Dynamics of Run-and-Tumble Particles in Dense Single-File Systems
Thibault Bertrand, Pierre Illien, Olivier Bénichou, Raphaël Voituriez
DDynamics of Run-and-Tumble Particles in Dense Single-File Systems
Thibault Bertrand, ∗ Pierre Illien, Olivier B´enichou, and Rapha¨el Voituriez
1, 3, † Laboratoire Jean Perrin, UMR 8237 CNRS, Sorbonne Universit´e, 75005 Paris, France EC2M, CNRS UMR7083 Gulliver, ESPCI Paris, PSL Research University, 75005 Paris, France Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee,UMR 7600 CNRS, Sorbonne Universit´e, 75005 Paris, France (Dated: July 12, 2018)We study a minimal model of self-propelled particle in a crowded single-file environment. We extend classicalmodels of exclusion processes (previously analyzed for diffusive and driven tracer particles) to the case wherethe tracer particle is a run-and-tumble particle (RTP), while all bath particles perform symmetric random walks.In the limit of high density of bath particles, we derive exact expressions for the full distribution P n ( X ) ofthe RTP position X and all its cumulants, valid for arbitrary values of the tumbling probability α and time n . Our results highlight striking effects of crowding on the dynamics: even cumulants of the RTP positionare increasing functions of α at intermediate timescales, and display a subdiffusive anomalous scaling ∝ √ n independent of α in the limit of long times n → ∞ . These analytical results set the ground for a quantitativeanalysis of experimental trajectories of real biological or artificial microswimmers in extreme confinement. Stemming from experimental observations of bacterial mo-tion [1], Run-and-tumble particles (RTPs) provide a canonicalmodel for the theoretical description of biological or artificialself-propelled entities such as janus particles, bacteria [1–5],algae [6], eukaryotic cells [7], or larger scale animals [8]. Inthese examples of so-called active particles, self-propulsionresults from the conversion of energy supplied by the envi-ronment into mechanical work [9, 10]. In its simplest form,RTP trajectories consists of a sequence of randomly oriented’runs’ — periods of persistent motion in straight line at con-stant speed — interrupted by instantaneous changes of direc-tion (also named polarity of the RTP), called ’tumbles’, occur-ring at random with constant rate.The interplay between active particles and their environ-ment has attracted significant interest recently [10]. Indeed,most motile biological systems such as bacteria or mammaliancells navigate disordered and complex natural environmentssuch as soils, soft gels ( e.g. mucus or agar) or tissues. Throughtheir interactions with the environment, RTPs display robustnon-equilibrium features, focus of many works. For example,recent simulations explored the dynamics of active particlesin the presence of quenched disorder as well as active baths[11–15]. In confined geometries, active particles were foundto accumulate at the boundaries, at odds with the equilibriumBoltzmann distribution [16–21]. Such non trivial interactionswith obstacles can lead to effective trapping and thus have im-portant consequences in the dynamics of self-propelled par-ticles in disordered environments [22]. For example, it wasrecently shown that the large scale diffusivity of RTPs mov-ing in a dynamic crowded environment is nonmonotonic in thetumbling rate for low enough obstacle mobility in dimension d ≥ [23].Effects of crowding are known to have particularly strongconsequences in one-dimensional systems. In the classical ex-ample of single-file diffusion, where identical passive parti-cles diffuse on a line, hard-core interactions impose conserva-tion of the ordering of particles, thereby inducing long livedcorrelations in the motion of a tagged particle and eventu- ally a subdiffusive scaling of its mean squared displacement, MSD ∝ √ n with time n [24–31]. Physically, this resultsfrom the fact that displacements on increasingly large dis-tances need to mobilize the motion of an increasingly largenumber of particles. Single file diffusion has been experimen-tally observed in a variety of natural and man-made materialsranging from passive rheology in zeolites [32, 33], transportof colloidal particles under confinement [34–38], to diffusionof water in carbon nanotubes [39, 40].Diffusion under extreme confinement is also seen in biolog-ical settings with examples ranging from DNA translocation[41, 42], transport of proteins in crowded fluids like the cyto-plasm [43], transport of ions in membrane channels [44, 45]and even migration of dendritic cells in lymphatic vessels [7].Theoretically, recent works have studied the dynamics of ac-tive and biased tracer particles in single-file systems [46–49].Exact predictions for the full distribution of particle positionswere derived in the case of a tracer particle driven out of equi-librium by external forcing in a dense single-file environment[50]. Despite few effort, analytical results for the dynamics ofself-propelled particles in complex and confined environmentsare still largely missing.In this Letter, we study a minimal model of self-propelledparticle in a crowded single-file environment. Despite thechallenge of the inherent coupling between the dynamic envi-ronment of the tracer and its polarity, we derive exact analyti-cal results for the dynamics of RTP in the limit of high densityof diffusive bath particles; in particular, we provide expres-sions for the full distribution P n ( X ) of the RTP position X and all its cumulants, valid for arbitrary values of the tumblingprobability α and time n . Our results highlight striking effectsof crowding on the dynamics. We show in particular that evencumulants of the RTP position are increasing functions of α at intermediate timescales, and display a subdiffusive anoma-lous scaling ∝ √ n with a prefactor independent of α in thelimit n → ∞ . We show a perfect agreement between our an-alytical predictions and the results of numerical simulations.We generalize to RTPs questions that have attracted attention a r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l for passively diffusing and externally driven tracers. Model and RTP polarity dynamics —
In what follows, weconsider a model discrete in space and in time. We take aone-dimensional lattice, infinite in both directions. The latticesites are occupied by particles with mean density ρ performingsymmetric random walks; the bath particles interact via hard-core repulsion, i.e. the occupancy number of each lattice siteis at most equal to one. At time n = 0 , we place at the origin aRTP. In absence of bath particles, the RTP moves along the di-rection of its polarity with probability 1. Thus, if its polarity isalong e (respectively, e − ), the RTP moves to its right withprobability p = 1 and to its left with probability p − = 0 (respectively, p = 0 and p − = 1 ). Further, the polarity di-rection is reversed at each timestep with tumbling probability α (see Supplemental Material [51] for details). As opposed toexternally driven tracers previously studied [50, 52], we treathere the case of self-propelled particles. Our model then cor-responds to the limit of a self-propelled particle with infiniteP´eclet number [10].
FIG. 1.
