Current fluctuations in noninteracting run-and-tumble particles in one dimension

Occupation time of a system of Brownian particles on the line with steplike initial condition

We consider a system of noninteracting Brownian particles on the line with steplike initial condition and study the statistics of the occupation time on the positive half-line. We demonstrate that even at large times, the behavior of the occupation time exhibits long-lasting memory effects of the initialization. Specifically, we calculate the mean and the variance of the occupation time, demonstrating that the memory effects in the variance are determined by a generalized compressibility (or Fano factor), associated with the initial condition. In the particular case of the uncorrelated uniform initial condition we conduct a detailed study of two probability distributions of the occupation time: annealed (averaged over all possible initial configurations) and quenched (for a typical configuration). We show that at large times both the annealed and the quenched distributions admit large deviation form and we compute analytically the associated rate functions. We verify our analytical predictions via numerical simulations using importance sampling Monte Carlo strategy.

Time-dependent properties of run-and-tumble particles. II. Current fluctuations

We investigate steady-state current fluctuations in two models of hardcore run-and-tumble particles (RTPs) on a periodic one-dimensional lattice of L sites, for arbitrary tumbling rate γ=τ_{p}^{-1} and density ρ; model I consists of standard hardcore RTPs, while model II is an analytically tractable variant of model I, called a long-ranged lattice gas (LLG). We show that, in the limit of L large, the fluctuation of cumulative current Q_{i}(T,L) across the ith bond in a time interval T≫1/D grows first subdiffusively and then diffusively (linearly) with T: 〈Q_{i}^{2}〉∼T^{α} with α=1/2 for 1/D≪T≪L^{2}/D and α=1 for T≫L^{2}/D, where D(ρ,γ) is the collective- or bulk-diffusion coefficient; at small times T≪1/D, exponent α depends on the details. Remarkably, regardless of the model details, the scaled bond-current fluctuations D〈Q_{i}^{2}(T,L)〉/2χL≡W(y) as a function of scaled variable y=DT/L^{2} collapse onto a universal scaling curve W(y), where χ(ρ,γ) is the collective particle mobility. In the limit of small density and tumbling rate, ρ,γ→0, with ψ=ρ/γ fixed, there exists a scaling law: The scaled mobility γ^{a}χ(ρ,γ)/χ^{(0)}≡H(ψ) as a function of ψ collapses onto a scaling curve H(ψ), where a=1 and 2 in models I and II, respectively, and χ^{(0)} is the mobility in the limiting case of a symmetric simple exclusion process; notably, the scaling function H(ψ) is model dependent. For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, W(y) and H(ψ). We also calculate spatial correlation functions for the current and compare our theory with simulation results of model I; for both models, the correlation functions decay exponentially, with correlation length ξ∼τ_{p}^{1/2} diverging with persistence time τ_{p}≫1. Overall, our theory is in excellent agreement with simulations and complements the prior findings [T. Chakraborty and P. Pradhan, Phys. Rev. E 109, 024124 (2024)1539-375510.1103/PhysRevE.109.024124].

From Particle Currents to Tracer Diffusion: Universal Correlation Profiles in Single-File Dynamics

Single-file transport refers to the motion of particles in a narrow channel, such that they cannot bypass each other. This constraint leads to strong correlations between the particles, described by correlation profiles, which measure the correlation between a generic observable and the density of particles at a given position and time. They have recently been shown to play a central role in single-file systems. Up to now, these correlations have only been determined for diffusive systems in the hydrodynamic limit. Here, we consider a model of reflecting point particles on the infinite line, with a general individual stochastic dynamics. We show that the correlation profiles take a simple universal form, at arbitrary time. We illustrate our approach by the study of the integrated current of particles through the origin, and apply our results to representative models such as Brownian particles, run-and-tumble particles and Lévy flights. We further emphasise the generality of our results by showing that they also apply beyond the 1D case, and to other observables.

Local time of a system of Brownian particles on the line with steplike initial condition

We consider a system of noninteracting Brownian particles on a line with a steplike initial condition, and we investigate the behavior of the local time at the origin at large times. We compute the mean and the variance of the local time, and we show that the memory effects are governed by the Fano factor associated with the initial condition. For the uniform initial condition, we show that the probability distribution of the local time admits a large deviation form, and we compute the corresponding large deviation functions for the annealed and quenched averaging schemes. The two resulting large deviation functions are very different. Our analytical results are supported by extensive numerical simulations.

