Long-time position distribution of an active Brownian particle in two dimensions

Importance sampling for counting statistics in one-dimensional systems

In this paper, we consider the problem of numerical investigation of the counting statistics for a class of one-dimensional systems. Importance sampling, the cornerstone technique usually implemented for such problems, critically hinges on selecting an appropriate biased distribution. While an exponential tilt in the observable stands as the conventional choice for various problems, its efficiency in the context of counting statistics may be significantly hindered by the genuine discreteness of the observable. To address this challenge, we propose an alternative strategy, which we call importance sampling with the local tilt. We demonstrate the efficiency of the proposed approach through the analysis of three prototypical examples: a set of independent Gaussian random variables, Dyson gas, and symmetric simple exclusion process with a steplike initial condition.

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.

Large deviations in statistics of the convex hull of passive and active particles: A theoretical study

We investigate analytically the distribution tails of the area A and perimeter L of a convex hull for different types of planar random walks. For N noninteracting Brownian motions of duration T we find that the large-L and -A tails behave as P(L)∼e^{-b_{N}L^{2}/DT} and P(A)∼e^{-c_{N}A/DT}, while the small-L and -A tails behave as P(L)∼e^{-d_{N}DT/L^{2}} and P(A)∼e^{-e_{N}DT/A}, where D is the diffusion coefficient. We calculated all of the coefficients (b_{N},c_{N},d_{N},e_{N}) exactly. Strikingly, we find that b_{N} and c_{N} are independent of N for N≥3 and N≥4, respectively. We find that the large-L (A) tails are dominated by a single, most probable realization that attains the desired L (A). The left tails are dominated by the survival probability of the particles inside a circle of appropriate size. For active particles and at long times, we find that large-L and -A tails are given by P(L)∼e^{-TΨ_{N}^{per}(L/T)} and P(A)∼e^{-TΨ_{N}^{area}(sqrt[A]/T)}, respectively. We calculate the rate functions Ψ_{N} exactly and find that they exhibit multiple singularities. We interpret these as DPTs of first order. We extended several of these results to dimensions d>2. Our analytic predictions display excellent agreement with existing results that were obtained from extensive numerical simulations.

Confined run-and-tumble particles with non-Markovian tumbling statistics

Confined active particles constitute simple, yet realistic, examples of systems that converge into a nonequilibrium steady state. We investigate a run-and-tumble particle in one spatial dimension, trapped by an external potential, with a given distribution g(t) of waiting times between tumbling events whose mean value is equal to τ. Unless g(t) is an exponential distribution (corresponding to a constant tumbling rate), the process is non-Markovian, which makes the analysis of the model particularly challenging. We use an analytical framework involving effective position-dependent tumbling rates to develop a numerical method that yields the full steady-state distribution (SSD) of the particle's position. The method is very efficient and requires modest computing resources, including in the large-deviation and/or small-τ regime, where the SSD can be related to the the large-deviation function, s(x), via the scaling relation P_{st}(x)∼e^{-s(x)/τ}.

Inertia suppresses signatures of activity of active Brownian particles in a harmonic potential

A harmonically trapped active Brownian particle exhibits two types of positional distributions-one has a single peak and the other has a single well-that signify steady-state dynamics with low and high activity, respectively. Adding inertia to the translational motion preserves this strict classification of either single-peak or single-well densities but shifts the dividing boundary between the states in the parameter space. We characterize this shift for the dynamics in one spatial dimension using the static Fokker-Planck equation for the full joint distribution of the state space. We derive local results analytically with a perturbation method for a small rotational velocity and then extend them globally with a numerical approach.

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.

Control of Active Brownian Particles: An Exact Solution

Control of stochastic systems is a challenging open problem in statistical physics, with a wealth of potential applications from biology to granulates. Unlike most cases investigated so far, we aim here at controlling a genuinely out-of-equilibrium system, the two dimensional active Brownian particles model in a harmonic potential, a paradigm for the study of self-propelled bacteria. We search for protocols for the driving parameters (stiffness of the potential and activity of the particles) bringing the system from an initial passivelike steady state to a final activelike one, within a chosen time interval. The exact analytical results found for this prototypical model of self-propelled particles brings control techniques to a wider class of out-of-equilibrium systems.