Model and Evolution Rules - The RTP (orange) movesthrough a bath of diffusive particles (blue), its polarity flips at eachtime step with probability α ; the RTP moves by exchanging positionswith vacancies. We focus here on the limit of dense systems, i.e. the limitof low vacancy densities ρ = 1 − ρ (cid:28) . Following [53],we formulate in this limit the dynamics of the vacancies ratherthan that of the particles which equivalently encodes the dy-namics of the whole system. We follow the evolution rulespreviously established by [50, 53–55] in which we assumethat each vacancy performs a nearest-neighbor symmetric ran-dom walk everywhere on the lattice except in the vicinity ofthe RTP. When surrounded by bath particles, a vacancy thusmoves to one of its neighboring sites with equal probability.However, when the RTP lies on one of its adjacent sites, weaccomodate the nature of the RTP dynamics by implementingthe following specific rules (as detailed in [51]): if the va-cancy occupies the site to the right (respectively, to the left) ofthe RTP, it has a probability q = 1 / (2 p + 1) (respectively, q − = 1 / (2 p − + 1) to jump to the right (respectively, to theleft) and − q (respectively, − q − ) to jump to the left (re-spectively, to the right). It is important to note that additionalrules for the cases where two vacancies are adjacent or havecommon neighbors would normally be needed to complete thedescription of the dynamics; however, these cases lead to cor-rections in O ( ρ ) and are thus unnecessary here. Althoughthese hopping probabilities implicitly depend on the RTP po-larity at a given time, the decoupling between the dynamicsof the polarity and the vacancies makes analytical calculationtractable. Systems with a single vacancy —
Following Brummelhuisand Hilhorst [54], we first consider an auxiliary problem: asystem containing a single vacancy. Trivially in the singlevacancy case, the RTP can only be in one of two positionswhich depend on the original location of the vacancy: if thevacancy starts in
Z > (respectively, Z < ), the RTP canbe found in X = 0 or X = 1 (respectively, in X = − or X = 0 ). The dynamics of the RTP is dictated by exchangesof positions with the vacancies, as such it is controlled by thefirst-passage statistics of vacancies to the position of the RTP.Clearly, there are four cases to consider in general, but sym-metry considerations reduce those cases to two: for a giveninitial RTP polarity, the vacancy can start either on the side ofthe RTP the polarity points to or not.We write p ( n ) ν ( X | Z ) the probability to find the RTP in po-sition X at time n , knowing that it started in X = 0 withpolarity ν and the vacancy at position Z . We can representthis quantity as a sum over all passage events of the vacancyto the RTP location through the following recurrence relation[50, 53, 55] p ( n ) ν ( X | Z ) = δ X, (cid:32) − n (cid:88) k =0 M ( k ) ν ( Z ) (cid:33) + n (cid:88) k =0 p ( n − k )sgn( Z ) ( X − sgn( Z ) | − sgn( Z )) M ( k ) ν ( Z ) (1)with M ( n ) ν ( Z ) the probability that the vacancy starting in Z exchanged position with the TP in time n knowing that itstarted with polarity ν . From Equation (1), we see that thesingle vacancy propagator only depends on M ( n ) ν ( Z ) and thepropagator in the case of a vacancy starting right next to theRTP. Denoting F ( n )sgn( µν ) the first-passage time (FPT) densityin X = 0 for a vacancy starting in µ = ± , knowing that theRTP started with a polarity ν at n = 0 , we notice that M ( n ) ν ( Z ) = n (cid:88) k =0 f ( k )sgn( Z ) Z − (cid:104) S ( k )+ F ( n − k )sgn( νZ ) + S ( k ) − F ( n − k ) − sgn( νZ ) (cid:105) (2)where f ( n ) X denotes the classical first-passage time density atthe origin at time n of a symmetrical one-dimensional P´olyawalk starting at n = 0 at position X [56] and S ( n )sgn( µν ) theprobability for the RTP to have a given polarity µ at timestep n , knowing that it was ν at n = 0 for which it is straightfor-ward to obtain exact expressions (see details in [51]). In theparticular case of a vacancy starting in µ = ± , this probabil-ity reads (see details in [51]) p ( n ) ν ( X | µ ) = δ X, − n (cid:88) j =1 F ( j )sgn( µν ) + ∞ (cid:88) j =1 ∞ (cid:88) m , ··· ,m j × δ (cid:80) ji =1 m k ,n δ X,µ [( − j +1] / F ( m )sgn( µν ) F ( m ) − · · · F ( m j − ) − × (cid:32) − m j (cid:88) i =1 F ( i ) − (cid:33) (3)with δ i,j the Kronecker delta. The first term in the right-handside of Equation (3) gives the probability that at time n theRTP was never visited by the vacancy. The second term rep-resents a partition over the number of visits by the vacancy j and the waiting times between successive visits m j . Here,the key step of our derivation is therefore to compute the first-passage time densities F ( n )sgn( µν ) which a priori couple the po-larity dynamics and vacancy dynamics. Thanks to the Marko-vian dynamics of the RTP polarity, we obtain exact full ex-pressions for these FPT densities and their generating func-tions as shown in [51]. For instance, we show that F ( n ) − obeysthe following recurrence relation, F ( n ) − = α (cid:40) (1 − q ) δ n, + q n (cid:88) k =1 (cid:104) f ( k − S ( k − F ( n − k )+ + f ( k − S ( k − − F ( n − k ) − (cid:105)(cid:111) + α n (cid:88) k =1 (cid:104) f ( k − S ( k − − F ( n − k )+ + f ( k − S ( k − − F ( n − k ) − (cid:105) . (4)Defining the generating function of any time-dependentfunction g ( n ) as (cid:98) g ( ξ ) ≡ (cid:80) ∞ n =0 g ( n ) ξ n , Equations (1)-(3) im-plie that the generating function of the single-vacancy prop-agator can be written in terms of the generating functions ofthe first-passage time densities above as (cid:98) p ν ( X | µ ; ξ ) = δ X, (cid:104) − δ sgn( µ ) ,ν ( (cid:98) F + − (cid:98) F − ) (cid:105) + δ X,µ (cid:98) F − (1 − ξ )(1 + (cid:98) F − ) (5)where we have used the short-hand notations (cid:98) F ± = (cid:98) F ± ( ξ ) and the original position of the vacancy, µ = ± . Systems with a small concentration of vacancies —
We nowconsider a lattice containing a low concentration of vacancies ρ ; we assume that the lattice of size L contains M vacan-cies such that ρ ≡ M/ L . First, we consider the case of afixed initial polarity for the RTP, ν . Following [53], we write P ( n ) ν ( X |{ Z j } ) the probability that the RTP with original po-larity ν is at position X at time n provided that the M vacan-cies started at positions { Z j } j ∈ [1 ,M ] . At the lowest order inthe vacancies density, we consider the contributions of eachvacancy to be independent and we write P ( n ) ν ( X |{ Z j } ) = (cid:88) Y ,...,Y M δ X,Y + ··· + Y M P ( n ) ν ( { Y j }|{ Z j } )= ρ → (cid:88) Y ,...,Y M δ X,Y + ··· + Y M M (cid:89) j =1 p ( n ) ν ( Y j | Z j ) (6)where P ( n ) ( { Y j }|{ Z j } ) is the conditional probability that intime n the RTP has performed a displacement Y due to in-teraction with the vacancy 1, Y due to interaction with thevacancy 2 etc. We define the Fourier transform in space as F{ X } ≡ X ∗ ( q ) = (cid:80) y e iqy X ( y ) , where the sum runs overall lattice sites. In Fourier space and averaged over the initial distribution of vacancies positions (assumed to be uniformlydistributed on the lattice, except for the origin which is occu-pied by the RTP and denoted ¯ · ), we obtain ¯ P ∗ ν ( q, n ) = [¯ p ∗ ν ( q, n )] M ∼ ρ → − ρ Ω ν ( q, n ) (7)where we impose L, M → ∞ (while ρ is kept constant) andwe define Ω ν ( q, n ) = n (cid:88) k =0 (cid:34) [1 − p ∗ + ( q | − n − k )e iq ] + ∞ (cid:88) Z =1 M ( k ) ν ( Z )+[1 − p ∗− ( q | + 1; n − k )e − iq ] − (cid:88) Z = −∞ M ( k ) ν ( Z ) (cid:35) . (8)Lastly, we average over the initial polarity of the RTP toobtain ¯ P ∗ ( q, n ) = 12 (cid:2) ¯ P ∗ + ( q, n ) + ¯ P ∗− ( q, n ) (cid:3) ∼ ρ → − ρ Ω( q, n ) (9)with Ω( q, n ) ≡ [Ω + ( q, n ) + Ω − ( q, n )] / . The generatingfunction of the second characteristic function, which is de-fined as ψ ( q, ξ ) ≡ (cid:80) ∞ n =0 ln (cid:10) e iqX n (cid:11) ξ n , satisfies the follow-ing relation lim ρ → ψ ( q, ξ ) ρ = − (cid:98) Ω( q, ξ ) (10)After lengthy but straightforward algebra (see details in [51]),we proceed to an expansion of (cid:98) Ω( q, ξ ) in power series of q toobtain (cid:98) Ω( q, ξ ) = − (cid:80) ∞ n =1 [1 + ( − n ] (cid:104) (cid:98) F + + (cid:98) F − (cid:105) ( iq ) n /n !2(1 − ξ )(1 − (1 − (cid:112) − ξ ) /ξ )(1 + (cid:98) F − ) . (11)We now have all the tools to derive our central analyticalresult which defines the exact (in the leading order in ρ ) cu-mulant generating function. The cumulants κ j of arbitrary or-der j are defined by ln (cid:2) ¯ P ∗ ( q, n ) (cid:3) ≡ (cid:80) ∞ j =1 κ ( n ) j ( iq ) j /j ! . Wecan identify equal order terms in both expressions and find (cid:98) κ j ( ξ ) = ρ → ρ (cid:2) − j (cid:3) (cid:104) (cid:98) F + ( ξ ) + (cid:98) F − ( ξ ) (cid:105) (1 − ξ )(1 − (1 − (cid:112) − ξ ) /ξ )(1 + (cid:98) F − ( ξ )) (12)Recalling that the FPT densities (cid:98) F ν ( ξ ) are explicitly givenin [51] in terms of the tumbling probability (appearing explic-itly through the Poissonian dynamics of the polarity), Equa-tion (12) provides an expression for all cumulants exact in theleading order in ρ in Laplace space. Strikingly, all cumulantsof same parity are equal and in particular, all odd cumulantsare identically equal to 0. While we easily understand whyodd cumulants in a process averaged over the initial polarityshould be zero, it is interesting to note that in the case wherewe fix the initial polarity of the RTP, the odd cumulants were -3 -2 -1 FIG. 2.
