Universal scaling in active single-file dynamics

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].

Active osmoticlike pressure on permeable inclusions

We use a standard minimal active Brownian model to investigate the osmotic-like effective pressure generated by active fluids on fixed hollow inclusions. These inclusions are enclosed by a permeable (albeit nonflexible) membrane, and the interior and exterior regions of the inclusions have different particle motility strengths. We consider both rectangular and disklike inclusions and analyze the effects of various system parameters, such as excluded volume interaction between active particles, hardness of membrane, and active particle density, on the effective pressure produced on the enclosing membrane. We focus on the range of intermediate to high motility strengths and analyze the effective pressure in the steady state. Our findings for the active pressure produced in the interior and exterior regions of the inclusion indicate that the pressure is higher in the region with lower motility due to the relatively stronger accumulation of active particles.

Exact fluctuation and long-range correlations in a single-file model under resetting

Resetting is a renewal mechanism in which a process is intermittently repeated after a random or fixed time. This simple act of stop and repeat profoundly influences the behavior of a system as exemplified by the emergence of nonequilibrium properties and expedition of search processes. Herein we explore the ramifications of stochastic resetting in the context of a single-file system called random average process (RAP) in one dimension. In particular, we focus on the dynamics of tracer particles and analytically compute the variance, equal time correlation, autocorrelation, and unequal time correlation between the positions of different tracer particles. Our study unveils that resetting gives rise to rather different behaviors depending on whether the particles move symmetrically or asymmetrically. For the asymmetric case, the system for instance exhibits a long-range correlation which is not seen in absence of the resetting. Similarly, in contrast to the reset-free RAP, the variance shows distinct scalings for symmetric and asymmetric cases. While for the symmetric case, it decays (towards its steady value) as ∼e^{-rt}/sqrt[t], we find ∼te^{-rt} decay for the asymmetric case (r being the resetting rate). Finally, we examine the autocorrelation and unequal time correlation in the steady state and demonstrate that they obey interesting scaling forms at late times. All our analytical results are substantiated by extensive numerical simulations.

Time-dependent properties of run-and-tumble particles: Density relaxation

We characterize collective diffusion of hardcore run-and-tumble particles (RTPs) by explicitly calculating the bulk-diffusion coefficient D(ρ,γ) for arbitrary density ρ and tumbling rate γ, in systems on a d-dimensional periodic lattice. We study two minimal models of RTPs: Model I is the standard version of hardcore RTPs introduced in [Phys. Rev. E 89, 012706 (2014)10.1103/PhysRevE.89.012706], whereas model II is a long-ranged lattice gas (LLG) with hardcore exclusion, an analytically tractable variant of model I. We calculate the bulk-diffusion coefficient analytically for model II and numerically for model I through an efficient Monte Carlo algorithm; notably, both models have qualitatively similar features. In the strong-persistence limit γ→0 (i.e., dimensionless ratio r_{0}γ/v→0), with v and r_{0} being the self-propulsion speed and particle diameter, respectively, the fascinating interplay between persistence and interaction is quantified in terms of two length scales: (i) persistence length l_{p}=v/γ and (ii) a "mean free path," being a measure of the average empty stretch or gap size in the hopping direction. We find that the bulk-diffusion coefficient varies as a power law in a wide range of density: D∝ρ^{-α}, with exponent α gradually crossing over from α=2 at high densities to α=0 at low densities. As a result, the density relaxation is governed by a nonlinear diffusion equation with anomalous spatiotemporal scaling. In the thermodynamic limit, we show that the bulk-diffusion coefficient-for ρ,γ→0 with ρ/γ fixed-has a scaling form D(ρ,γ)=D^{(0)}F(ρav/γ), where a∼r_{0}^{d-1} is particle cross section and D^{(0)} is proportional to the diffusion coefficient of noninteracting particles; the scaling function F(ψ) is calculated analytically for model II (LLG) and numerically for model I. Our arguments are independent of dimensions and microscopic details.