Chiral active matter in external potentials

We investigate the interplay between chirality and confinement induced by the presence of an external potential. For potentials having radial symmetry, the circular character of the trajectories induced by the chiral motion reduces the spatial fluctuations of the particle, thus providing an extra effective confining mechanism, that can be interpreted as a lowering of the effective temperature. In the case of non-radial potentials, for instance, with an elliptic shape, chirality displays a richer scenario. Indeed, the chirality can break the parity symmetry of the potential that is always fulfilled in the non-chiral system. The probability distribution displays a strong non-Maxwell-Boltzmann shape that emerges in cross-correlations between the two Cartesian components of the position, that vanishes in the absence of chirality or when radial symmetry of the potential is restored. These results are obtained by considering two popular models in active matter, i.e. chiral Active Brownian particles and chiral active Ornstein-Uhlenbeck particles.

Nonequilibrium steady state of trapped active particles

We consider an overdamped particle with a general physical mechanism that creates noisy active movement (e.g., a run-and-tumble particle or active Brownian particle, etc.), that is confined by an external potential. Focusing on the limit in which the correlation time τ of the active noise is small, we find the nonequilibrium steady-state distribution P_{st}(X) of the particle's position X. While typical fluctuations of X follow a Boltzmann distribution with an effective temperature that is not difficult to find, the tails of P_{st}(X) deviate from a Boltzmann behavior: In the limit τ→0, they scale as P_{st}(X)∼e^{-s(X)/τ}. We calculate the large-deviation function s(X) exactly for arbitrary trapping potential and active noise in dimension d=1, by relating it to the rate function that describes large deviations of the position of the same active particle in absence of an external potential at long times. We then extend our results to d>1 assuming rotational symmetry.

Stationary states of an active Brownian particle in a harmonic trap

We study the stationary states of an overdamped active Brownian particle (ABP) in a harmonic trap in two dimensions via mathematical calculations and numerical simulations. In addition to translational diffusion, the ABP self-propels with a certain velocity, whose magnitude is constant, but its direction is subject to Brownian rotation. In the limit where translational diffusion is negligible, the stationary distribution of the particle's position shows a transition between two different shapes, one with maximum and the other with minimum density at the center, as the trap stiffness is increased. We show that this nonintuitive behavior is captured by the relevant Fokker-Planck equation, which, under minimal assumptions, predicts a continuous phase transition-like change between the two different shapes. As the translational diffusion coefficient is increased, both these distributions converge into the equilibrium, Boltzmann form. Our simulations support the analytical predictions and also show that the probability distribution of the orientation angle of the self-propulsion velocity undergoes a transition from unimodal to bimodal forms in this limit. We also extend our simulations to a three-dimensional trap and find similar behavior.

Improving estimation of entropy production rate for run-and-tumble particle systems by high-order thermodynamic uncertainty relation

Entropy production plays an important role in the regulation and stability of active matter systems, and its rate quantifies the nonequilibrium nature of these systems. However, entropy production is hard to experimentally estimate even in some simple active systems like molecular motors or bacteria, which may be modeled by the run-and-tumble particle (RTP), a representative model in the study of active matters. Here we resolve this problem for an asymmetric RTP in one dimension, first constructing a finite-time thermodynamic uncertainty relation (TUR) for a RTP, which works well in the short observation time regime for entropy production estimation. Nevertheless, when the activity dominates, i.e., the RTP is far from equilibrium, the lower bound for entropy production from TUR turns out to be trivial. We address this issue by introducing a recently proposed high-order thermodynamic uncertainty relation (HTUR), in which the cumulant generating function of current serves as a key ingredient. To exploit the HTUR, we adopt a method to analytically obtain the cumulant generating function of the current we study, with no need to explicitly know the time-dependent probability distribution. The HTUR is demonstrated to be able to estimate the steady state energy dissipation rate accurately because the cumulant generating function covers higher-order statistics of the current, including rare and large fluctuations besides its variance. Compared to the conventional TUR, the HTUR could give significantly improved estimation of energy dissipation, which can work well even in the far from equilibrium regime. We also provide a strategy based on the improved bound to estimate the entropy production from a moderate amount of trajectory data for experimental feasibility.

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.

Analytic Solution of an Active Brownian Particle in a Harmonic Well

We provide an analytical solution for the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle trapped in an isotropic harmonic potential. Using the passive Brownian particle as basis states we show that the Fokker-Planck operator becomes lower diagonal, implying that the eigenvalues are unaffected by the activity. The propagator is then expressed as a combination of the equilibrium eigenstates with weights obeying exact iterative relations. We show that for the low-order correlation functions, such as the positional autocorrelation function, the recursion terminates at finite order in the Péclet number, allowing us to generate exact compact expressions and derive the velocity autocorrelation function and the time-dependent diffusion coefficient. The nonmonotonic behavior of latter quantities serves as a fingerprint of the nonequilibrium dynamics.