Mean square displacement of the RTP - shown for varioustumbling probabilities α ∈ [0 .
01; 0 . showing an α -dependent tran-sient to a subdiffusive long-time scaling independent of α ; at longtimes, MSD( n ) → (cid:112) n/π (dashed black line). non-zero but decaying as / √ n (see details in [51]). As a sidenote, we show in [51] that our predictions retrieve the resultsfor the infinitely biased tracer particle in the α → limit [50]. Cumulants in the long-time limit —
We expand the gener-ating function of the even cumulants in power series of − ξ (which is equivalent to a long-time limit expansion) lim ρ → (cid:98) κ even ( ξ ) ρ = ξ → √ − ξ ) / − √ α √ − α − ξ + O (1) . (13)Using Tauberian theorems [57], we can invert term by termthis expression and we obtain in the time domain lim ρ → κ ( n )even ρ = n →∞ (cid:114) nπ − √ α √ − α + o ( n − / ) . (14)From Equation (14), we first notice that the leading asymp-totics are surprisingly independent of the tumbling probability α . In Figure 2, we show that the mean-square displacementof the RTP converges in the long-time limit to the anomaloussubdiffusive scaling observed for passive single-file diffusion.This comes from the fact that, for all finite values of α , thestatistics of the waiting times between successive steps of theRTP is asymptotically independent of α . Secondly, by obtain-ing higher order terms in the expansion of the cumulants, werealize that α enters in the first subleading order term with theexpected monotonicity. As shown in Figure 3, as the tumblingprobability decreases ( i.e. the more persistent the RTP be-comes), the importance of the higher order terms in the expan-sion decreases and the dynamics of the RTP converges fasterto the classical single-file scaling.Surprisingly, we observe that the time-averaged MSD de-creases faster for smaller α for short timescales t (cid:28) /α (see Figure 2). This counterintuitive result is opposite to themonotonicity expected in the absence of obstacles. After thistransient, one recovers the expected monotonicity of the MSDwith respect to tumbling probability for t (cid:29) /α . Thus, thetime-averaged stationary mean-square displacement (Figure FIG. 3.
Reduced cumulants - (
Top ) Reduced cumulants ˜ κ ( n )even = κ ( n )even / ( ρ (cid:112) n/π ) vs time n for several tumbling probabilities,showing transients dependent on α (increasing α from yellow toblue) obtained from the inversion of Equation (12); ( Bottom ) Re-duced cumulant for a tumbling probability α = 0 . and a vacan-cies density of ρ = 0 . , from inversion of Equation (12) (solidline), numerical simulations (symbols) and first higher order term inthe Taylor series of the cumulants as in Equation (14) (dashed line), − ˜ κ ( n )even = Γ( α ) / √ n . The α -dependance of the higher-order termis shown in inset.
2) shows an inversion of monotonicity of the second cumulantcompared to that of the ensemble-averaged cumulants (Figure3) in the n → limit, a common signature of aging phenom-ena. Namely, this result suggests that the stationary distribu-tion of vacancy positions is different from the Poissonian ini-tial conditions we consider in the derivation leading to Figure3. Indeed, the bath particles density displays fluctuations inthe vicinity of the RTP (with an increase of particles densityin front of the RTP and a decrease at the back). This impliesthat a higher persistence can lead to a slowing down of the dy-namics at short timescales, a counterintuitive idea reminiscentof the concept of negative differential mobility observed forbiased tracers and run-and-tumble particles in crowded envi-ronments [23, 52, 58]. Full statistics of the RTP position —
Finally, the equalityof same parity cumulants (to leading order in ρ ) implies thatthe associated distribution is of Skellam type [59]. As a conse-quence, our results provide the complete distribution of RTPpositions P n ( X ) for any time n , which in our case simplifiesto P n ( X ) ∼ ρ → exp (cid:16) − κ ( n )even (cid:17) I X (cid:16)(cid:12)(cid:12)(cid:12) κ ( n )even (cid:12)(cid:12)(cid:12)(cid:17) , (15) -10 -8 -6 -4 -2 0 2 4 6 8 1000.250.50.751 FIG. 4.
Distribution of positions of a RTP with random initial po-larity P n ( X ) , tumbling probability α = 0 . and vacancy density ρ = 0 . for various times n = 10 , , and (fromyellow to blue), for numerical simulations (symbols) and theoreticalpredictions obtained from Equation 15 (solid lines). where I n is a modified Bessel function of the first kind [60].Figure 4 shows a perfect agreement between the predicted dis-tributions and our numerical simulations. In the long timelimits, we recover the results derived by Levitt [25] and vanBeijeren et al. [61]. Importantly, we find that independently ofthe tumbling probability the rescaled variable X/ (2 ρ n/π ) / is asymptotically distributed according to a normal law. Summary —
Using a lattice model, we derived expressionsfor the full statistics of positions of run-and-tumble particleswith arbitrary tumbling probability α in dense single-file en-vironment. Our predictions are exact to the leading order invacancy density ρ → . We have shown that the asymp-totic dynamics of the RTP displays an anomalous subdiffusivescaling ∝ √ n with prefactor independent of α . Further, wehighlighted the presence of aging in this system for which thestationary distribution of bath particles is not Poissonian butdisplays density fluctuations in the vicinity of the RTP leadingstrikingly to a slowing down of the short timescale dynam-ics for more persistent RTPs. Run-and-tumble particles are acanonical model of natural and artificial self-propelled parti-cles; as a consequence, we believe that these exact results willfind a wealth of applications; they set the ground for a quan-titative analysis of experimental trajectories of real biologicalor artificial microswimmers in extreme confinement. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] H. C. Berg,
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HOPPING PROBABILITIES
