Hard-core Exclusion Research Articles

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 equationwith 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 sectionand 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 Ligand Binding on Polymer Diffusiophoresis

Diffusiophoresis is the migration of a macromolecule in response to a concentration gradient of a cosolute in liquids. Diffusiophoresis of polyethylene glycol (PEG) in water occurs from high to low concentration of trimethylamine-N-oxide (TMAO). This is consistent with the preferential hydration of PEG observed in the presence of TMAO. In other words, PEG migrates in the direction in which it lowers its chemical potential. On the other hand, although PEG is found to preferentially bind urea in water, PEG diffusiophoresis still occurs from high to low urea concentration. Thus, PEG migrates in the direction that increases its chemical potential in the urea case. Here, a ligand-binding model for polymer diffusiophoresis in the presence of a cosolute that preferentially binds to polymer is developed. It includes both actual polymer–ligand binding and the effect of the polymer on cosolute thermodynamic activity. This model shows that polymer–cosolute binding has a marginal effect on polymer diffusiophoresis and indicates that weak repulsive interactions, such as hard-core exclusion forces, are the main factor responsible for the observed PEG diffusiophoresis from high to low urea concentration. This work contributes to a better understanding of diffusiophoresis of macromolecules in response to gradients of nonelectrolytes.

Tracer dynamics in one dimensional gases of active or passive particles

We consider one-dimensional systems comprising either active run-and-tumble particles (RTPs) or passive Brownian random walkers. These particles are either noninteracting or have hardcore exclusions. We study the dynamics of a single tracer particle embedded in such a system—this tracer may be either active or passive, with hardcore exclusion from environmental particles. In an active hardcore environment, both active and passive tracers show long-time subdiffusion: displacements scale as t 1/4 with a density-dependent prefactor that is independent of tracer type, and differs from the corresponding result for passive-in-passive subdiffusion. In an environment of noninteracting active particles, the passive-in-passive results are recovered at low densities for both active and passive tracers, but transient caging effects slow the tracer motion at higher densities, delaying the onset of any t 1/4 regime. For an active tracer in a passive environment, we find more complex outcomes, which depend on details of the dynamical discretization scheme. We interpret these results by studying the density distribution of environmental particles around the tracer. In particular, sticking of environment particles to the tracer cause it to move more slowly in noninteracting than in interacting active environments, while the anomalous behaviour of the active-in-passive cases stems from a ‘snowplough’ effect whereby a large pile of diffusive environmental particles accumulates in front of an RTP tracer during a ballistic run.

Binary lattice-gases of particles with soft exclusion: exact phase diagrams for tree-like lattices

We study equilibrium properties of binary lattice-gases comprising A and B particles, which undergo continuous exchanges with their respective reservoirs, maintained at chemical potentials μ A = μ B = μ. The particles interact via on-site hard-core exclusion and also between the nearest-neighbours: there are a soft repulsion between AB pairs and also interactions of arbitrary strength J, positive or negative, for AA and BB pairs. For tree-like Bethe and Husimi lattices we determine the full phase diagram of such a ternary mixture of particles and voids. We show that for J being above a lattice-dependent threshold value, the critical behaviour is similar: the system undergoes a transition at μ = μ c from a phase with equal mean densities of species into a phase with a spontaneously broken symmetry, in which the mean densities are no longer equal. Depending on the value of J, this transition can be either continuous or of the first order. For sufficiently large negative J, the behaviour on the two lattices becomes markedly different: on the Bethe lattice there exist two separate phases with different kinds of structural order, which are absent on the Husimi lattice, due to stronger frustration effects.

Jamming of multiple persistent random walkers in arbitrary spatial dimension

We consider the persistent exclusion process in which a set of persistent random walkers interact via hard-core exclusion on a hypercubic lattice in d dimensions. We work within the ballistic regime whereby particles continue to hop in the same direction over many lattice sites before reorienting. In the case of two particles, we find the mean first-passage time to a jammed state where the particles occupy adjacent sites and face each other. This is achieved within an approximation that amounts to embedding the one-dimensional system in a higher-dimensional reservoir. Numerical results demonstrate the validity of this approximation, even for small lattices. The results admit a straightforward generalization to dilute systems comprising more than two particles. A self-consistency condition on the validity of these results suggest that clusters may form at arbitrarily low densities in the ballistic regime, in contrast to what has been found in the diffusive limit.

