We propose a model of run-and-tumble particles (RTPs) on a line with a fertile site at the origin. After going through the fertile site, a run-and-tumble particle gives rise to new particles until it flips direction. The process of creation of new particles is modelled by a fertility function (of the distance to the fertile site), multiplied by a fertility rate. If the initial conditions correspond to a single RTP with even probability density, the system is parity-invariant. The equations of motion can be solved in the Laplace domain, in terms of the density of right-movers at the origin. At large time, this density is shown to grow exponentially, at a rate that depends only on the fertility function and fertility rate. Moreover, the total density of RTPs (divided by the density of right-movers at the origin), reaches a stationary state that does not depend on the initial conditions, and presents a local minimum at the fertile site.
Run-and-tumble motion is an example of active motility where particles move at constant speed and change direction at random times. In this work we study run-and-tumble motion with diffusion in a harmonic potential in one dimension via a path integral approach. We derive a Doi-Peliti field theory and use it to calculate the entropy production and other observables in closed form. All our results are exact.
Thermodynamic uncertainty relations have emerged as universal bounds on current fluctuations in non-equilibrium systems. Here we derive a new bound for a particular class of run-and-tumble type processes using the mathematical framework of renewal-reward theory which can be applied to both Markovian and non-Markovian systems. We demonstrate the results for selected single-particle models as well as a variant of the asymmetric simple exclusion process with collective tumbles. Our bound is relatively tight for a broad parameter regime and only requires knowledge of the statistics of run lengths …
We analyze the case of run-and-tumble particles pushed through a rugged channel both in the continuum and on the lattice. The current characteristic is non-monotone in the external field with (1) the appearance of a current and nontrivial density profile even at zero field for asymmetric obstacles, (2) the current decreasing with persistence at small field and increasing with persistence at large field. Activity in terms of self-propulsion increases the maximal current and postpones dying. We give an effective theoretical description with wider validity.
We study the statistical properties of the convex hull of a planar run-and-tumble particle (RTP), also known as the "persistent random walk", where the particle/walker runs ballistically between tumble events at which it changes its direction randomly. We consider two different statistical ensembles where we either fix (i) the total number of tumblings $n$ or (ii) the total duration $t$ of the time interval. In both cases, we derive exact expressions for the average perimeter of the convex hull and then compare to numerical estimates finding excellent agreement. Further, we numerically compute…
Planar run-and-tumble walks with orthogonal directions of motion are considered. After formulating the problem with generic transition probabilities among the orientational states, we focus on the symmetric case, giving general expressions of the probability distribution function (in the Laplace-Fourier domain), the mean square displacement and the effective diffusion constant in terms of transition rate parameters. As case studies we treat and discuss two classes of motion, alternate/forward and isotropic/backward, obtaining, when possible, analytic expressions of probability distribution fun…
The motion of a tagged degree of freedom can give important insight in the interactions present in a complex environment. We investigate the dynamics of a tagged particle in two non-equilibrium systems that consist of interacting run-and-tumble particles. The first one is an exactly solvable polymer model, the second is a two-dimensional lattice model, which is studied through simulations. We find that in both cases a tagged particle shows anomalous dynamics and non-Gaussian behaviour for times below the persistence time of the run-and-tumble motion. For later times, the dynamics of the tagged…
Active Brownian particles (ABPs, such as self-phoretic colloids) swim at fixed speed $v$ along a body-axis ${\bf u}$ that rotates by slow angular diffusion. Run-and-tumble particles (RTPs, such as motile bacteria) swim with constant $\u$ until a random tumble event suddenly decorrelates the orientation. We show that when the motility parameters depend on density $ρ$ but not on ${\bf u}$, the coarse-grained fluctuating hydrodynamics of interacting ABPs and RTPs can be mapped onto each other and are thus strictly equivalent. In both cases, a steeply enough decreasing $v(ρ)$ causes phase separati…