While previous studies examined the dynamics of externally driven tracers [50, 52], we interested here in a model of internallydriven self-propelled particles. We consider a one-dimensional lattice infinite in both directions. The lattice sites are occupiedby hard-core particles present at mean density ρ performing symmetrical random walks, with the restriction that the occupancynumber for each site is at most equal to one. The mean density of vacancies is thus equal to ρ = 1 − ρ . The tracer particle(TP) performs a biased random walk and is initially placed at the origin. The jump direction is chosen with probability / forthe gas particles in the bulk. While the TP chooses to hop along the biasing direction with probability p + , and in the oppositedirection with probability p − , such that p + + p − = 1 . At each time step, the TP reverses its polarity (direction of the bias) withprobability α . This model is an example of Run-and-Tumble motion [1].We consider that the bias originates from the application of a constant internal drive F = F e applied to the run-and-tumbleparticle (RTP) in the direction of its polarity. In this case, we know that the probability for the RTP to be in position x in steadystate is given by a Boltzmann distribution: P stat ( x ) = 1 Z e βF x , (S1)where Z is a normalization constant and β = 1 / ( k B T ) is the inverse temperature. Let us consider for instance that the polarityof the RTP points towards the positive integers ( e = ˆ x ). If we denote, p (1 → the probability for the RTP to go from x = 1 to x = 0 in one step, we know that the detailed balance condition imposes that: P stat (0) p (0 →
1) = P stat (1) p (1 → (S2)We know that by definition: p (0 →
1) = p + and p (1 →
0) = p − . Thus, one gets that: p + p − = e βF (S3)In order to fulfill the detailed balance condition, we finally have that the hopping probabilities are given by: p + = e βF e βF + e − βF , p − = e − βF e βF + e − βF (S4)In the high density limit, the TP only moves when a vacancy visits one of its neighboring sites. Hence, it is interesting to lookat the dynamics of the vacancies rather than the dynamics of the particles. In our one-dimensional problem, a vacancy moves ateach step according to the following evolution rules: • if the vacancy occupies the site to the right of the TP, it has a probability q = 1 / (2 p + 1) to jump to the right and − q to jump to the left; • if the vacancy occupies the site to the left of the TP, it has a probability q − = 1 / (2 p − + 1) to jump to the left and − q − to jump to the right; • in any other case, the vacancy has a probability 1/2 to jump to the left and 1/2 to jump to the right.with by definition p = p + and p − = p − (respectively, p = p + and p − = p − ) if the polarity points in the positive direction(respectively, in the negative direction). According to those definitions, the vacancies perform a P´olya walk everywhere on thelattice except in the vicinity of the RTP [56].In this Letter, we consider only the case of an infinitely biased RTP ( F → ∞ ) along a polarity direction which flips onexponentially distributed times. In the case the polarity of the RTP points in the positive direction, the evolution rules thus read: • if the vacancy occupies the site to the right of the RTP, it has a probability q = 1 / to jump to the right and − q = 2 / to jump to the left (exchange its position with the RTP); • if the vacancy occupies the site to the left of the RTP, it has a probability q − = 1 to jump to the left and − q − = 0 tojump to the right (exchange its position with the RTP); • in any other case, the vacancy has a probability 1/2 to jump to the left and 1/2 to jump to the right.We can easily obtain the evolutions rules in the case the polarity points in the reverse direction by symmetry. POLARITY DYNAMICS OF THE RUN-AND-TUMBLE PARTICLE
The polarity dynamics of the RTP is decoupled from its motion and is solely controlled by its flipping probability α . We knowthat the polarity of our RTP can be positive ( + ) or negative ( − ). We will denote S ( k ) µν the probability for the TP to have a givenpolarity µ at time step k , knowing that it was ν at k = 0 . For all time k assuming that we started with a positive polarity, we canwrite the following master equation: S ( k )++ = αS ( k − − + + (1 − α ) S ( k − (S5) S ( k ) − + = 1 − S ( k )++ (S6)Using the fixed-points method to solve this reccurence relation, we obtain that for all times k , S ( k )++ = 12 (cid:2) − α ) k (cid:3) (S7) S ( k ) − + = 1 − S ( k )++ (S8)We can similarly obtain the following probabilities in the case we start with a negative polarity. In all generality, we can writefor a polarity µ = {±} that: S ( k ) µµ = 12 (cid:2) − α ) k (cid:3) = 1 − S ( k ) − µµ . (S9)For the sake of simplicity, we will sometimes write S ( k ) µµ = S ( k )+ and S ( k ) − µµ = S ( k ) − in what follows. SINGLE VACANCY PROPAGATOR
Following Brummelhuis and Hilhorst [54], we first consider an auxiliary problem: a system containing a single vacancy. Weconsider a lattice on the integers l ∈ [ − L ; L ] , the vacancy can occupy any site but the origin. As we saw in the main text, wecan write the probability to find the RTP in position X at time n , knowing it started with polarity ν at time n = 0 as an averageover the initial condition for the position of the vacancy Z , ¯ p ( n ) ν ( X ) = 12 L (cid:32) − (cid:88) Z = − L p ( n ) ν ( X | Z ) + L (cid:88) Z =1 p ( n ) ν ( X | Z ) (cid:33) (S10)with p ( n ) ν ( X | Z ) defined (as in the main text) as the probability to find the RTP in position X at a time n knowing its polaritywas initially ν and that the vacancy started in Z . In this particular case, the RTP can only be found in two positions dependingon the location of the vacancy at n = 0 : • if the vacancy starts on the right of the RTP, the RTP can be in 0 or 1; • if the vacancy starts on the left of the RTP, the RTP can be in -1 or 0.First, we consider the case where Z > and the RTP started with a positive polarity at n = 0 . We can write the followingrecurrence relation: p ( n )+ ( X | Z ) = δ X, (cid:32) − n (cid:88) k =0 M ( k )+ ( Z ) (cid:33) + n (cid:88) k =0 p ( n − k )+ ( X − | − M ( k )+ ( Z ) (S11)with M ( n )+ ( Z ) the probability that the vacancy starting in Z exchanged positions with the TP in time n knowing that its polaritywas + at n = 0 . In particular, we notice that M ( n )+ ( Z ) can be written as the convolution of the first-passage time density ofa P´olya walk to the adjacent site to the RTP and the first-passage time density of exchange of RTP and vacancy position, for avacancy starting on one of the lattice sites adjacent to the RTP knowing the polarity of the RTP. M ( n )+ ( Z ) = (cid:40) (cid:80) nk =0 f ( k ) Z − S ( k )++ F ( n − k )++ + (cid:80) nk =0 f ( k ) Z − S ( k ) − + F ( n − k )+ − for Z > (cid:80) nk =0 f ( k ) − − Z S ( k )++ F ( n − k ) − + + (cid:80) nk =0 f ( k ) Z − S ( k ) − + F ( n − k ) −− for Z < (S12)where f ( n ) X denotes the first-passage time density at the origin at time n of a symmetrical one-dimensional P´olya walk startingin position X at n = 0 and F ( n ) µν is the first-passage time density of exchange of the positions of the RTP and the vacancy intime n , knowing that the RTP started with an initial polarity ν and the vacancy started in Z = µ . Full expressions for thoseFTP densities are provided in Section . We note that we have similarly M ( n ) − ( Z ) = (cid:40) (cid:80) nk =0 f ( k ) Z − S ( k )+ − F ( n − k )++ + (cid:80) nk =0 f ( k ) Z − S ( k ) −− F ( n − k )+ − for Z > (cid:80) nk =0 f ( k ) − − Z S ( k )+ − F ( n − k ) − + + (cid:80) nk =0 f ( k ) Z − S ( k ) −− F ( n − k ) −− for Z < (S13)In Equation (S11), the first term corresponds to the probability that the RTP has not moved and the second term is composedof a convolution between the probability to have exchanged positions in k steps and the probability that the TP travels X − steps in time n − k , granted that it had a polarity + at n = 0 and that the vacancy started now in − (vacancy and RTP exchangedpositions).Similarly in the case Z < , we have: p ( n )+ ( X | Z ) = δ X, (cid:32) − n (cid:88) k =0 M ( k )+ ( Z ) (cid:33) + n (cid:88) k =0 p ( n − k ) − ( X + 1 | + 1) M ( k )+ ( Z ) (S14)(in this expression the second term contains the probability p ( n − k ) − ( X + 1 | + 1) because we know that the propagator needsto be shifted following interaction with the vacancy coming from the left of the RTP and the fact that the polarity had to flip inorder for the RTP and the vacancy to exchange positions.)By symmetry, we obtain similarly, p ( n ) − ( X | Z ) = δ X, (cid:16) − (cid:80) nk =0 M ( k ) − ( Z ) (cid:17) + (cid:80) nk =0 p ( n − k )+ ( X − | − M ( k ) − ( Z ) , if Z > δ X, (cid:16) − (cid:80) nk =0 M ( k ) − ( Z ) (cid:17) + (cid:80) nk =0 p ( n − k ) − ( X + 1 | + 1) M ( k ) − ( Z ) , if Z < (S15)As we can see from Equations (S11), (S14) and (S15), finding the general single vacancy propagator reduces to finding anexpression for the single vacancy propagator in the case of a vacancy adjacent to the RTP. With some care, the general singlevacancy propagator can be expressed using the first-passage statistics the vacancy starting on a site adjacent to the RTP to itsposition. For instance, the probability to find the RTP in position X in time n knowing that the TP started in X = 0 with apositive polarity and a vacancy in Z = +1 reads: p ( n )+ ( X | + 1) = δ X, − n (cid:88) j =0 F ( j )++ + δ X, ∞ (cid:88) j =1 ∞ (cid:88) m =0 . . . ∞ (cid:88) m j +1 =0 δ (cid:80) j +1 i =1 m i ,n F ( m )++ F ( m ) − + F ( m )+ − . . . F ( m j ) − + (cid:32) − m j +1 (cid:88) k =0 F ( k )+ − (cid:33) + δ X, ∞ (cid:88) j =1 ∞ (cid:88) m =0 . . . ∞ (cid:88) m j =0 δ (cid:80) ji =1 m i ,n F ( m )++ F ( m ) − + F ( m )+ − . . . F ( m j − )+ − (cid:32) − m j (cid:88) k =0 F ( k ) − + (cid:33) (S16)Equation (S16) is composed of three terms:1. The RTP is in 0, it was never visited by the vacancy;2. The RTP is in 0, it was visited an even number of times; this term contains j visits and last non-visiting event;3. The RTP is in 1, it was visited an odd number of times; this term contains j + 1 visits and last non-visiting event.Taking the discrete Laplace transform of Equation (S16) and denoting the FPT densities (cid:98) F µν ( ξ ) = (cid:98) F sgn( µν ) for the sake ofsimplicity, we find (cid:98) p + ( X | + 1; ξ ) = 11 − ξ δ X, (1 − (cid:98) F + ) + δ X, (1 − (cid:98) F − ) ∞ (cid:88) j =1 (cid:98) F + (cid:98) F j − − + δ X, (1 − (cid:98) F − ) ∞ (cid:88) j =1 (cid:98) F + (cid:98) F j − − (S17)which finally reduces to, (cid:98) p + ( X | + 1; ξ ) = δ X, (1 − (cid:98) F + + (cid:98) F − ) + δ X, (cid:98) F + (1 − ξ )(1 + (cid:98) F − ) . (S18)Similarly, we find that p ( n )+ ( X | −
1) = δ X, − n (cid:88) j =0 F ( j ) − + + δ X, ∞ (cid:88) j =1 ∞ (cid:88) m =0 . . . ∞ (cid:88) m j +1 =0 δ (cid:80) j +1 i =1 m i ,n F ( m ) − + F ( m )+ − F ( m ) − + . . . F ( m j )+ − (cid:32) − m j +1 (cid:88) k =0 F ( k ) − + (cid:33) + δ X, − ∞ (cid:88) j =1 ∞ (cid:88) m =0 . . . ∞ (cid:88) m j =0 δ (cid:80) ji =1 m i ,n F ( m ) − + F ( m )+ − F ( m ) − + . . . F ( m j − ) − + (cid:32) − m j (cid:88) k =0 F ( k )+ − (cid:33) (S19)Taking the discrete Laplace transform of Equation (S19), we obtain: (cid:98) p + ( X | − ξ ) = δ X, + δ X, − (cid:98) F − (1 − ξ )(1 + (cid:98) F − ) (S20)By symmetry, we can write the remaining two cases, (cid:98) p − ( X | + 1; ξ ) = δ X, + δ X, (cid:98) F − (1 − ξ )(1 + (cid:98) F − ) (S21) (cid:98) p − ( X | − ξ ) = δ X, (1 − (cid:98) F + + (cid:98) F − ) + δ X, − (cid:98) F + (1 − ξ )(1 + (cid:98) F − ) (S22)Finally, we realize that the single vacancy propagator only depends on the FPT densities F ( n ) µν , ¯ p ( n ) ν ( X ) = δ X, − L (cid:88) Z (cid:54) =0 n (cid:88) k =0 M ( k ) ν ( Z ) + 12 L n (cid:88) k =0 p ( n − k ) − ( X + 1 | + 1) − (cid:88) Z = − L M ( k ) ν ( Z ) + 12 L n (cid:88) k =0 p ( n − k )+ ( X − | − L (cid:88) Z =1 M ( k ) ν ( Z ) . (S23)We provide a full derivation of those quantities in Section . FIRST-PASSAGE TIME DENSITIES FOR VACANCIES ADJACENT TO THE RTP
Knowing that the RTP polarity is in a given state at n = 0 and that a vacancy is adjacent to the TP at that time, we want tocalculate the probability that a first interaction ( i.e. exchange of positions) will happen at time n . In all generality, there are fourpossible configurations to consider at n = 0 assuming that the RTP is in X = 0 : • RTP polarity is ( + ) and the vacancy is in X = +1 ; • RTP polarity is ( + ) and the vacancy is in X = − ; • RTP polarity is ( − ) and the vacancy is in X = +1 ; • RTP polarity is ( − ) and the vacancy is in X = − .We denote F ( n ) µν the first-passage time density in the case where the RTP polarity is originally ν and the vacancy started in Z = µ . We will detail here the derivation on an example. We will consider that the polarity is positive at n = 0 . Knowing thata vacancy is next to the RTP at n = 0 , we want to calculate the probability of first interaction (exchange of positions) at time n .In this particular case, the vacancy has a chance to interact with the RTP at the first time step, and we can express this quantityvia the First-Passage Time (FPT) density at the origin at time n of a symmetrical one-dimensional P´olya walk starting at n = 0 at position l and denoted f ( n ) l . Thus, we write the FPT density as the following convolution: F ( n )++ = (1 − α ) (cid:40) (1 − q ) δ n, + q n (cid:88) k =1 (cid:104) f ( k − S ( k − F ( n − k )++ + f ( k − S ( k − − + F ( n − k )+ − (cid:105)(cid:41) + α n (cid:88) k =1 (cid:104) f ( k − S ( k − − F ( n − k )++ + f ( k − S ( k − −− F ( n − k )+ − (cid:105) . (S24)At each time step, the RTP (i) flips its polarity with probability α and (ii) attempts to make a step in the direction of its polarity.The first term in equation (S24) corresponds to the case where the RTP does not flip its polarity at n = 1 . In this case, we knowthat the RTP has a chance to exchange positions with the vacancy at n = 1 (this is the first term in the curly brackets), the secondterm is based on the probability that the vacancy did not exchange positions with the RTP at n = 1 but came back in k − stepswhile the polarity is still positive, and