Coarse graining of a Fokker–Planck equation with excluded volume effects preserving the gradient flow structure

The propagation of gradient flow structures from microscopic to macroscopic models is a topic of high current interest. In this paper, we discuss this propagation in a model for the diffusion of particles interacting via hard-core exclusion or short-range repulsive potentials. We formulate the microscopic model as a high-dimensional gradient flow in the Wasserstein metric for an appropriate free-energy functional. Then we use the JKO approach to identify the asymptotics of the metric and the free-energy functional beyond the lowest order for single particle densities in the limit of small particle volumes by matched asymptotic expansions. While we use a propagation of chaos assumption at far distances, we consider correlations at small distance in the expansion. In this way, we obtain a clear picture of the emergence of a macroscopic gradient structure incorporating corrections in the free-energy functional due to the volume exclusion.

Chaotic study on a multibody interacting particle system with trajectory of variable curvature radius

Multibody interacting particle system is one of the most important driven-diffusive systems, which can well describe stochastic dynamics of self-driven particles unidirectionally updating along one-dimensional discrete lattices controlled by hard-core exclusions. Such determined nonlinear dynamic system shows the chaos, a complicated motion that is similar to random and can’t determine future states according to given initial conditions. Derived from the study of randomness in the framework of determinism, chaos effectively unifies determinacy and randomness of nonlinear physics, which also embodies the randomness and uncertainty of multi-body particle motions. Besides, chaos often presents complex ordered phenomena like aperiodic ordered motion state etc., which indicates it’s a stable coordination of local random and global mode. Different with previous work, a mesoscopic multibody interacting particle system considering the trajectory of variable curvature radius is proposed, which is more reliable to depict true driven-diffusive systems. Nonlinear dynamical master equations are established. Linear and nonlinear stability analyses are performed to test system robustness, which lead to linear stable region, metastable region, unstable region, triangular wave, solitary wave and kink-antikink wave. Complete numerical simulations under quantitatively changing characteristic order parameters are performed to reveal intrinsic chaotic dynamics mechanisms. By comparing fruitful chaotic patterns, double periodic convergence in stable limit cycles is observed due to attractors folding. System states tend to be limit cycles with the passage of time. The longer time is, the higher density is near attractors. The number of main components of attractors and the shape of them are found to drastically change with these parameters. Hereafter, the chaotic formation mechanism is explained by performing Fourier spectrums analyses. High and low frequency elements are obtained, whose module lengths are found to change periodically with time and nonperiodically with space. Our results show the sensitivity of the chaotic system to initial conditions, which mean slight changes of initial states in one part of system can lead to disproportionate consequences in other parts. The sensitivity is directly related to uncertainty and unpredictability. Moreover, calculations also show that three kinds of attractors (namely, point attractor, limit-cycle attractor and strange attractor) exist in the proposed chaotic system, which control particle motions. The first two attractors are convergence attractors playing a limiting role, which lead to static and balanced features of system. Oppositely, strange attractor makes the system deviate from regions of convergent attractors and creates unpredictability by inducing the vitality of the system and turning it into a non-preset mode. Furthermore, interactions between convergence attractors and strange attractors trigger such locally fruitful modes. This work will be helpful to understand mesoscopic dynamics, stochastic dynamics and non-equilibrium dynamical behaviors of multibody interacting particle systems.

Directing Liquid Crystalline Self-Organization of Rodlike Particles through Tunable Attractive Single Tips.