Effect of contact inhibition locomotion on confined cellular organization

Experiments performed using micro-patterned one dimensional collision assays have allowed a precise quantitative analysis of the collective manifestation of contact inhibition locomotion (CIL) wherein, individual migrating cells reorient their direction of motion when they come in contact with other cells. Inspired by these experiments, we present a discrete, minimal 1D Active spin model that mimics the CIL interaction between cells in one dimensional channels. We analyze the emergent collective behaviour of migrating cells in such confined geometries, as well as the sensitivity of the emergent patterns to driving forces that couple to cell motion. In the absence of vacancies, akin to dense cell packing, the translation dynamics is arrested and the model reduces to an equilibrium spin model which can be solved exactly. In the presence of vacancies, the interplay of activity-driven translation, cell polarity switching, and CIL results in an exponential steady cluster size distribution. We define a dimensionless Péclet number Q-the ratio of the translation rate and directional switching rate of particles in the absence of CIL. While the average cluster size increases monotonically as a function of Q, it exhibits a non-monotonic dependence on CIL strength, when the Q is sufficiently high. In the high Q limit, an analytical form of average cluster size can be obtained approximately by effectively mapping the system to an equivalent equilibrium process involving clusters of different sizes wherein the cluster size distribution is obtained by minimizing an effective Helmholtz free energy for the system. The resultant prediction of exponential dependence on CIL strength of the average cluster size and [Formula: see text] dependence of the average cluster size is borne out to reasonable accuracy as long as the CIL strength is not very large. The consequent prediction of a single scaling function of Q, particle density and CIL interaction strength, characterizing the distribution function of the cluster sizes and resultant data collapse is observed for a range of parameters.

Kinetic temperature and pressure of an active Tonks gas

Using computer simulation and analytical theory, we study an active analog of the well-known Tonks gas, where active Brownian particles are confined to a periodic one-dimensional (1D) channel. By introducing the notion of a kinetic temperature, we derive an accurate analytical expression for the pressure and clarify the paradoxical behavior where active Brownian particles confined to 1D exhibit anomalous clustering but no motility-induced phase transition. More generally, this work provides a deeper understanding of pressure in active systems as we uncover a unique link between the kinetic temperature and swim pressure valid for active Brownian particles in higher dimensions.

Rescaling invariance and anomalous energy transport in a small vertical column of grains

It is well known that energy dissipation and finite size can deeply affect the dynamics of granular matter, often making usual hydrodynamic approaches problematic. Here we report on the experimental investigation of a small model system, made of ten beads constrained into a 1D geometry by a narrow vertical pipe and shaken at the base by a piston excited by a periodic wave. Recording the beads motion with a high frame rate camera allows to investigate in detail the microscopic dynamics and test hydrodynamic and kinetic models. Varying the energy, we explore different regimes from fully fluidized to the edge of condensation, observing good hydrodynamic behavior down to the edge of fluidization, despite the small system size. Density and temperature fields for different system energies can be collapsed by suitable space and time rescaling, and the expected constitutive equation holds very well when the particle diameter is considered. At the same time, the balance between dissipated and fed energy is not well described by commonly adopted dependence due to the up-down symmetry breaking. Our observations, supported by the measured particle velocity distributions, show a different phenomenological temperature dependence, which yields equation solutions in agreement with experimental results.

Scaling laws for single-file diffusion of adhesive particles

Single-file diffusion refers to the Brownian motion in narrow channels where particles cannot pass each other. In such processes, the diffusion of a tagged particle is typically normal at short times and becomes subdiffusive at long times. For hard-sphere interparticle interaction, the time-dependent mean squared displacement of a tracer is well understood. Here we develop a scaling theory for adhesive particles. It provides a full description of the time-dependent diffusive behavior with a scaling function that depends on an effective strength of adhesive interaction. Particle clustering induced by the adhesive interaction slows down the diffusion at short times, while it enhances subdiffusion at long times. The enhancement effect can be quantified in measurements irrespective of how tagged particles are injected into the system. Combined effects of pore structure and particle adhesiveness should speed up translocation of molecules through narrow pores.

Attractor-driven matter

The state of a classical point-particle system may often be specified by giving the position and momentum for each constituent particle. For non-pointlike particles, the center-of-mass position may be augmented by an additional coordinate that specifies the internal state of each particle. The internal state space is typically topologically simple, in the sense that the particle's internal coordinate belongs to a suitable symmetry group. In this paper, we explore the idea of giving internal complexity to the particles, by attributing to each particle an internal state space that is represented by a point on a strange (or otherwise) attracting set. It is, of course, very well known that strange attractors arise in a variety of nonlinear dynamical systems. However, rather than considering strange attractors as emerging from complex dynamics, we may employ strange attractors to drive such dynamics. In particular, by using an attractor (strange or otherwise) to model each particle's internal state space, we present a class of matter coined "attractor-driven matter." We outline the general formalism for attractor-driven matter and explore several specific examples, some of which are reminiscent of active matter. Beyond the examples studied in this paper, our formalism for attractor-driven dynamics may be applicable more broadly, to model complex dynamical and emergent behaviors in a variety of contexts.