Self-propulsion with speed and orientation fluctuation: Exact computation of moments and dynamical bistabilities in displacement

We consider the influence of active speed fluctuations on the dynamics of a d-dimensional active Brownian particle performing a persistent stochastic motion. The speed fluctuation brings about a dynamical anisotropy even in the absence of shape anisotropy. We use the Laplace transform of the Fokker-Planck equation to obtain exact expressions for time-dependent dynamical moments. Our results agree with direct numerical simulations and show several dynamical crossovers determined by the activity, persistence, and speed fluctuation. The dynamical anisotropy leads to a subdiffusive scaling in the parallel component of displacement fluctuation at intermediate times. The kurtosis remains positive at short times determined by the speed fluctuation, crossing over to a negative minimum at intermediate times governed by the persistence before vanishing asymptotically. The probability distribution of particle displacement obtained from numerical simulations in two dimensions shows two crossovers between compact and extended trajectories via two bimodal distributions at intervening times. While the speed fluctuation dominates the first crossover, the second crossover is controlled by persistence like in the wormlike chain model of semiflexible polymers.

Microscopic Theory for the Diffusion of an Active Particle in a Crowded Environment

We calculate the diffusion coefficient of an active tracer in a schematic crowded environment, represented as a lattice gas of passive particles with hardcore interactions. Starting from the master equation of the problem, we put forward a closure approximation that goes beyond trivial mean field and provides the diffusion coefficient for an arbitrary density of crowders in the system. We show that our approximation is accurate for a very wide range of parameters, and that it correctly captures numerous nonequilibrium effects, which are the signature of the activity in the system. In addition to the determination of the diffusion coefficient of the tracer, our approach allows us to characterize the perturbation of the environment induced by the displacement of the active tracer. Finally, we consider the asymptotic regimes of low and high densities, in which the expression of the diffusion coefficient of the tracer becomes explicit, and which we argue to be exact.

Stationary nonequilibrium bound state of a pair of run and tumble particles

We study two interacting identical run-and-tumble particles (RTPs) in one dimension. Each particle is driven by a telegraphic noise and, in some cases, also subjected to a thermal white noise with a corresponding diffusion constant D. We are interested in the stationary bound state formed by the two RTPs in the presence of a mutual attractive interaction. The distribution of the relative coordinate y indeed reaches a steady state that we characterize in terms of the solution of a second-order differential equation. We obtain the explicit formula for the stationary probability P(y) of y for two examples of interaction potential V(y). The first one corresponds to V(y)∼|y|. In this case, for D=0 we find that P(y) contains a δ function part at y=0, signaling a strong clustering effect, together with a smooth exponential component. For D>0, the δ function part broadens, leading instead to weak clustering. The second example is the harmonic attraction V(y)∼y^{2} in which case, for D=0, P(y) is supported on a finite interval. We unveil an interesting relation between this two-RTP model with harmonic attraction and a three-state single-RTP model in one dimension, as well as with a four-state single-RTP model in two dimensions. We also provide a general discussion of the stationary bound state, including examples where it is not unique, e.g., when the particles cannot cross due to an additional short-range repulsion.

Direction reversing active Brownian particle in a harmonic potential

We study the two-dimensional motion of an active Brownian particle of speed v₀, with intermittent directional reversals in the presence of a harmonic trap of strength μ. The presence of the trap ensures that the position of the particle eventually reaches a steady state where it is bounded within a circular region of radius v₀/μ, centered at the minimum of the trap. Due to the interplay between the rotational diffusion constant D_R, reversal rate γ, and the trap strength μ, the steady state distribution shows four different types of shapes, which we refer to as active-I & II, and passive-I & II phases. In the active-I phase, the weight of the distribution is concentrated along an annular region close to the circular boundary, whereas in active-II, an additional central diverging peak appears giving rise to a Mexican hat-like shape of the distribution. The passive-I is marked by a single Boltzmann-like centrally peaked distribution in the large D_R limit. On the other hand, while the passive-II phase also shows a single central peak, it is distinguished from passive-I by a non-Boltzmann like divergence near the origin. We characterize these phases by calculating the exact analytical forms of the distributions in various limiting cases. In particular, we show that for D_R ≪ γ, the shape transition of the two-dimensional position distribution from active-II to passive-II occurs at μ = γ. We compliment these analytical results with numerical simulations beyond the limiting cases and obtain a qualitative phase diagram in the (D_R, γ, μ^-1) space.