Dispersions of rodlike colloidal particles exhibit a plethora of liquid crystalline states, including nematic, smectic A, smectic B, and columnar phases. This phase behavior can be explained by presuming the predominance of hard-core volume exclusion between the particles. We show here how the self-organization of rodlike colloids can be controlled by introducing a weak and highly localized directional attractive interaction between one of the ends of the particles. This has been performed by functionalizing the tips of filamentous viruses by means of regioselectively grafting fluorescent dyes onto them, resulting in a hydrophobic patch whose attraction can be tuned by varying the number of bound dye molecules. We show, in agreement with our computer simulations, that increasing the single tip attraction stabilizes the smectic phase at the expense of the nematic phase, leaving all other liquid crystalline phases invariant. For a sufficiently strong tip attraction, the nematic state may be suppressed completely to get a direct isotropic liquid-to-smectic phase transition. Our findings provide insights into the rational design of building blocks for functional structures formed at low densities.

Exact spectral solution of two interacting run-and-tumble particles on a ring lattice

Exact solutions of interacting random walk models, such as 1D lattice gases, offer precise insight into the origin of nonequilibrium phenomena. Here, we study a model of run-and-tumble particles on a ring lattice interacting via hardcore exclusion. We present the exact solution for one and two particles using a generating function technique. For two particles, the eigenvectors and eigenvalues are explicitly expressed using two parameters reminiscent of Bethe roots, whose numerical values are determined by polynomial equations which we derive. The spectrum depends in a complicated way on the ratio of direction reversal rate to lattice jump rate, . For both one and two particles, the spectrum consists of separate real bands for large , which mix and become complex-valued for small . At exceptional values of , two or more eigenvalues coalesce such that the Markov matrix is non-diagonalizable. A consequence of this intricate parameter dependence is the appearance of dynamical transitions: non-analytic minima in the longest relaxation times as functions of (for a given lattice size). Exceptional points are theoretically and experimentally relevant in, e.g. open quantum systems and multichannel scattering. We propose that the phenomenon should be a ubiquitous feature of classical nonequilibrium models as well, and of relevance to physical observables in this context.

Analytical and simulation studies of driven diffusive system with asymmetric heterogeneous interactions

Totally asymmetric simple exclusion process (namely, TASEP) is one of the most vital driven diffusive systems, which depicts stochastic dynamics of self-driven particles unidirectional updating along one-dimensional discrete lattices controlled by hard-core exclusions. Different with pre-existing results, driven diffusive system composed by multiple TASEPs with asymmetric heterogeneous interactions under two-dimensional periodic boundaries is investigated. By using detailed balance principle, particle configurations are extensively studied to obtain universal laws of characteristic order parameters of such stochastic dynamic system. By performing analytical analyses and Monte-Carlo simulations, local densities are found to be monotone increase with global density and spatially homogeneous to site locations. Oppositely, local currents are found to be non-monotonically increasing against global density and proportional to forward rate. Additionally, by calculating different cases of topologies, changing transition rates are found to have greater effects on particle configurations in adjacent subsystems. By intuitively comparing with pre-existing results, the improvement of our work also shows that introducing and considering totally heterogeneous interactions can improve the total current in such multiple TASEPs and optimize the overall transport of such driven-diffusive system. Our research will be helpful to understand microscopic dynamics and non-equilibrium dynamical behaviors of interacting particle systems.

Density profiles of a self-gravitating lattice gas in one, two, and three dimensions.

We consider a lattice gas in spaces of dimensionality D=1,2,3. The particles are subject to a hardcore exclusion interaction and an attractive pair interaction that satisfies Gauss' law as do Newtonian gravity in D=3, a logarithmic potential in D=2, and a distance-independent force in D=1. Under mild additional assumptions regarding symmetry and fluctuations we investigate equilibrium states of self-gravitating material clusters, in particular radial density profiles for closed and open systems. We present exact analytic results in several instances and high-precision numerical data in others. The density profile of a cluster with finite mass is found to exhibit exponential decay in D=1 and power-law decay in D=2 with temperature-dependent exponents in both cases. In D=2 the gas evaporates in a continuous transition at a nonzero critical temperature. We describe clusters of infinite mass in D=3 with a density profile consisting of three layers (core, shell, halo) and an algebraic large-distance asymptotic decay. In D=3 a cluster of finite mass can be stabilized at T>0 via confinement to a sphere of finite radius. In some parameter regime, the gas thus enclosed undergoes a discontinuous transition between distinct density profiles. For the free energy needed to identify the equilibrium state we introduce a construction of gravitational self-energy that works in all D for the lattice gas. The decay rate of the density profile of an open cluster is shown to transform via a stretched exponential for 1<D<2, whereas it crosses over from one power-law at intermediate distances to a different power-law at larger distances for 2<D<3.