the third term is based on the probability that the vacancy did not exchange positions withthe RTP at n = 1 but came back in k − steps while the polarity flipped in the meantime. The second term in equation (S24)corresponds to the case where the RTP does flip its polarity at n = 1 . This case is similar to the first term, with the exceptionthat we know that the RTP and the adjacent vacancy cannot interact at the first time step.Similarly, for a vacancy starting on the left of the RTP with a positive polarity, we write F ( n ) − + = α (cid:40) (1 − q ) δ n, + q n (cid:88) k =1 (cid:104) f ( k − S ( k − −− F ( n − k ) −− + f ( k − S ( k − − F ( n − k ) − + (cid:105)(cid:41) + (1 − α ) n (cid:88) k =1 (cid:104) f ( k − S ( k − − + F ( n − k ) −− + f ( k − S ( k − F ( n − k ) − + (cid:105) . (S25)For a given vacancy position, the polarity of the TP can be pointing towards the vacancy or not. By symmetry, we only needto compute these two cases. We denote F ( n )+ the First-Time Passage density for the case where the vacancy is on the right sideof the RTP given its polarity, while we denote F ( n ) − the inverse case and we have, F ( n )++ = F ( n ) −− = F ( n )+ (S26) F ( n ) − + = F ( n )+ − = F ( n ) − (S27)Furthermore, by definition, we can also write that f ( n ) l = f ( n ) − l . For the sake of simplicity, we will define : g ( n )+ = f ( n − S ( n − = f ( n − S ( n − −− (S28) g ( n ) − = f ( n − S ( n − − + = f ( n − S ( n − − (S29)We can study the discrete Laplace transform of these two quantities, defined as: (cid:98) F ( ξ ) = (cid:80) ∞ n =0 ξ n F ( n ) . Using the convolutiontheorem, we obtain: (cid:98) F + ( ξ ) = (1 − α ) (cid:110) (1 − q ) ξ + q (cid:104)(cid:98) g + ( ξ ) (cid:98) F + ( ξ ) + (cid:98) g − ( ξ ) (cid:98) F − ( ξ ) (cid:105)(cid:111) + α (cid:104)(cid:98) g − ( ξ ) (cid:98) F + ( ξ ) + (cid:98) g + ( ξ ) (cid:98) F − ( ξ ) (cid:105) (S30) (cid:98) F − ( ξ ) = α (cid:110) (1 − q ) ξ + q (cid:104)(cid:98) g + ( ξ ) (cid:98) F + ( ξ ) + (cid:98) g − ( ξ ) (cid:98) F − ( ξ ) (cid:105)(cid:111) + (1 − α ) (cid:104)(cid:98) g − ( ξ ) (cid:98) F + ( ξ ) + (cid:98) g + ( ξ ) (cid:98) F − ( ξ ) (cid:105) (S31)We can combine equations (S30) and (S31) to obtain finally (cid:98) F + ( ξ ) = (1 − q ) [1 − α − (1 − α ) (cid:98) g + ( ξ ))] ξ − (1 + q ) [(1 − α ) (cid:98) g + ( ξ ) + α (cid:98) g − ( ξ )] + q (1 − α ) (cid:2)(cid:98) g ( ξ ) − (cid:98) g − ( ξ ) (cid:3) (S32) (cid:98) F − ( ξ ) = (1 − q ) [ α + (1 − α ) (cid:98) g − ( ξ ))] ξ − (1 + q ) [(1 − α ) (cid:98) g + ( ξ ) + α (cid:98) g − ( ξ )] + q (1 − α ) (cid:2)(cid:98) g ( ξ ) − (cid:98) g − ( ξ ) (cid:3) (S33)(S34)Going back to the definition of the discrete Laplace transform, we obtain the following expressions (cid:98) g + ( ξ ) = ∞ (cid:88) n =1 ξ n f ( n − (1 + (1 − α ) n − ) / ξ (cid:104) (cid:98) f ( ξ ) + (cid:98) f ((1 − α ) ξ ) (cid:105) (S35) (cid:98) g − ( ξ ) = ∞ (cid:88) n =1 ξ n f ( n − (1 − (1 − α ) n − ) / ξ (cid:104) (cid:98) f ( ξ ) − (cid:98) f ((1 − α ) ξ ) (cid:105) (S36)(S37)with by definition of a P´olya walk: (cid:98) f x ( ξ ) = (cid:32) − (cid:112) − ξ ξ (cid:33) | x | (S38)As a conclusion, we have obtained exact expressions for the First-Passage Time densities we needed to complete our expres-sion of the single vacancy propagator. SINGLE FILE WITH A SMALL CONCENTRATION OF VACANCIES
In this section, we consider now the case of a small but finite concentration of vacacancies, we assume that the system contains M vacancies such that ρ = M/ L . We will start by deriving expression for the cumulants, exact in the linear order in the densityof vacancies. We will then show that our results are consistent with the results in the case of a biased tracer particle (derived inReference [50]). Finally, we will generalize our derivation to the case of a random initial polarity to obtain the cumulants andfull statistics for the RTP position. Case of a fixed initial polarity
First, we consider the case where we fix the initial polarity of the RTP. Following Brummelhuis and Hilhorst [53, 54], wewrite in general P ( n ) ν ( X |{ Z j } ) the probability that the RTP is at position X at time n provided that the M vacancies were atpositions { Z j } j ∈ [1 ,M ] and the RTP polarity was ν at n = 0 . We can write: P ( n ) ν ( X |{ Z j } ) = (cid:88) Y ,...,Y M δ X,Y + ...Y M P ( n ) ν ( { Y j }|{ Z j } ) (S39)where P ( n ) ν ( { Y j }|{ Z j } ) is the conditional probability that, within the time interval n , the RTP has performed a displacement Y due to interaction with the vacancy 1, Y due to interaction with the vacancy 2 etc. In the lowest order of the vacancy density ρ ,the vacancies contribute independently to the displacement: P ( n ) ν ( { Y j }|{ Z j } ) = ρ → M (cid:89) j =1 p ( n ) ν ( Y j | Z j ) (S40)Thus, we can express this probability as a function of the single vacancy propagator: P ( n ) ν ( X |{ Z j } ) = ρ → (cid:88) Y ,...,Y M δ X,Y + ...Y M M (cid:89) j =1 p ( n ) ν ( Y j | Z j ) (S41)If we suppose that the vacancies are uniformly distributed, we can average P ( n ) ν ( X |{ Z j } ) over the initial distribution ofvacancies: ¯ P ( n ) ν ( X ) = (cid:88) Y ,...,Y M δ X,Y + ...Y M M (cid:89) j =1 p ( n ) ν ( Y j | Z j ) (S42) = (cid:88) Y ,...,Y M δ X,Y + ...Y M M (cid:89) j =1 p ( n ) ν ( Y j | Z j ) (S43) = ρ → (cid:88) Y ,...,Y M δ X,Y + ...Y M M (cid:89) j =1 ¯ p ( n ) ν ( Y j ) (S44)By definition, the Fourier transform in space is written as: F{ X } = X ∗ ( q ) = + ∞ (cid:88) y = −∞ e iqy X ( y ) (S45)where the sum runs over all lattice sites.For the sake of simplicity and without loss of generality, we can assume that the spin is always positive in n = 0 . As aconsequence of Equation (S44), the probability averaged over initial conditions for the vacancies reduces in Fourier transformto ¯ P ∗ + ( q, n ) = (cid:2) ¯ p ∗ + ( q, n ) (cid:3) M , (S46)where we see that the total contribution of the M vacancies reduces to a superposition of the contributions of single vacancies.This expression gives formally a relationship between the general propagator and the single vacancy propagator. From Equation(S23), we write in Fourier space: ¯ p ∗ + ( q, n ) = 1 − L n (cid:88) k =0 (cid:34) [1 − p ∗− ( q | + 1; n − k )e − iq ] − (cid:88) Z = − L M ( k )+ ( Z ) + [1 − p ∗ + ( q | − n − k )e iq ] L (cid:88) Z =1 M ( k )+ ( Z ) (cid:35) (S47)We can rewrite Equation (S46) as: ¯ P ∗ + ( q ; n ) = (cid:20) − L Ω L + ( q, n ) (cid:21) M , (S48)where we define the following quantity Ω L + ( q, n ; L ) = n (cid:88) k =0 (cid:34) [1 − p ∗− ( q | + 1; n − k )e − iq ] − (cid:88) Z = − L M ( k )+ ( Z ) + [1 − p ∗ + ( q | − n − k )e iq ] + L (cid:88) Z =1 M ( k )+ ( Z ) (cid:35) . (S49)By definition of the Fourier transform for a random variable X n : ¯ P ∗ + ( q, n ) = (cid:10) e iqX n (cid:11) (S50)The cumulant generating function, defined as ψ n ( q ) = ln (cid:10) e iqX n (cid:11) , reads in the limit of low vacancies density ( ρ (cid:28) ): ψ n ( q ) ≡ ln ¯ P ∗ + ( q, n ) ∼ ρ → − ρ Ω ∞ + ( q, n ) (S51)with ρ = M/ L and M, L → ∞ . This leads to the Z-transform relation: lim ρ → ψ ( q, ξ ) ρ = − ∞ (cid:88) k =0 Ω ∞ + ( q, n ) ξ k = − (cid:98) Ω + ( q, ξ ) (S52)By discrete Laplace transform, we obtain: (cid:98) Ω + ( q, ξ ) = (cid:20) − ξ − (cid:98) p ∗− ( q | + 1 , ξ )e − iq (cid:21) h − ( ξ ) + (cid:20) − ξ − (cid:98) p ∗ + ( q | − , ξ )e iq (cid:21) h + ( ξ ) (S53)with h µ = (cid:80) µ ∞ Z = µ (cid:99) M + ( ξ ) , we provide full calculation and expressions for these quantitites in Section . Calculation of h µ We have already noticed that: M ( n )+ ( Z ) = (cid:40) (cid:80) nk =0 f ( k ) Z − S ( k )++ F ( n − k )++ + (cid:80) nk =0 f ( k ) Z − S ( k ) − + F ( n − k )+ − for Z > (cid:80) nk =0 f ( k ) − − Z S ( k )++ F ( n − k ) − + + (cid:80) nk =0 f ( k ) − − Z S ( k ) − + F ( n − k ) −− for Z < (S54)The associated generating functions are thus simply: (cid:99) M + ( Z, ξ ) = (cid:40) (cid:92) f Z − S ++ ( ξ ) (cid:98) F + ( ξ ) + (cid:92) f Z − S − + ( ξ ) (cid:98) F − ( ξ ) for Z > (cid:92) f − − Z S ++ ( ξ ) (cid:98) F − ( ξ ) + (cid:92) f − − Z S − + ( ξ ) (cid:98) F + ( ξ ) for Z < (S55)In particular, we have (cid:92) f X S ++ ( ξ ) = 12 (cid:104) (cid:98) f X ( ξ ) + (cid:98) f X ( ξ (1 − α )) (cid:105) (S56) (cid:92) f X S − + ( ξ ) = 12 (cid:104) (cid:98) f X ( ξ ) − (cid:98) f X ( ξ (1 − α )) (cid:105) (S57)Finally, we can write that: h + ( ξ ) = ∞ (cid:88) Z =1 (cid:99) M + ( Z, ξ )= 12 (cid:34) (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )1 − (cid:98) f ( ξ ) + (cid:98) F + ( ξ ) − (cid:98) F − ( ξ )1 − (cid:98) f ( ξ (1 − α )) (cid:35) (S58)We can also write: h − ( ξ ) = − (cid:88) Z = −∞ (cid:99) M + ( Z, ξ )= 12 (cid:34) (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )1 − (cid:98) f ( ξ ) − (cid:98) F + ( ξ ) − (cid:98) F − ( ξ )1 − (cid:98) f ( ξ (1 − α )) (cid:35) (S59)In conclusion, in general, we have: h µ ( ξ ) = 12 (cid:34) (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )1 − (cid:98) f ( ξ ) + sgn( µ ) (cid:98) F + ( ξ ) − (cid:98) F − ( ξ )1 − (cid:98) f ( ξ (1 − α )) (cid:35) (S60) Expression for the cumulants with positive initial polarity
From Equations (S20) and (S21), the Laplace-Fourier transform of the single vacancy propagator is given by: (cid:98) p ∗ + ( q | − ξ ) = 1 + e − iq (cid:98) F − ( ξ )(1 − ξ )(1 + (cid:98) F − ( ξ )) (S61) (cid:98) p ∗− ( q | + 1; ξ ) = 1 + e iq (cid:98) F − ( ξ )(1 − ξ )(1 + (cid:98) F − ( ξ )) (S62)Combining Equations (S53), (S60), (S61) and (S62), we finally obtain: (cid:98) Ω + ( q, ξ ) = 1 − e − iq − ξ ][1 + (cid:98) F − ( ξ )] (cid:34) (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )1 − (cid:98) f ( ξ ) − (cid:98) F + ( ξ ) − (cid:98) F − ( ξ )1 − (cid:98) f ( ξ (1 − α )) (cid:35) +1 − e iq − ξ ][1 + (cid:98) F − ( ξ )] (cid:34) (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )1 − (cid:98) f ( ξ ) + (cid:98) F + ( ξ ) − (cid:98) F − ( ξ )1 − (cid:98) f ( ξ (1 − α )) (cid:35) (S63)On one hand, we can proceed to the expansion of (cid:98) Ω + ( q, ξ ) in power series of q : (cid:98) Ω + ( q, ξ ) = − (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )2[1 − ξ ][1 + (cid:98) F − ( ξ )][1 − (cid:98) f ( ξ )] ∞ (cid:88) j =1 ( iq ) j j ! (cid:2) − j (cid:3) − (cid:98) F + ( ξ ) − (cid:98) F − ( ξ )2[1 − ξ ][1 + (cid:98) F − ( ξ )][1 − (cid:98) f ( ξ (1 − α ))] ∞ (cid:88) j =1 ( iq ) j j ! (cid:2) − ( − j (cid:3) (S64)Recalling the definition of the generating functions of the cumulants κ ( n ) j of arbitrary order j , we can write ψ n ( q ) = ln ¯ P ∗ + ( q, n ) ≡ ∞ (cid:88) j =1 κ ( n ) j j ! ( iq ) j = − ρ Ω + ( q, n ) (S65)So we can identify same order terms and write that: (cid:98) κ j ( ξ ) = ρ → ρ − ξ ][1 + (cid:98) F − ( ξ )] (cid:40) (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )1 − (cid:98) f ( ξ ) (cid:2) − j (cid:3) + (cid:98) F + ( ξ ) − (cid:98) F − ( ξ )1 − (cid:98) f ( ξ (1 − α )) (cid:2) − ( − j (cid:3)(cid:41) (S66)Equation (S66) provides an exact expression of the cumulants in the Fourier-Laplace space. Recalling that the functions (cid:98) F ν ( ξ ) are explicitly given in Section in terms of the tumbling probability, this equation gives an expression of the cumulants ofarbitrary order. Cumulants in the long-time limit
From Equation (S66), we see that all odd cumulants have the same generating function (cid:98) κ odd ( ξ ) and all even cumulants havethe same generating function (cid:98) κ even ( ξ ) . We recall that the expression for the (cid:98) F ν ( ξ ) is given by Equations (S32) and (S33). Wecan thus proceed to an expansion in power series of − ξ (which is equivalent to a long-time expansion in the time domain) ofthe generating function of the cumulants and using the fact that q = 1 / , we find that: lim ρ → (cid:98) κ odd ( ξ ) ρ = ξ → √ − α ) (cid:112) α (1 − α ) (cid:104)(cid:112) α (1 − α ) − α (cid:105) √ − ξ + O (1) (S67) lim ρ → (cid:98) κ even ( ξ ) ρ = ξ → √ − ξ ) / + O (cid:18) √ − ξ (cid:19) (S68)At this point, it is useful to remember the Tauberian theorem [57]. For a time-dependent function φ ( n ) and its associatedgenerating function (cid:98) φ ( ξ ) = (cid:80) ∞ n =1 φ ( n ) ξ n , if the expansion of (cid:98) φ ( ξ ) in powers of (1 − ξ ) has the form (cid:98) φ ( ξ ) ∼ ξ → − ξ ) χ Φ (cid:18) − ξ (cid:19) , (S69)Then, the long time behavior of φ ( n ) is given by φ ( t ) ∼ n →∞ χ ) n χ − Φ( n ) , (S70)where Γ is the usual gamma function. This relation holds if χ > , φ ( n ) > , φ ( n ) is monotonic and Φ is slowly varying inthe sense that lim x →∞ Φ( λx )Φ( x ) = 1 (S71)for any λ > .Using the Tauberian theorem, we find that the long-time behavior of the odd cumulants is given by lim ρ → κ odd ( n ) ρ = n →∞ (1 − α ) √ πn (cid:112) α (1 − α ) (cid:104)(cid:112) α (1 − α ) − α (cid:105) + o (1 /n ) . (S72)and a long-time behavior of the even cumulants given by lim ρ → κ even ( n ) ρ = n →∞ (cid:114) nπ + o (1) . (S73)We note that remarkably the leading order in time of the even cumulants does not depend on α , while the leading order intime of the odd cumulants does and decays to zero in the long time limit. The asymptotic behavior of κ even ( n ) tells us that thevariance of the RTP position grows as √ n , this subdiffusive behavior is the one obtained for the classical symmetric single filedynamics. Retrieving the case of a biased TP in the α → limit For sanity, we can check our calculation against the result for a biased TP [50]. In this section, we will check that ourderivation for a run-and-tumble tracer particle in the limit α → gives the