We calculate the large deviation function of the end-to-end distance and the corresponding extension-versus-force relation for (isotropic) random walks, on and off-lattice, with and without persistence, and in any spatial dimension. For off-lattice random walks with persistence, the large deviation function undergoes a first order phase transition in dimension $d> 5$. In the corresponding force-versus-extension relation, the extension becomes independent of the force beyond a critical value. The transition is anticipated in dimensions $d=4$ and $d=5$, where full extension is reached at a finit…
We investigate the statistics of the convex hull for a single run-and-tumble particle in two dimensions. Run-and-tumble particle, also known as persistent random walker, has gained significant interest in the recent years due to its biological application in modelling the motion of bacteria. We consider two different statistical ensembles depending on whether (i) the total number of tumbles $n$ or (ii) the total observation time $t$ is kept fixed. Benchmarking the results on perimeter, we study the statistical properties of the area of the convex hull for RTP. Exploiting the connections to ext…
We study a set of Run-and-tumble particle (RTP) dynamics in two spatial dimensions. In the first case of the orientation θ of the particle can assume a set of n possible discrete values while in the second case θ is a continuous variable. We calculate exactly the marginal position distributions for n = 3,4 and the continuous case and show that in all the cases the RTP shows a cross-over from a ballistic to diffusive regime. The ballistic regime is a typical signature of the active nature of the systems and is characterized by non-trivial position distributions which depends on the specific mod…
Microscopic thermal machines that are of the dimensions of around few hundred nanometers have been the subject of intense study over the last two decades. Recently, it has been shown that the efficiency of such thermal engines can be enhanced by using active Ornstein-Uhlenbeck particles (AOUP). In this work, we numerically study the behaviour of tiny engines and refrigerators that use an active run-and-tumble particle (RTP) as the working system. We find that the results for the engine mode are in sharp contrast with those of engines using AOUP, thus showing that the nature of activity has a s…
Run-and-Tumble particles, mimicking the behaviour of microorganisms like E. coli, are a paradigmatic model of active matter. Due to self-propulsion, their random and undirected motion can be rectified in a ratchet potential. Using perturbative field theory, we determine the shape of the potential that produces the maximum particle current as a function of the particles' parameters.
We study the transport of self-propelled particles in dynamic complex environments. To obtain exact results, we introduce a model of run-and-tumble particles (RTPs) moving in discrete time on a $d$-dimensional cubic lattice in the presence of diffusing hard core obstacles. We derive an explicit expression for the diffusivity of the RTP, which is exact in the limit of low density of fixed obstacles. To do so, we introduce a generalization of Kac's theorem on the mean return times of Markov processes, which we expect to be relevant for a large class of lattice gas problems. Our results show the …
We study a simple run-and-tumble random walk whose switching frequency from run mode to tumble mode and the reverse depend on a stochastic signal. We consider a particularly sharp, step-like dependence, where the run to tumble switching probability jumps from zero to one as the signal crosses a particular value (say y_1 ) from below. Similarly, tumble to run switching probability also shows a jump like this as the signal crosses another value (y_2 < y_1 ) from above. We are interested in characterizing the effect of signaling noise on the long time behavior of the random walker. We consider tw…
We study a model of bacterial dynamics where two interacting random walkers perform run-and-tumble motion on a one-dimensional lattice under mutual exclusion and find an exact expression for the probability distribution in the steady state. This stationary distribution has a rich structure comprising three components: a jammed component, where the particles are adjacent and block each other; an attractive component, where the probability distribution for the distance between particles decays exponentially; and an extended component in which the distance between particles is uniformly distribut…