Collective motion of run-and-tumble repulsive and attractive particles in one-dimensional systems

Active matter deals with systems whose particles consume energy at the individual level in order to move. To unravel features such as the emergence of collective structures, several models have been suggested, such as the on-lattice model of run-and-tumble particles implemented via the persistent exclusion process (PEP). In our work, we study a one-dimensional system of run-and-tumble repulsive or attractive particles, both on-lattice and off-lattice. Additionally, we implement cluster motility dynamics in the on-lattice case (since in the off-lattice case, cluster motility arises from the individual particle dynamics). While we observe important differences between discrete and continuous dynamics, few common features are of particular importance. Increasing particle density drives aggregation across all different systems explored. For non-attractive particles, the effects of particle activity on aggregation are largely independent of the details of the dynamics. In contrast, once attractive interactions are introduced, the steady-state, which is completely determined by the interplay between these and the particles' activity, becomes highly dependent on the details of the dynamics.

Diversity of self-propulsion speeds reduces motility-induced clustering in confined active matter

Self-propelled swimmers such as bacteria agglomerate into clusters as a result of their persistent motion. In 1D, those clusters do not coalesce macroscopically and the stationary cluster size distribution (CSD) takes an exponential form. We develop a minimal lattice model for active particles in narrow channels to study how clustering is affected by the interplay between self-propulsion speed diversity and confinement. A mixture of run-and-tumble particles with a distribution of self-propulsion speeds is simulated in 1D. Particles can swap positions at rates proportional to their relative self-propulsion speed. Without swapping, we find that the average cluster size L_c decreases with diversity and follows a non-arithmetic power mean of the single-component L_c's, unlike the case of tumbling-rate diversity previously studied. Effectively, the mixture is thus equivalent to a system of identical particles whose self-propulsion speed is the harmonic mean self-propulsion speed of the mixture. With swapping, particles escape more quickly from clusters. As a consequence, L_c decreases with swapping rates and depends less strongly on diversity. We derive a dynamical equilibrium theory for the CSDs of binary and fully polydisperse systems. Similarly to the clustering behaviour of one-component models, our qualitative results for mixtures are expected to be universal across active matter. Using literature experimental values for the self-propulsion speed diversity of unicellular swimmers known as choanoflagellates, which naturally differentiate into slower and faster cells, we predict that the error in estimating their L_cvia one-component models which use the conventional arithmetic mean self-propulsion speed is around 30%.

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.

Active mixtures in a narrow channel: motility diversity changes cluster sizes

The persistent motion of bacteria produces clusters with a stationary cluster size distribution (CSD). Here we develop a minimal model for bacteria in a narrow channel to assess the relative importance of motility diversity (i.e. polydispersity in motility parameters) and confinement. A mixture of run-and-tumble particles with a distribution of tumbling rates (denoted generically by α) is considered on a 1D lattice. Particles facing each other cross at constant rate, rendering the lattice quasi-1D. To isolate the role of diversity, the global average α stays fixed. For a binary mixture with no particle crossing, the average cluster size (Lc) increases with the diversity as lower-α particles trap higher-α ones for longer. At finite crossing rate, particles escape from the clusters sooner, making Lc smaller and the diversity less important, even though crossing can enhance demixing of particle types between the cluster and gas phases. If the crossing rate is increased further, the clusters become controlled by particle crossing. We also consider an experiment-based continuous distribution of tumbling rates, revealing similar physics. Using parameters fitted from experiments with Escherichia coli bacteria, we predict that the error in estimating Lc without accounting for polydispersity is around 60%. We discuss how to find a binary system with the same CSD as the fully polydisperse mixture. An effective theory is developed and shown to give accurate expressions for the CSD, the effective α, and the average fraction of mobile particles. We give reasons why our qualitative results are expected to be valid for other active matter models and discuss the changes that would result from polydispersity in the active speed rather than in the tumbling rate.

Hard core run and tumble particles on a one-dimensional lattice

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 rates. We also calculate the hydrodynamic coefficients of diffusivity and conductivity for both large and small spin-flip rates and show that the Einstein relation is violated in both regimes. We also show how the nongradient nature of the process can be taken into account in a systematic manner to calculate the hydrodynamic coefficients.