Generalized disorder averages and current fluctuations in run and tumble particles

We present exact results for the fluctuations in the number of particles crossing the origin up to time t in a collection of noninteracting run and tumble particles in one dimension. In contrast to passive systems, such active particles are endowed with two inherent degrees of freedom, positions and velocities, which can be used to construct density and magnetization fields. We introduce generalized disorder averages associated with both these fields and perform annealed and quenched averages over various initial conditions. We show that the variance σ^{2} of the current in annealed versus quenched magnetization situations exhibits a surprising difference at short times, σ^{2}∼t vs σ^{2}∼t^{2}, respectively, with a sqrt[t] behavior emerging at large times. Our analytical results demonstrate that in the strictly quenched scenario, where both the density and magnetization fields are initially frozen, the fluctuations in the current are strongly suppressed. Importantly, these anomalous fluctuations cannot be obtained solely by freezing the density field.

Work Fluctuations for a Harmonically Confined Active Ornstein-Uhlenbeck Particle

We study the active work fluctuations of an active Ornstein-Uhlenbeck particle in the presence of a confining harmonic potential. We tackle the problem analytically both for stationary and generic uncorrelated initial states. Our results show that harmonic confinement can induce singularities in the active work rate function, with linear stretches at large positive and negative active work, at sufficiently large active and harmonic force constants. These singularities originate from big jumps in the displacement and in the active force, occurring at the initial or ending points of trajectories and marking the relevance of boundary terms in this problem.

Current fluctuations in stochastically resetting particle systems

We consider a system of noninteracting particles on a line with initial positions distributed uniformly with density ρ on the negative half-line. We consider two different models: (i) Each particle performs independent Brownian motion with stochastic resetting to its initial position with rate r and (ii) each particle performs run-and-tumble motion, and with rate r its position gets reset to its initial value and simultaneously its velocity gets randomized. We study the effects of resetting on the distribution P(Q,t) of the integrated particle current Q up to time t through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution P_{an}(Q,t) approaches its stationary limit uniformly for all Q. In contrast, the quenched distribution P_{qu}(Q,t) attains its stationary form for QQ_{crit}(t). We show that Q_{crit}(t) increases linearly with t for large t. On the scale where Q∼Q_{crit}(t), we show that P_{qu}(Q,t) has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as 10^{-14000}, we were able to compute numerically this nonanalytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction.

Role of initial conditions in one-dimensional diffusive systems: Compressibility, hyperuniformity, and long-term memory

We analyze the long-lasting effects of initial conditions on dynamical fluctuations in one-dimensional diffusive systems. We consider the mean-squared displacement of tracers in homogeneous systems with single-file diffusion, and current fluctuations for noninteracting diffusive particles. In each case we show analytically that the long-term memory of initial conditions is mediated by a single static quantity: a generalized compressibility that quantifies the density fluctuations of the initial state. We thereby identify a universality class of hyperuniform initial states whose dynamical variances coincide with the quenched cases studied previously, alongside a continuous family of other classes among which equilibrated (or annealed) initial conditions are but one member. We verify our predictions through extensive Monte Carlo simulations.

Exact position distribution of a harmonically confined run-and-tumble particle in two dimensions

We consider an overdamped run-and-tumble particle in two dimensions, with self-propulsion in an orientation that stochastically rotates by 90^{∘} at a constant rate, clockwise or counterclockwise with equal probabilities. In addition, the particle is confined by an external harmonic potential of stiffness μ, and possibly diffuses. We find the exact time-dependent distribution P(x,y,t) of the particle's position, and in particular, the steady-state distribution P_{st}(x,y) that is reached in the long-time limit. We also find P(x,y,t) for a "free" particle, μ=0. We achieve this by showing that, under a proper change of coordinates, the problem decomposes into two statistically independent one-dimensional problems, whose exact solution has recently been obtained. We then extend these results in several directions, to two such run-and-tumble particles with a harmonic interaction, to analogous systems of dimension three or higher, and by allowing stochastic resetting.

Active random walks in one and two dimensions

We investigate active lattice walks: biased continuous time random walks which perform orientational diffusion between lattice directions in one and two spatial dimensions. We study the occupation probability of an arbitrary site on the lattice in one and two dimensions and derive exact results in the continuum limit. Next, we compute the large deviation free-energy function in both one and two dimensions, which we use to compute the moments and the cumulants of the displacements exactly at late times. Our exact results demonstrate that the cross-correlations between the motion in the x and y directions in two dimensions persist in the large deviation function. We also demonstrate that the large deviation function of an active particle with diffusion displays two regimes, with differing diffusive behaviors. We verify our analytic results with kinetic Monte Carlo simulations of an active lattice walker in one and two dimensions.