Out-of-equilibrium systems continuously generate entropy, with its rate of production being a fingerprint of non-equilibrium conditions. In small-scale dissipative systems subject to thermal noise, fluctuations of entropy production are significant. Hitherto, mean and variance have been abundantly studied, even if higher moments might be important to fully characterize the system of interest. Here, we introduce a graphical method to compute any moment of entropy production for a generic discrete-state system. Then, we focus on a paradigmatic model of active particles, i.e., run-and-tumble dyna…
We compare the fluctuations in the velocity and in the fraction of time spent at a given position for minimal models of a passive and an active particle: an asymmetric random walker and a run-and-tumble particle in continuous time and on a 1D lattice. We compute rate functions and effective dynamics conditioned on large deviations for these observables. While generally different, for a unique and non-trivial choice of rates (up to a rescaling of time) the velocity rate functions for the two models become identical, whereas the effective processes generating the fluctuations remain distinct. Th…
We investigate the motion of a run-and-tumble particle (RTP) in one dimension. We find the exact probability distribution of the particle with and without diffusion on the infinite line, as well as in a finite interval. In the infinite domain, this probability distribution approaches a Gaussian form in the long-time limit, as in the case of a regular Brownian particle. At intermediate times, this distribution exhibits unexpected multi-modal forms. In a finite domain, the probability distribution reaches a steady state form with peaks at the boundaries, in contrast to a Brownian particle. We al…
We study a minimal model of self-propelled particle in a crowded single-file environment. We extend classical models of exclusion processes (previously analyzed for diffusive and driven tracer particles) to the case where the 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 $\mathcal{P}_n(X)$ of the RTP position $X$ and all its cumulants, valid for arbitrary values of the tumbling probability $α$ and time $n$. Our results highlight st…
We study a model of interacting run-and-tumble random walkers operating under mutual hardcore exclusion on a one-dimensional lattice with periodic boundary conditions. We incorporate a finite, Poisson-distributed, tumble duration so that a particle remains stationary whilst tumbling, thus generalising the persistent random walker model. We present the exact solution for the nonequilibrium stationary state of this system in the case of two random walkers. We find this to be characterised by two lengthscales, one arising from the jamming of approaching particles, and the other from one particle …
We consider a particle undergoing run and tumble dynamics, in which its velocity stochastically reverses, in one dimension. We study the addition of a Poissonian resetting process occurring with rate $r$. At a reset event the particle's position is returned to the resetting site $X_r$ and the particle's velocity is reversed with probability $η$. The case $η= 1/2$ corresponds to position resetting and velocity randomization whereas $η=0$ corresponds to position-only resetting. We show that, beginning from symmetric initial conditions, the stationary state does not depend on $η$ i.e. it is indep…
Run-and-tumble dynamics is a wide-spread mechanism of swimming bacteria. The accumulation of run-and-tumble microswimmers near impermeable surfaces is studied theoretically and numerically in the low-density limit in two and three spatial dimensions. Both uni-modal and exponential distributions of the run lengths are considered. Constant run lengths lead to {peaks and depletions regions} in the density distribution of particles near the surface, in contrast to {exponentially-distributed run lengths}. Finally, we present a universal accumulation law for large channel widths, which applies not o…
We study the probability distribution $P(X_N=X,N)$ of the total displacement $X_N$ of an $N$-step run and tumble particle on a line, in presence of a constant nonzero drive $E$. While the central limit theorem predicts a standard Gaussian form for $P(X,N)$ near its peak, we show that for large positive and negative $X$, the distribution exhibits anomalous large deviation forms. For large positive $X$, the associated rate function is nonanalytic at a critical value of the scaled distance from the peak where its first derivative is discontinuous. This signals a first-order dynamical phase transi…
Absorption problems of run-and-tumble particles, described by the telegrapher's equation, are analyzed in one space dimension considering partially reflecting boundaries. Exact expressions for the probability distribution function in the Laplace domain and for the mean time to absorption are given, discussing some interesting limits (Brownian and wave limit, large volume limit) and different case studies (semi-infinite segment, equal and symmetric boundaries, totally/partially reflecting boundaries).