Analytical and simulation studies of 2D asymmetric simple exclusion process

Two species of particles driven perpendicularly and interacting through hardcore exclusion on the 2D lattice is studied. Each particle has three moving directions without the back step. Under periodic boundary conditions, an intermediate phase has been found at which some particles could move along the border of jamming stripes. We have performed mean field analysis for the moving and intermediate phase respectively. The analytical results agree with the simulation results well. The empty site moves along the interface of jamming stripes when the system only has one empty site. The average movement of empty site in one Monte Carlo step (MCS) has been analyzed through the master equation. Under open boundary conditions, the system exhibits moving and jamming phases. The critical injection probability αc shows non-monotonically against the forward moving probability q. The analytical results of average velocity, the density and the flow rate against the injection probability in the moving phase also agree with simulation results well.

Matrix product states for interacting particles without hardcore constraints

We construct matrix product steady states for a class of interacting particle systems where particles do not obey hardcore exclusion, meaning each site can occupy any number of particles subjected to the global conservation of the total number of particles in the system. To represent the arbitrary occupancy of the sites, the matrix product ansatz here requires an infinite set of matrices which in turn leads to an algebra involving an infinite number of matrix equations. We show that these matrix equations, in fact, can be reduced to a single functional relation when the matrices are parametric functions of the representative occupation number. We demonstrate this matrix formulation in a class of stochastic particle hopping processes on a one dimensional periodic lattice where hop rates depend on the occupation numbers of the departure site and its neighbors within a finite range; this includes some well known stochastic processes like, the totally or partially asymmetric zero range process, misanthrope process and finite range process.

Exact solution of two interacting run-and-tumble random walkers with finite tumble duration

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 moving when the other is tumbling. The first of these lengthscales vanishes in a scaling limit where the continuous-space dynamics is recovered whilst the second remains finite. Thus the nonequilibrium stationary state reveals a rich structure of attractive, jammed and extended pieces.

Large Deviations of a Tracer in the Symmetric Exclusion Process.

The one-dimensional symmetric exclusion process, the simplest interacting particle process, is a lattice gas made of particles that hop symmetrically on a discrete line respecting hard-core exclusion. The system is prepared on the infinite lattice with a step initial profile with average densities ρ_{+} and ρ_{-} on the right and on the left of the origin. When ρ_{+}=ρ_{-}, the gas is at equilibrium and undergoes stationary fluctuations. When these densities are unequal, the gas is out of equilibrium and will remain so forever. A tracer, or a tagged particle, is initially located at the boundary between the two domains; its position X_{t} is a random observable in time that carries information on the nonequilibrium dynamics of the whole system. We derive an exact formula for the cumulant generating function and the large deviation function of X_{t} in the long-time limit and deduce the full statistical properties of the tracer's position. The equilibrium fluctuations of the tracer's position, when the density is uniform, are obtained as an important special case.

Kinetically constrained lattice gases: Tagged particle diffusion

Les gaz reticulaires avec contrainte cinetique (KCLG) sont des systemes de particules en interaction sur le reseau $\mathbb{Z}^{d}$ avec au plus une particule par site et une dynamique de type Kawasaki. Leur particularite est que les sauts ne sont autorises que si la configuration satisfait une contrainte exigeant suffisamment de sites vides dans un certain voisinage local. Les KCLG ont ete introduits et massivement etudies dans la litterature physique comme modeles pour les systemes vitreux. Nous nous concentrons sur l’une des classes de KCLG les plus etudiees : les modeles de Kob–Andersen (KA). Nous analysons le comportement d’un traceur (c.-a-d. une particule marquee) a l’equilibre. Pour toute dimension $d\geq2$ et toute densite de particules, nous montrons qu’a l’echelle diffusive la trajectoire du traceur converge vers un mouvement brownien $d$-dimensionnel avec matrice de diffusion non degeneree. Par consequent nous demontrons la non-existence d’une transition (qui avait ete conjecturee dans la litterature physique) entre un regime diffusif et un regime non diffusif. Notre technique est assez flexible et peut etre etendue pour analyser le comportement du traceur sous d’autres choix de contraintes.

