System Of Brownian Particles Research Articles

Small systems of laser-driven active Brownian particles: Evolution and dynamic entropy

We investigate the evolution of small systems of strongly interacting active Brownian particles. Metal-coated grains of micron size levitating in gas-discharge plasma are irradiated by laser inducing their active motion. We present the experimentally obtained functions of mean first-passage time dynamic entropy (MFPT) for each grain in a two-dimensional system for various values of their kinetic temperature, and also the values of the fractal dimension of trajectories of these systems and their localization area and match them to the point of transformation of the system.

Brownian ensemble of random-radius buoyancy vortices and Maxwell velocity distribution in a turbulent convective mixed-layer

The stochastic ensemble of convective thermals (vortices), forming the fine structure of a turbulent convective atmospheric layer, is considered. The proposed ensemble model assumes all thermals in the mixed-layer to have the same determinate buoyancies and considers them as solid spheres of variable volumes. The values of radii and vertical velocities of the thermals are assumed random. The motion of the stochastic system of convective vortices is described by the nonlinear Langevin equation with a linear drift coefficient and a random force, whose structure is known for a system of Brownian particles. The probability density of the thermal ensemble in velocity phase space is shown to satisfy an associated K-form of the Fokker-Planck equation with variable coefficients. Maxwell velocity distribution of convective thermals is constructed as a steady-state solution of a simplified Fokker-Planck equation. The obtained Maxwell velocity distribution is shown to give a good approximation of experimental distributions in a turbulent convective mixed-layer.

Collective Motion of Repulsive Brownian Particles in Single-File Diffusion with and without Overtaking.

Subdiffusion is commonly observed in liquids with high density or in restricted geometries, as the particles are constantly pushed back by their neighbors. Since this “cage effect” emerges from many-body dynamics involving spatiotemporally correlated motions, the slow diffusion should be understood not simply as a one-body problem but as a part of collective dynamics, described in terms of space–time correlations. Such collective dynamics are illustrated here by calculations of the two-particle displacement correlation in a system of repulsive Brownian particles confined in a (quasi-)one-dimensional channel, whose subdiffusive behavior is known as the single-file diffusion (SFD). The analytical calculation is formulated in terms of the Lagrangian correlation of density fluctuations. In addition, numerical solutions to the Langevin equation with large but finite interaction potential are studied to clarify the effect of overtaking. In the limiting case of the ideal SFD without overtaking, correlated motion with a diffusively growing length scale is observed. By allowing the particles to overtake each other, the short-range correlation is destroyed, but the long-range weak correlation remains almost intact. These results describe nested space–time structure of cages, whereby smaller cages are enclosed in larger cages with longer lifetimes.

Mode-Coupling theory for glass transition of active-passive binary mixture

Collective behaviours of active particle systems have gained great research attentions in recent years. Here we present a mode-coupling theory (MCT) framework to study the glass transition of a mixture system of active and passive Brownian particles. The starting point is an effective Smoluchowski equation, which governs the dynamics of the probability distribution function in the position phase space. With the assumption of the existence of a nonequilibrium steady state, we are able to obtain dynamic equations for the intermediate scattering functions (ISFs), wherein an irreducible memory function is introduced which in turn can be written as functions of the ISFs based on standard mode-coupling approximations. The effect of particle activity is included through an effective diffusion coefficient which can be obtained via short time simulations. By calculating the long-time limit of the ISF, the Debye-Waller (DW) factor, one can determine the critical packing fraction ηc of glass transition. We find that for active-passive (AP) mixtures with the same particle sizes, ηc increases as the partial fraction of active particle xA increases, which is in agreement with previous simulation works. For system with different active/passive particle sizes, we find an interesting reentrance behaviour of glass transition, i.e., ηc shows a non-monotonic dependence on xA. In addition, such a reentrance behaviour would disappear if the particle activity is large enough. Our results thus provide a useful theoretical scheme to study glass transition behaviour of active-passive mixture systems in a promising way.

