(cid:21) == √ r / (cid:20) (cid:18) π (cid:112) r − (cid:19)(cid:21) (B.25)In order to draw explicitly the function χ ( z ), e.g. with the help of Mathematica , onecan plug the expression of θ ( r = z/z l ) from Eq. (B.25) into the following formula: χ ( z ) = √ z √ E θ ( z/z l ) + 3 θ ( z/z l ) , (B.26) Appendix B.3. The critical value z c We show here how to compute the critical value z c at which χ ( z ) equals z / (2 σ ),i.e., the value at which the two branches in Fig. 2 cross each other. To make thecomputations easier, it is convenient to work with dimensionless variables. Using z l = (3 / σ /E ) / from Eq. (B.5), we express z in units of z l , i.e., we define r = zz l = 2 z (cid:18) Eσ (cid:19) / . (B.27)In terms of r , one can rewrite b ( z ) in Eq. (B.12) as (using the shorthand notation b z = b ( z )): b z = 12 (cid:18) r (cid:19) . (B.28)Consequently, Eq. (B.11) reduces to − √ (cid:18) (cid:19) / r − / θ ( r ) + θ ( r ) − , (B.29)where θ ( r ) = θ z = rz l is dimensionless. Quite remarkably, it turns out that to determinethe critical value z c , rather conveniently we do not need to solve the above cubicequation, Eq. (B.29). Indeed, at z = z c , i.e., r = r c , equating χ ( z c ) = z c / σ , we get √ z c √ E (cid:20) θ ( r c ) + 3 θ ( r c ) (cid:21) = z c σ . (B.30) First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle r c , Eq. (B.30) simplifies to θ ( r c ) + 3 θ ( r c ) = 3 / r / c . (B.31)Consider now Eq. (B.29) evaluated at r = r c . In this equation, we replace r c by itsexpression in Eq. (B.31). This immediately gives θ ( r c ) = 3 / θ ( r c ) = (cid:114) . (B.32)Using this exact θ ( r c ) in Eq. (B.31) gives r c = z c z l = 2 / = 1 . . . . (B.33)It is now straightforward to check that the expression of θ ( r ) written in Eq. (B.25) isconsistent with the result just found, i.e., from it we retrieve θ ( r c = 2 / ) = (cid:112) / θ ( r c = 2 / ) = √ r / c (cid:20) (cid:18) π (cid:112) r c − (cid:19)(cid:21) == (cid:114) (cid:20) (cid:18) π (cid:19)(cid:21) = (cid:114) (cid:104) (cid:16) π (cid:17)(cid:105) == (cid:114) , (B.34)as expected.For comparison to numerical simulations, we chose E = 2, for which σ =2 + 5 E = 22. We get z l = (3 / σ /E ) / = 9 . . . . , which gives z c = r c z l =(1 . . . . ) z l = 11 . . . . . This is represented by a black dotted vertical line inFig. 4. [1] M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B. Liverpool, J. Prost, M. Rao, A. Simha, Rev.Mod. Phys. , 1143 (2013).[2] C. Bechinger, R. Di Leonardo, H. Lowen, C. Reichhardt, G. Volpe, G. Volpe, Rev. Mod. Phys. , 045006 (2016).[3] S. Ramaswamy, J. Stat. Mech. P054002 (2017).[4] E. Fodor, M. C. 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