Pair correlation of dilute active Brownian particles: From low-activity dipolar correction to high-activity algebraic depletion wings

Large deviations in statistics of the local time and occupation time for a run and tumble particle

We investigate the statistics of the local time T=∫_{0}^{T}δ(x(t))dt that a run and tumble particle (RTP) x(t) in one dimension spends at the origin, with or without an external drift. By relating the local time to the number of times the RTP crosses the origin, we find that the local time distribution P(T) satisfies the large deviation principle P(T)∼e^{-TI(T/T)} in the large observation time limit T→∞. Remarkably, we find that in the presence of drift the rate function I(ρ) is nonanalytic: we interpret its singularity as a dynamical phase transition of first order. We then extend these results by studying the statistics of the amount of time R that the RTP spends inside a finite interval (i.e., the occupation time), with qualitatively similar results. In particular, this yields the long-time decay rate of the probability P(R=T) that the particle does not exit the interval up to time T. We find that the conditional end-point distribution exhibits an interesting change of behavior from unimodal to bimodal as a function of the size of the interval. To study the occupation time statistics, we extend the Donsker-Varadhan large-deviation formalism to the case of RTPs, for general dynamical observables and possibly in the presence of an external potential.

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}.

When an active bath behaves as an equilibrium one

Active scalar baths consisting of active Brownian particles are characterized by a non-Gaussian velocity distribution, a kinetic temperature, and a diffusion coefficient that scale with the square of the active velocity v_{0}. While these results hold in overdamped active systems, inertial effects lead to normal velocity distributions, with kinetic temperature and diffusion coefficient increasing as ∼v_{0}^{α} with 1<α<2. Remarkably, the late-time diffusivity and mobility decrease with mass. Moreover, we show that the equilibrium Einstein relation is asymptotically recovered with inertia. In summary, the inertial mass restores an equilibriumlike behavior.

On analytical theories for conductivity and self-diffusion in concentrated electrolytes

Describing analytically the transport properties of electrolytes, such as their conductivity or the self-diffusion of the ions, has been a central challenge of chemical physics for almost a century. In recent years, this question has regained some interest in light of Stochastic Density Field Theory (SDFT) - an analytical framework that allows the approximate determination of density correlations in fluctuating systems. In spite of the success of this theory to describe dilute electrolytes, its extension to concentrated solutions raises a number of technical difficulties, and requires simplified descriptions of the short-range repulsion between the ions. In this article, we discuss recent approximations that were proposed to compute the conductivity of electrolytes, in particular truncations of Coulomb interactions at short distances. We extend them to another observable (the self-diffusion coefficient of the ions) and compare them to earlier analytical approaches, such as the mean spherical approximation and mode-coupling theory. We show how the treatment of hydrodynamic effects in SDFT can be improved, that the choice of the modified Coulomb interactions significantly affects the determination of the properties of the electrolytes, and that comparison with other theories provides a guide to extend SDFT approaches in this context.

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.

Quasi-two-dimensional bacterial swimming around pillars: Enhanced trapping efficiency and curvature dependence

Microswimmers exhibit more diverse behavior in quasi-two dimensions than in three dimensions. Such behavior remains elusive due to the analytical difficulty of dealing with two parallel solid boundaries. The existence of additional obstacles in quasi-two dimensional systems further complicates the analysis. Combining experiments and hydrodynamic simulations, we investigate how the spatial dimension affects the interactions between microswimmers and obstacles. We fabricated microscopic pillars in quasi-two dimensions by etching glass coverslips and observed bacterial swimming among the pillars. Bacteria got trapped around the circular pillars and the trapping efficiency increased as the quasi-two-dimensionality was increased or as the curvature of the pillars was decreased. Numerical simulations of the simplest situation of a confined squirmer showed anomalous increase of hydrodynamic attractions, establishing that the enhanced interaction is a universal property of quasi-two-dimensional microhydrodynamics. We also demonstrated that the local curvature of the obstacle controls the trapping efficiency by experiments with elliptic pillars.