Reentrant disorder-disorder transitions in generalized multicomponent Widom-Rowlinson models.

In the lattice version of the multicomponent Widom-Rowlinson (WR) model, each site can be either empty or singly occupied by one of M different particles, all species having the same fugacity z. The only nonzero interaction potential is a nearest-neighbor hard-core exclusion between unlike particles. For M<M(0) with some minimum M(0) dependent on the lattice structure, as z increases from 0 to ∞ there is a direct transition from the disordered (gas) phase to a demixed (liquid) phase with one majority component at z>z(d) (M). If M≥M(0), there is an intermediate ordered "crystal phase" (composed of two nonequivalent even and odd sublattices) for z lying between z(c)(M) and z(d)(M) which is driven by entropy. We generalize the multicomponent WR model by replacing the hard-core exclusion between unlike particles by more realistic large (but finite) repulsion. The model is solved exactly on the Bethe lattice with an arbitrary coordination number. The numerical calculations, based on the corner transfer matrix renormalization group, are performed for the two-dimensional square lattice. The results for M=4 indicate that the second-order phase transitions from the disordered gas to the demixed phase become of first order, for an arbitrarily large finite repulsion. The results for M≥M(0) show that, as the repulsion weakens, the region of the crystal phase diminishes itself. For weak enough repulsions, the direct transition between the crystal and demixed phases changes into a separate pair of crystal-gas and gas-demixed transitions; this is an example of a disorder-disorder reentrant transition via an ordered crystal phase. If the repulsion between unlike species is too weak, the crystal phase disappears from the phase diagram. It is shown that the generalized WR model belongs to the Ising universality class.

Monodisperse hard rods in external potentials.

We consider linear arrays of cells of volume V(c) populated by monodisperse rods of size σV(c),σ=1,2,..., subject to hardcore exclusion interaction. Each rod experiences a position-dependent external potential. In one application we also examine effects of contact forces between rods. We employ two distinct methods of exact analysis with complementary strengths and different limits of spatial resolution to calculate profiles of pressure and density on mesoscopic and microscopic length scales at thermal equilibrium. One method uses density functionals and the other statistically interacting vacancy particles. The applications worked out include gravity, power-law traps, and hard walls. We identify oscillations in the profiles on a microscopic length scale and show how they are systematically averaged out on a well-defined mesoscopic length scale to establish full consistency between the two approaches. The continuum limit, realized as V(c)→0,σ→∞ at nonzero and finite σV(c), connects our highest-resolution results with known exact results for monodisperse rods in a continuum. We also compare the pressure profiles obtained from density functionals with the average microscopic pressure profiles derived from the pair distribution function.

Interacting particles in disordered flashing ratchets

We study the steady state properties of a system of particles interacting via hard core exclusion and moving in a discrete flashing disordered ratchet potential. Quenched disorder is introduced by breaking the periodicity of the ratchet potential through changing shape of the potential across randomly chosen but fixed periods. We show that the effects of quenched disorder can be broadly classified as strong or weak with qualitatively different behaviour of the steady state particle flux as a function of overall particle density. We further show that most of the effects including a density driven nonequilibrium phase transition observed can be understood by constructing an effective asymmetric simple exclusion process with quenched disorder in the hop rates.

A Classical WR Model with $$q$$ q Particle Types

A version of the Widom–Rowlinson model is considered, where particles of $$q$$ types coexist, subject to pairwise hard-core exclusions. For $$q\le 4$$ , in the case of large equal fugacities, we give a complete description of the pure phase picture based on the theory of dominant ground states.