Heat fluctuations in a harmonic chain of active particles

One of the major challenges in stochastic thermodynamics is to compute the distributions of stochastic observables for small-scale systems for which fluctuations play a significant role. Hitherto much theoretical and experimental research has focused on systems composed of passive Brownian particles. In this paper, we study the heat fluctuations in a system of interacting active particles. Specifically we consider a one-dimensional harmonic chain of N active Ornstein-Uhlenbeck particles, with the chain ends connected to heat baths of different temperatures. We compute the moment-generating function for the heat flow in the steady state. We employ our general framework to explicitly compute the moment-generating function for two example single-particle systems. Further, we analytically obtain the scaled cumulants for the heat flow for the chain. Numerical Langevin simulations confirm the long-time analytical expressions for first and second cumulants for the heat flow for a two-particle chain.

Extinctions of coupled populations, and rare event dynamics under non-Gaussian noise

The survival of natural populations may be greatly affected by environmental conditions that vary in space and time. We look at a population residing in two locations (patches) coupled by migration, in which the local conditions fluctuate in time. We report on two findings. First, we find that, unlike rare events in many other systems, here the histories leading to a rare extinction event are not dominated by a single path. We develop the appropriate framework, which turns out to be a hybrid of the standard saddle-point method, and the Donsker-Varadhan formalism which treats rare events of atypical averages over a long time. It provides a detailed description of the statistics of histories leading to the rare event and the mean time to extinction. The framework applies to rare events in a broad class of systems driven by non-Gaussian noise. Second, applying this framework to the population-dynamics model, we find a phase transition in its extinction behavior. Strikingly, a patch which is a sink (where individuals die more than are born) can nonetheless reduce the probability of extinction, even if it lowers the average population's size and growth rate.

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}.

Active Brownian motion with directional reversals

Active Brownian motion with intermittent direction reversals is common in bacteria like Myxococcus xanthus and Pseudomonas putida. We show that, for such a motion in two dimensions, the presence of the two timescales set by the rotational diffusion constant D_{R} and the reversal rate γ gives rise to four distinct dynamical regimes: (I) t≪min(γ^{-1},D_{R}^{-1}), (II) γ^{-1}≪t≪D_{R}^{-1}, (III) D_{R}^{-1}≪t≪γ^{-1}, and (IV) t≫max(γ^{-1}, D_{R}^{-1}), showing distinct behaviors. We characterize these behaviors by analytically computing the position distribution and persistence exponents. The position distribution shows a crossover from a strongly nondiffusive and anisotropic behavior at short times to a diffusive isotropic behavior via an intermediate regime, II or III. In regime II, we show that, the position distribution along the direction orthogonal to the initial orientation is a function of the scaled variable z∝x_{⊥}/t with a nontrivial scaling function, f(z)=(2π^{3})^{-1/2}Γ(1/4+iz)Γ(1/4-iz). Furthermore, by computing the exact first-passage time distribution, we show that a persistence exponent α=1 emerges due to the direction reversal in this regime.

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.

Fractal analysis of recurrence networks constructed from the two-dimensional fractional Brownian motions

In this study, we focus on the fractal property of recurrence networks constructed from the two-dimensional fractional Brownian motion (2D fBm), i.e., the inter-system recurrence network, the joint recurrence network, the cross-joint recurrence network, and the multidimensional recurrence network, which are the variants of classic recurrence networks extended for multiple time series. Generally, the fractal dimension of these recurrence networks can only be estimated numerically. The numerical analysis identifies the existence of fractality in these constructed recurrence networks. Furthermore, it is found that the numerically estimated fractal dimension of these networks can be connected to the theoretical fractal dimension of the 2D fBm graphs, because both fractal dimensions are piecewisely associated with the Hurst exponent H in a highly similar pattern, i.e., a linear decrease (if H varies from 0 to 0.5) followed by an inversely proportional-like decay (if H changes from 0.5 to 1). Although their fractal dimensions are not exactly identical, their difference can actually be deciphered by one single parameter with the value around 1. Therefore, it can be concluded that these recurrence networks constructed from the 2D fBms must inherit some fractal properties of its associated 2D fBms with respect to the fBm graphs.