same prediction as in the case of an infinitely biasedtracer particle. In particular, we recall that the cumulants of all order in the biased case are given by: lim ρ → (cid:98) κ bj ( ξ ) ρ = (cid:98) F (1 − (cid:98) F − ) + ( − j (cid:98) F − (1 − (cid:98) F )(1 − ξ )(1 − (1 − (cid:112) − ξ ) /ξ )(1 − (cid:98) F (cid:98) F − ) (S74)0where (cid:98) F ± = (1 − q ± ξ ) / (1 − q ± (1 − (cid:112) − ξ )) and q ± defined as in the RTP case. In the case of an infinite bias, we knowthat q = 1 / and q − = 1 leading to lim ρ → (cid:98) κ bj ( ξ ) ρ = 2 ξ (1 − ξ )(1 − (1 − (cid:112) − ξ ) /ξ )(2 + (cid:112) − ξ ) (S75)In the Run-and-Tumble case, we need to first take the limit of low tumbling rate, α → , before taking the limit of long-times.Using Equations (S32) and (S33) with q = 1 / , we can write that the FTP densities as: (cid:98) F + ( ξ ) = 2 ξ (cid:112) − ξ (S76) (cid:98) F − ( ξ ) = 0 (S77)Injecting this result in Equation (S66), we obtain that for all orders: lim ρ → (cid:98) κ j ( ξ ) ρ = 2 ξ (1 − ξ )(1 − (1 − (cid:112) − ξ ) /ξ )(2 + (cid:112) − ξ ) (S78)We notice that: (1) cumulants in the infinitely biased case and the RTP case in the limit α → are equal and (2) expressionsfor the cumulants for all orders are equal. This means that in all cases, all cumulants are equal and in particular, an expansion inpower series of − ξ gives: lim ρ → (cid:98) κ j ( ξ ) ρ = ξ → √ − ξ ) / + O (cid:18) √ − ξ (cid:19) (S79)(S80)Hence, we conveniently retrieve the biased case in the zero tumbling rate limit ( α → ). GENERAL CASE: RANDOM INITIAL SPIN
We now generalize the results of the previous section to the more general case, where the polarity is not fixed to a particulardirection at n = 0 . For that, we need to average over trajectories conditioned with positive and negative initial polarity. Cumulants in the case of a negative initial polarity
It is rather straightforward to calculate the specific case of a fixed negative polarity and check that it gives us the expectedresult considering the derivation in section . Similarly to the previous derivation, we write in this case, ¯ P ∗− ( q, n ) = (cid:2) ¯ p ∗− ( q, n ) (cid:3) M (S81)From Equation (S23), we can write: ¯ p ∗− ( q, n ) = 1 − L n (cid:88) k =0 (cid:34) [1 − p ∗− ( q | + 1; n − k )e iq ] − (cid:88) Z = − L M ( k ) − ( Z ) + [1 − p ∗ + ( q | − n − k )e − iq ] L (cid:88) Z =1 M ( k ) − ( Z ) (cid:35) (S82)Following the same procedure and definition as in Section , the previous expression expanded reads (cid:98) Ω − ( q, ξ ) = − (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )2[1 − ξ ][1 + (cid:98) F − ( ξ )][1 − (cid:98) f ( ξ )] ∞ (cid:88) j =1 ( iq ) j j ! (cid:2) − j (cid:3) + (cid:98) F + ( ξ ) − (cid:98) F − ( ξ )2[1 − ξ ][1 + (cid:98) F − ( ξ )][1 − (cid:98) f ( ξ (1 − α ))] ∞ (cid:88) j =1 ( iq ) j j ! (cid:2) − ( − j (cid:3) (S83)Finally the cumulants are given by identification of the n -th order terms in the development, and we obtain: (cid:98) κ j ( ξ ) = ρ → ρ − ξ ][1 + (cid:98) F − ( ξ )] (cid:40) (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )1 − (cid:98) f ( ξ ) (cid:2) − j (cid:3) − (cid:98) F + ( ξ ) − (cid:98) F − ( ξ )1 − (cid:98) f ( ξ (1 − α )) (cid:2) − ( − j (cid:3)(cid:41) (S84)Comparing Equations (S66) and (S84), it is easy to see that these expressions will yield the same cumulants for all even orders,and cumulants with opposite signs for all odd orders.1 Expression of the cumulants in the general case
The final step is now to average over the initial spin, we condition on the spin right after the averaging over initial conditionsfor a small concentration of vacancies. As such, we write: ¯ P ∗ ( q, n ) = 12 (cid:2) ¯ P ∗ + ( q, n ) + ¯ P ∗− ( q, n ) (cid:3) (S85)Reinjecting in this expression equations (S46) and (S81), we obtain ¯ P ∗ ( q, n ) = 12 (cid:104)(cid:2) ¯ p ∗ + ( q, n ) (cid:3) M + (cid:2) ¯ p ∗− ( q, n ) (cid:3) M (cid:105) ∼ ρ → − ρ Ω L ( q, n ) (S86)with Ω ∞ ( q, n ) = (cid:2) Ω ∞ + ( q, n ) + Ω ∞− ( q, n ) (cid:3) / when L, M → ∞ . In the Laplace domain, we obtain then (cid:98) Ω( q, ξ ) = 12 (cid:104)(cid:98) Ω + ( q, ξ ) + (cid:98) Ω − ( q, ξ ) (cid:105) (S87) i.e. (cid:98) Ω( q, ξ ) = − (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )2[1 − ξ ][1 + (cid:98) F − ( ξ )][1 − (cid:98) f ( ξ )] ∞ (cid:88) j =1 ( iq ) j j ! (cid:2) − j (cid:3) (S88)As usual, the expression for the cumulants is given by identification of the terms in the expansion and we get (cid:98) κ j ( ξ ) = ρ → ρ (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )(1 − ξ )(1 − (cid:98) f ( ξ ))(1 + (cid:98) F − ( ξ )) (cid:2) − j (cid:3) (S89)As a conclusion, we can easily see that all even cumulants are equal, as are all odd cumulants. We obtain the following finalcumulants lim ρ → (cid:98) κ even ( ξ ) ρ = (cid:98) F + ( ξ ) + (cid:98) F − ( ξ )(1 − ξ )(1 − (1 − (cid:112) − ξ ) /ξ )(1 + (cid:98) F − ( ξ )) (S90) lim ρ → (cid:98) κ odd ( ξ ) ρ = 0 (S91)We realize that all odd cumulants are identically equal to 0. As for the even cumulants, they present the same form as theeven cumulants in the case of a positive conditioning, which was expected as the initial polarity only affects the dynamics untilthe first polarity flip. We can thus proceed to an expansion in power series of − ξ the generating function of the cumulant andusing the fact that q = 1 / , we can proceed to a Taylor expansion of the cumulants at long time to higher orders and we obtain: lim ρ → (cid:98) κ even ( ξ ) ρ = ξ → √ − ξ ) / − √ α √ − α − ξ + O (1) (S92)This expression can be inverted term by term and we obtain in time domain lim ρ → κ even ( n ) ρ = n →∞ (cid:114) nπ − √ α √ − α + o (cid:18) √ n (cid:19) (S93)In particular, the second cumulant is by definition the mean-square displacement of the TP. Although we can see that thetransient depends on the tumbling probability, the MSDs display a √ n long-time scaling characteristic of the original single fileprocess that is in particular independent of α .2 Full distribution of positions
The simple fact that to leading order in ρ cumulants of the same parity are equal tells us that distribution associated to thesecumulants is a Skellam distribution [59]. In this case, we know that the full distribution function P n ( X ) for any time n is givenby P n ( X ) ∼ ρ → exp ( − κ even ( n )) (cid:18) κ even ( n ) + κ odd ( n ) κ even ( n ) − κ odd ( n ) (cid:19) X/ I X (cid:16)(cid:112) κ even ( n ) − κ odd ( n ) (cid:17) , (S94)where I X is a modified Bessel function of the first kind [60]. We show in the main text a perfect agreement between thisdistribution and the results of our numerical simulations.Here, in the long time limit, the distribution reads P n ( X ) ∼ ρ → exp (cid:16) − ρ (cid:112) n/π (cid:17) I X (cid:16) ρ (cid:112) n/π (cid:17) ..