We solve the problem of first-passage time for run-and-tumble particles in one dimension. Exact expression is derived for the mean first-passage time in the general case, considering external force-fields and chemotactic-fields, giving rise to space dependent swim-speed and tumble rate. Agreement between theoretical formulae and numerical simulations is obtained in the analyzed case studies -- constant and sinusoidal force fields, constant gradient chemotactic field. Reported findings can be useful to get insights into very different phenomena involving active particles, such as bacterial moti…
This work considers an extension of the Kuramoto model with run-and-tumble dynamics -- a type of self-propelled motion. The difference between the extended and the original model is that in the extended version angular velocity of individual particles is no longer fixed but can change sporadically with a new velocity drawn from a distribution g(w). Because the Kuramoto model undergoes a phase transition, it offers a simple case study for investigating phase transition for a system with self-propelled particles.
We present an analysis of the stationary distributions of run-and-tumble particles trapped in external potentials in terms of a thermophoretic potential, that emerges when trapped active motion is mapped to trapped passive Brownian motion in a fictitious inhomogeneous thermal bath. We elaborate on the meaning of the non-Boltzmann-Gibbs stationary distributions that emerge as a consequence of the persistent motion of active particles. These stationary distributions are interpreted as a class of distributions in nonequilibrium statistical mechanics known as superstatistics. Our analysis provides…
In this paper, Doi-Peliti field theory is used to describe the motion of free Run and Tumble particles in arbitrary dimensions. After deriving action and propagators, the mean square displacement and the corresponding entropy production at stationarity are calculated in this framework. We further derive the field theory of free Active Brownian Particles in two dimensions for comparison.
We consider an active run-and-tumble particle (RTP) in $d$ dimensions, starting from the origin and evolving over a time interval $[0,t]$. We examine three different models for the dynamics of the RTP: the standard RTP model with instantaneous tumblings, a variant with instantaneous runs and a general model in which both the tumblings and the runs are non-instantaneous. For each of these models, we use the Sparre Andersen theorem for discrete-time random walks to compute exactly the probability that the $x$ component does not change sign up to time $t$, showing that it does not depend on $d$. …
We consider self-propelled particles undergoing run-and-tumble dynamics (as exhibited by E. coli) in one dimension. Building on previous analyses at drift-diffusion level for the one-particle density, we add both interactions and noise, enabling discussion of domain formation by "self-trapping", and other collective phenomena. Mapping onto detailed-balance systems is possible in certain cases.
The persistent character of the motion of active particles gives rise to accumulation at boundaries. I investigate the problem of run-and-tumble swimmers confined in a 1D box with hard walls, reporting expressions for the particles probability distribution and wall pressure. A crossover box length value is found below which the initial value of the pressure turns out to be higher than the asymptotic one, indicating a bounce effect of the active "wave" of swimmers. The case of attracting and repelling boundaries are also investigated using two different tumble rates for particles in the bulk an…
We use a first-passage time approach to study the statistics of the trapping times induced by persistent motion of active particles colliding with flat boundaries. The angular first-passage time distribution and mean first-passage time is calculated exactly for active Brownian and run-and-tumble particles and the results are compared. Theoretical predictions are in excellent agreement with Langevin simulations. Our results shed further light onto how active particles with different dynamics may be equivalent in the bulk, yet behave differently near boundaries or obstacles.