Diffusion of a tracer in a dense mixture of soft particles connected to different thermostats

We study the dynamics of a tracer in a dense mixture of particles connected to different thermostats. Starting from the overdamped Langevin equations that describe the evolution of the system, we derive the expression of the self-diffusion coefficient of a tagged particle in the suspension, in the limit of soft interactions between the particles. Our derivation, which relies on the linearization of the Dean-Kawasaki equations obeyed by the density fields and on a path-integral representation of the dynamics of the tracer, extends previous derivations that held for tracers in contact with a single bath. Our analytical result is confronted to results from Brownian dynamics simulations. The agreement with numerical simulations is very good even for high densities. We show how the diffusivity of tracers can be affected by the activity of a dense environment of soft particles that may represent polymer coils-a result that could be of relevance in the interpretation of measurements of diffusivity in biological media. Finally, our analytical result is general and can be applied to the diffusion of tracers coupled to different types of fluctuating environments, provided that their evolution equations are linear and that the coupling between the tracer and the bath is weak.

Trapped-particle microrheology of active suspensions

In microrheology, the local rheological properties of a complex fluid are inferred from the free or forced motion of embedded colloidal probes. Theoretical machinery developed for forced-probe microrheology of colloidal suspensions focused on either constant-force (CF) or constant-velocity (CV) probes while in experiments neither the force nor the kinematics of the probe is fixed. More importantly, the constraint of CF or CV introduces a difficulty in the meaningful quantification of the fluctuations of the probe due to a thermodynamic uncertainty relation. It is known that for a Brownian particle trapped in a harmonic potential well, the product of the standard deviations of the trap force and the particle position is $dk_BT$ in $d$ dimensions with $k_BT$ being the thermal energy. As a result, if the force (position) is not allowed to fluctuate, the position (force) fluctuation becomes infinite. To allow the measurement of fluctuations, we consider a microrheology model in which the probe is dragged along by a moving harmonic potential so that both its position and the trap force are allowed to fluctuate. Starting from the full Smoluchowski equation governing the dynamics of $N$ hard active Brownian particles, we derive a pair equation describing the dynamics of the probe as it interacts with one bath particle in the dilute limit. From this, we determine the mean and the variance (i.e., fluctuation) of the probe position in terms of the pair probability distribution. We then characterize the behavior of the system in the limits of both weak and strong traps.

Unified analysis of topological defects in 2D systems of active and passive disks

We provide a comprehensive quantitative analysis of localized and extended topological defects in the steady state of 2D passive and active repulsive Brownian disk systems. We show that, both in and out-of-equilibrium, the passage from the solid to the hexatic is driven by the unbinding of dislocations, in quantitative agreement with the KTHNY singularity. Instead, extended clusters of defects largely dominate below the solid-hexatic critical line. The latter percolate in the liquid phase very close to the hexatic-liquid transition, both for continuous and discontinuous transitions, in the homogeneous liquid regime. At critical percolation the clusters of defects are fractal with statistical and geometric properties that are independent of the activity and compatible with the universality class of uncorrelated critical percolation. We also characterize the spatial organization of point-like defects and we show that the disclinations are not free, but rather always very near more complex defect structures. At high activity, the bulk of the dense phase generated by Motility-Induced Phase Separation is characterized by a density of point-like defects, and statistics and morphology of defect clusters, set by the amount of activity and not the packing fraction. Hexatic domains within the dense phase are separated by grain-boundaries along which a finite network of topological defects resides, interrupted by gas bubbles in cavitation. This structure is dynamic in the sense that the defect network allows for an unzipping mechanism that leaves free space for gas bubbles to appear, close, and even be released into the dilute phase.

Motility-induced inter-particle correlations and dynamics: a microscopic approach for active Brownian particles

Amongst the theoretical approaches towards the dynamics and phase behaviour of suspensions of active Brownian particles (ABPs), no attempt has been made to specify the motility-induced inter-particle correlations as quantified by the pair-correlation function. Here, we derive expressions for the pair-correlation function for ABPs with very short-ranged direct interactions for small and large swimming velocities and low concentrations. The pair-correlation function is the solution of a differential equation that is obtained from the Fokker-Planck equation for the probability density function of the positions and orientations of the ABPs. For large swimming Peclet numbers, λ, the pair-correlation function is highly asymmetric. The pair-correlation function attains a large value, ∼λ, within a small region of spatial extent, ∼1/λ, near contact of the ABPs when the ABPs approach each other. The pair-correlation function is small within a large region of spatial extent, ∼λ1/3, when the ABPs move apart, with a contact value that is essentially zero. From the explicit expressions for the pair-correlation function, Fick's diffusion equation is generalized to include motility. It is shown that mass transport, in case of large swimming velocities, is dominated by a preferred swimming direction that is induced by concentration gradients. The expression for the pair-correlation function derived in this paper could serve as a starting point to obtain approximate results for high concentrations, which could then be employed in a first-principles analysis of the dynamics and phase behaviour of ABPs at higher concentrations.