Active Brownian motion in two dimensions under stochastic resetting

We study the position distribution of an active Brownian particle (ABP) in the presence of stochastic resetting in two spatial dimensions. We consider three different resetting protocols: (1) where both position and orientation of the particle are reset, (2) where only the position is reset, and (3) where only the orientation is reset with a certain rate r. We show that in the first two cases, the ABP reaches a stationary state. Using a renewal approach, we calculate exactly the stationary marginal position distributions in the limiting cases when the resetting rate r is much larger or much smaller than the rotational diffusion constant D_{R} of the ABP. We find that, in some cases, for a large resetting rate, the position distribution diverges near the resetting point; the nature of the divergence depends on the specific protocol. For the orientation resetting, there is no stationary state, but the motion changes from a ballistic one at short times to a diffusive one at late times. We characterize the short-time non-Gaussian marginal position distributions using a perturbative approach.

Toward the full short-time statistics of an active Brownian particle on the plane

We study the position distribution of a single active Brownian particle (ABP) on the plane. We show that this distribution has a compact support, the boundary of which is an expanding circle. We focus on a short-time regime and employ the optimal fluctuation method to study large deviations of the particle position coordinates x and y. We determine the optimal paths of the ABP, conditioned on reaching specified values of x and y, and the large deviation functions of the marginal distributions of x and of y. These marginal distributions match continuously with "near tails" of the x and y distributions of typical fluctuations, studied earlier. We also calculate the large deviation function of the joint x and y distribution P(x,y,t) in a vicinity of a special "zero-noise" point, and show that lnP(x,y,t) has a nontrivial self-similar structure as a function of x, y, and t. The joint distribution vanishes extremely fast at the expanding circle, exhibiting an essential singularity there. This singularity is inherited by the marginal x- and y-distributions. We argue that this fingerprint of the short-time dynamics remains there at all times.

Run-and-tumble particles in two dimensions: Marginal position distributions

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 cases the RTP shows a crossover from a ballistic to diffusive regime. The ballistic regime is a typical signature of the active nature of the systems and is characterized by nontrivial position distributions which depend on the specific model. We also show that the signature of activity at long times can be found in the atypical fluctuations, which we also characterize by computing the large deviation functions explicitly.

Active Brownian particles: mapping to equilibrium polymers and exact computation of moments

It is well known that the path probabilities of Brownian motion correspond to the equilibrium configurational probabilities of flexible Gaussian polymers, while those of active Brownian motion correspond to in-extensible semiflexible polymers. Here we investigate the properties of the equilibrium polymer that corresponds to the trajectories of particles acted on simultaneously by both Brownian and active noise. Through this mapping we can see interesting crossovers in the mechanical properties of the polymer with changing contour length. The polymer end-to-end distribution exhibits Gaussian behaviour for short lengths, which changes to the form of semiflexible filaments at intermediate lengths, to finally go back to a Gaussian form for long contour lengths. By performing a Laplace transform of the governing Fokker-Planck equation of the active Brownian particle, we discuss a direct method to derive exact expressions for all the moments of the relevant dynamical variables, in arbitrary dimensions. These are verified via numerical simulations and used to describe interesting qualitative features such as, for example, dynamical crossovers. Finally we discuss the kurtosis of the ABP's position, which we compute exactly, and show that it can be used to differentiate between active Brownian particles and the active Ornstein-Uhlenbeck process.

Collective forces in scalar active matter

Large-scale collective behavior in suspensions of active particles can be understood from the balance of statistical forces emerging beyond the direct microscopic particle interactions. Here we review some aspects of the collective forces that can arise in suspensions of self-propelled active Brownian particles: wall forces under confinement, interfacial forces, and forces on immersed bodies mediated by the suspension. Even for non-aligning active particles, these forces are intimately related to a non-uniform polarization of particle orientations induced by walls and bodies, or inhomogeneous density profiles. We conclude by pointing out future directions and promising areas for the application of collective forces in synthetic active matter, as well as their role in living active matter.

How a local active force modifies the structural properties of polymers

We study the dynamics of a polymer, described as a variant of a Rouse chain, driven by an active terminal monomer (head). The local active force induces a transition from a globule-like to an elongated state, as revealed by the study of the end-to-end distance, the variance of which is analytically predicted under suitable approximations. The change in the relaxation times of the Rouse-modes produced by the local self-propulsion is consistent with the transition from globule to elongated conformations. Moreover, also the bond-bond spatial correlation for the chain head are affected by the self-propulsion and a gradient of over-stretched bonds along the chain is observed. We compare our numerical results both with the phenomenological stiff-polymer theory and several analytical predictions in the Rouse-chain approximation.