We investigate the local time $(T_{loc})$ statistics for a run and tumble particle in an one dimensional inhomogeneous medium. The inhomogeneity is introduced by considering the position dependent rate of the form $R(x) = γ\frac{|x|^α}{l^α}$ with $α\geq 0$. For $α=0$, we derive the probability distribution of $T_{loc}$ exactly which is expressed as a series of $δ$-functions in which the coefficients can be interpreted as the probability of multiple revisits of the particle to the origin starting from the origin. For general $α$, we show that the typical fluctuations of $T_{loc}$ scale with tim…
Run-and-tumble particles, frequently considered today for modeling bacterial locomotion, naturally appear outside a biological context as well, e.g. for producing waves in the telegraph process. Here, we use a wave function to drive their propulsion and tumbling. Such quantum-active motion realizes a jittery motion of Dirac electrons (as in the famous Zitterbewegung): the Dirac electron is a run-and-tumble particle, where the tumbling is between chiralities. We visualize the trajectories in diffraction and double slit experiments for electrons. In particular, that yields the time-of-arrival st…
{\it E. coli} bacteria swim in straight runs interrupted by sudden reorientation events called tumbles. The resulting random walks give rise to density fluctuations that can be derived analytically in the limit of non interacting particles or equivalently of very low concentrations. However, in situations of practical interest, the concentration of bacteria is always large enough to make interactions an important factor. Using molecular dynamics simulations, we study the dynamic structure factor of a model bacterial bath for increasing values of densities. We show that it is possible to reprod…
We study the positive occupation time of a run-and-tumble particle (RTP) subject to stochastic resetting. Under the resetting protocol, the position of the particle is reset to the origin at a random sequence of times that is generated by a Poisson process with rate $r$. The velocity state is reset to $\pm v$ with fixed probabilities $ρ_1$ and $ρ_{-1}=1-ρ_1$, where $v$ is the speed. We exploit the fact that the moment generating functions with and without resetting are related by a renewal equation, and the latter generating function can be calculated by solving a corresponding Feynman-Kac equ…
We study the long-time asymptotic behavior of the position distribution of a run-and-tumble particle (RTP) in two dimensions and show that the distribution at a time $t$ can be expressed as a perturbative series in $(γt)^{-1}$, where $γ^{-1}$ is the persistence time of the RTP. We show that the higher order corrections to the leading order Gaussian distribution generically satisfy an inhomogeneous diffusion equation where the source term depends on the previous order solutions. The explicit solution of the inhomogeneous equation requires the position moments, and we develop a recursive formali…
The 'Arcsine' laws of Brownian particles in one dimension describe distributions of three quantities: the time $t_m$ to reach maximum position, the time $t_r$ spent on the positive side and the time $t_\ell$ of the last visit to the origin. Interestingly, the cumulative distribution of all the three quantities are same and given by Arcsine function. In this paper, we study distribution of these three times $t_m,~t_r$ and $t_\ell$ in the context of single run-and-tumble particle in one dimension, which is a simple non-Markovian process. We compute exact distributions of these three quantities f…
We study the large scale behavior of a collection of hard core run and tumble particles on a one dimensional lattice with periodic boundary conditions. Each particle has persistent motion in one direction decided by an associated spin variable until the direction of spin is reversed. We map the run and tumble model to a mass transfer model with fluctuating directed bonds. We calculate the steady state single site mass distribution in the mass model within a mean field approximation for larger spin-flip rates and by analyzing an appropriate coalescence fragmentation model for small spin-flip ra…
A one-dimensional run-and-tumble particle (RTP) switches randomly between a left and right moving state of constant speed $v$. This type of motion arises in a wide range of applications in cell biology, including the unbiased growth and shrinkage of microtubules or cytonemes, the bidirectional motion of molecular motors, and the "run-and-tumble" motion of bacteria such as {\em E. coli}. RTPs are also of more general interest within the non-equilibrium statistical physics community, both at the single particle level and at the interacting population level, where it provides a simple example of …
We simulate by lattice Boltzmann the nonequilibrium steady states of run-and-tumble particles (inspired by a minimal model of bacteria), interacting by far-field hydrodynamics, subject to confinement. Under gravity, hydrodynamic interactions barely perturb the steady state found without them, but for particles in a harmonic trap such a state is quite changed if the run length is larger than the confinement length: a self-assembled pump is formed. Particles likewise confined in a narrow channel show a generic upstream flux in Poiseuille flow: chiral swimming is not required.
Exact solutions of interacting random walk models, such as 1D lattice gases, offer precise insight into the origin of nonequilibrium phenomena. Here, we study a model of run-and-tumble particles on a ring lattice interacting via hardcore exclusion. We present the exact solution for one and two particles using a generating function technique. For two particles, the eigenvectors and eigenvalues are explicitly expressed using two parameters reminiscent of Bethe roots, whose numerical values are determined by polynomial equations which we derive. The spectrum depends in a complicated way on the ra…
