In this paper we extend the encounter-based model of diffusion-mediated surface absorption to the case of an unbiased run-and-tumble particle (RTP) confined to a finite interval $[0,L]$ and switching between two constant velocity states $\pm v$ at a rate $α$. The encounter-based formalism is motivated by the observation that various surface-based reactions are better modeled in terms of a reactivity that is a function of the amount of time that a particle spends in a neighborhood of an absorbing surface, which is specified by a functional known as the boundary local time. The effects of surface reactions are taken into account by identifying the first passage time (FPT) for absorption with the event that the local time crosses some random threshold $\widehat{\ell}$. In the case of a Brownian particle, the local time $\ell(t)$ is a continuous non-decreasing function of the time $t$. Taking $Ψ(\ell)\equiv ¶[\widehat{\ell}>\ell]$ to be an exponential distribution, $Ψ[\ell]=\e^{-κ_0\ell}$, is equivalent to imposing a Robin boundary condition with a constant rate of absorption $κ_0$. One major difference in the encounter-based model of an RTP is that the boundary local time $\ell(t)$ is a now a discrete random variable that counts the number of collisions of the RTP with the boundary. Given this modification, we show that in the case of a geometric distribution $Ψ(\ell)=z^{\ell}$, $z=1/(1+κ_0/v)$, we recover the RTP analog of the Robin boundary condition. This allows us to solve the boundary value problem (BVP) for the joint probability density for particle position and the local time, and thus incorporate more general models of absorption based on non-geometric distributions $Ψ(\ell)$.
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 …
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 …
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 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 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.
This paper introduces a run-and-tumble model with self-reinforcing directionality and rests. We derive a single governing hyperbolic partial differential equation for the probability density of random walk position, from which we obtain the second moment in the long time limit. We find the criteria for the transition between superdiffusion and diffusion caused by the addition of a rest state. The emergence of superdiffusion depends on both the parameter representing the strength of self-reinforcement and the ratio between mean running and resting times. The mean running time must be at least $…
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 …
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.
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).
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…
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 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…
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 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…
The run-and-tumble walk, consisting in randomly reoriented ballistic excursions, models phenomena ranging from gas kinetics to bacteria motility. We evaluate the mean time required for this walk to find a fixed target within a 2D or 3D spherical confinement. We find that the mean search time admits a minimum as a function of the mean run duration for various types of boundary conditions and run duration distributions (exponential, power-law, deterministic). Our result stands in sharp contrast to the pure ballistic motion, which is predicted to be the optimal search strategy in the case of Pois…
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…
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…
We study the stochastic dynamics of a particle with two distinct motility states. Each one is characterized by two parameters: one represents the average speed and the other represents the persistence quantifying the tendency to maintain the current direction of motion. We consider a run-and-tumble process, which is a combination of an active fast motility mode (persistent motion) and a passive slow mode (diffusion). Assuming stochastic transitions between the two motility states, we derive an analytical expression for the time evolution of the mean square displacement. The interplay of the ke…
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 …
We propose a method to exactly generate bridge run-and-tumble trajectories that are constrained to start at the origin with a given velocity and to return to the origin after a fixed time with another given velocity. The method extends the concept of effective Langevin equations, valid for Markovian stochastic processes such as Brownian motion, to a non-Markovian stochastic process driven by a telegraphic noise, with exponentially decaying correlations. We obtain effective space-time dependent tumbling rates that implicitly accounts for the bridge constraint. We extend the method to other type…
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 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 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…
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…
In our article we present a computational model for the simulation of self-propelled anisotropic bacteria. To this end we use a self-propelled particle model and augment it with a statistical algorithm for the run-and-tumble motion. We derive an equation for the distribution of reorientations of the bacteria that we use to analyze the statistics of the random walk and that allows us to tune the behavior of our model to the characteristics of an E.coli bacterium. We validate our implementation in terms of a single swimmer and demonstrate that our model is capable of reproducing E.coli's run-and…
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…
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…