Exact closure and solution for spatial correlations in single-file diffusion

In single-file transport particles diffuse in narrow channels while not overtaking each other. it is a fundamental model for the tracer subdiffusion observed in confined systems, such as zeolites or carbon nanotubes. This anomalous behavior originates from strong bath-tracer correlations in one dimension. Despite extensive effort, these remained elusive, because they involve an infinite hierarchy of equations. For the symmetric exclusion process, a paradigmatic model of single-file diffusion, we break the hierarchy to unveil and solve a closed exact equation satisfied by these correlations. Beyond quantifying the correlations, the role of this key equation as a tool for interacting particle systems is further demonstrated by its application to out-of-equilibrium situations, other observables, and other representative single-file systems.

Condensation transition in the late-time position of a run-and-tumble particle

We study the position distribution P(R[over ⃗],N) of a run-and-tumble particle (RTP) in arbitrary dimension d, after N runs. We assume that the constant speed v>0 of the particle during each running phase is independently drawn from a probability distribution W(v) and that the direction of the particle is chosen isotropically after each tumbling. The position distribution is clearly isotropic, P(R[over ⃗],N)→P(R,N) where R=|R[over ⃗]|. We show that, under certain conditions on d and W(v) and for large N, a condensation transition occurs at some critical value of R=R_{c}∼O(N) located in the large-deviation regime of P(R,N). For R<R_{c} (subcritical fluid phase), all runs are roughly of the same size in a typical trajectory. In contrast, an RTP trajectory with R>R_{c} is typically dominated by a "condensate," i.e., a large single run that subsumes a finite fraction of the total displacement (supercritical condensed phase). Focusing on the family of speed distributions W(v)=α(1-v/v_{0})^{α-1}/v_{0}, parametrized by α>0, we show that, for large N, P(R,N)∼exp[-Nψ_{d,α}(R/N)], and we compute exactly the rate function ψ_{d,α}(z) for any d and α. We show that the transition manifests itself as a singularity of this rate function at R=R_{c} and that its order depends continuously on d and α. We also compute the distribution of the condensate size for R>R_{c}. Finally, we study the model when the total duration T of the RTP, instead of the total number of runs, is fixed. Our analytical predictions are confirmed by numerical simulations, performed using a constrained Markov chain Monte Carlo technique, with precision ∼10^{-100}.

Local time for run and tumble particle

We investigate the local time T_{loc} statistics for a run and tumble particle (RTP) in one dimension, which is the quintessential model for the motion of bacteria. In random walk literature, the RTP dynamics is studied as the persistent Brownian motion. We consider the inhomogeneous version of this model where the inhomogeneity is introduced by considering the position-dependent rate of the form R(x)=γ|x|^{α}/l^{α} with α≥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 RTP to the origin starting from the origin. For general α, we show that the typical fluctuations of T_{loc} scale with time as T_{loc}∼t^{1+α/2+α} for large t and their probability distribution possesses a scaling behavior described by a scaling function which we have computed analytically. Second, we study the statistics of T_{loc} until the RTP makes a first passage to x=M(>0). In this case, we also show that the probability distribution can be expressed as a series sum of δ functions for all values of α(≥0) with coefficients originating from appropriate exit problems. All our analytical findings are supported with numerical simulations.

Position distribution in a generalized run-and-tumble process

We study a class of stochastic processes of the type d^{n}x/dt^{n}=v_{0}σ(t) where n>0 is a positive integer and σ(t)=±1 represents an active telegraphic noise that flips from one state to the other with a constant rate γ. For n=1, it reduces to the standard run-and-tumble process for active particles in one dimension. This process can be analytically continued to any n>0, including noninteger values. We compute exactly the mean-squared displacement at time t for all n>0 and show that at late times while it grows as ∼t^{2n-1} for n>1/2, it approaches a constant for n<1/2. In the marginal case n=1/2, it grows very slowly with time as ∼lnt. Thus, the process undergoes a localization transition at n=1/2. We also show that the position distribution p_{n}(x,t) remains time-dependent even at late times for n≥1/2, but approaches a stationary time-independent form for n<1/2. The tails of the position distribution at late times exhibit a large deviation form, p_{n}(x,t)∼exp[-γtΦ_{n}(x/x^{*}(t))], where x^{*}(t)=v_{0}t^{n}/Γ(n+1). We compute the rate function Φ_{n}(z) analytically for all n>0 and also numerically using importance sampling methods, finding excellent agreement between them. For three special values n=1, n=2, and n=1/2 we compute the exact cumulant-generating function of the position distribution at all times t.

Universal properties of a run-and-tumble particle in arbitrary dimension

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 noninstantaneous. 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. As a consequence of this result, we compute exactly other x-component properties, namely, the distribution of the time of the maximum and the record statistics, showing that they are universal, i.e., they do not depend on d. Moreover, we show that these universal results hold also if the speed v of the particle after each tumbling is random, drawn from a generic probability distribution. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 124, 090603 (2020)10.1103/PhysRevLett.124.090603].