Finite-particle approximations for interacting Brownian particles with logarithmic potentials

We prove the convergence of $N$-particle systems of Brownian particles with logarithmic interaction potentials onto a system described by the infinite-dimensional stochastic differential equation (ISDE). For this proof we present two general theorems on the finite-particle approximations of interacting Brownian motions. In the first general theorem, we present a sufficient condition for a kind of tightness of solutions of stochastic differential equations (SDE) describing finite-particle systems, and prove that the limit points solve the corresponding ISDE. This implies, if in addition the limit ISDE enjoy a uniqueness of solutions, then the full sequence converges. We treat non-reversible case in the first main theorem. In the second general theorem, we restrict to the case of reversible particle systems and simplify the sufficient condition. We deduce the second theorem from the first. We apply the second general theorem to $\mathrm{Airy}_\beta$ interacting Brownian motion with $\beta=1, 2, 4$, and the Ginibre interacting Brownian motion. The former appears in the soft-edge limit of Gaussian (orthogonal/unitary/symplectic) ensembles in one spatial dimension, and the latter in the bulk limit of Ginibre ensemble in two spatial dimensions, corresponding to a quantum statistical system for which the eigen-value spectra belong to non-Hermitian Gaussian random matrices. The passage from the finite-particle stochastic differential equation (SDE) to the limit ISDE is a sensitive problem because the logarithmic potentials are long range and unbounded at infinity. Indeed, the limit ISDEs are not easily detectable from those of finite dimensions. Our general theorems can be applied straightforwardly to the grand canonical Gibbs measures with Ruelle-class potentials such as Lennard-Jones 6-12 potentials and and Riesz potentials.

Flow induced crystallisation of penetrable particles

For a system of Brownian particles interacting via a soft exponential potential we investigate the interaction between equilibrium crystallisation and spatially varying shear flow. For thermodynamic state points within the liquid part of the phase diagram, but close to the crystallisation phase boundary, we observe that imposing a Poiseuille flow can induce nonequilibrium crystalline ordering in regions of low shear gradient. The physical mechanism responsible for this phenomenon is shear-induced particle migration, which causes particles to drift preferentially towards the center of the flow channel, thus increasing the local density in the channel center. The method employed is classical dynamical density functional theory.

Static structure of active Brownian hard disks

We explore the changes in static structure of a two-dimensional system of active Brownian particles (ABP) with hard-disk interactions, using event-driven Brownian dynamics simulations. In particular, the effect of the self-propulsion velocity and the rotational diffusivity on the orientationally-averaged fluid structure factor is discussed. Typically activity increases structural ordering and generates a structure factor peak at zero wave vector which is a precursor of motility-induced phase separation. Our results provide reference data to test future statistical theories for the fluid structure of active Brownian systems.This manuscript was submitted for the special issue of the Journal of Physics: Condensed Matter associated with the Liquid Matter Conference 2017.

Active cluster crystals

We study the appearance and properties of cluster crystals (solids in which the unit cell is occupied by a cluster of particles) in a two-dimensional system of self-propelled active Brownian particles with repulsive interactions. Self-propulsion deforms the clusters by depleting particle density inside, and for large speeds it melts the crystal. Continuous field descriptions at several levels of approximation allow us to identify the relevant physical mechanisms.

Non-equilibrium surface tension of the vapour-liquid interface of active Lennard-Jones particles.

We study a three-dimensional system of self-propelled Brownian particles interacting via the Lennard-Jones potential. Using Brownian dynamics simulations in an elongated simulation box, we investigate the steady states of vapour-liquid phase coexistence of active Lennard-Jones particles with planar interfaces. We measure the normal and tangential components of the pressure tensor along the direction perpendicular to the interface and verify mechanical equilibrium of the two coexisting phases. In addition, we determine the non-equilibrium interfacial tension by integrating the difference of the normal and tangential components of the pressure tensor and show that the surface tension as a function of strength of particle attractions is well fitted by simple power laws. Finally, we measure the interfacial stiffness using capillary wave theory and the equipartition theorem and find a simple linear relation between surface tension and interfacial stiffness with a proportionality constant characterized by an effective temperature.

Large Deviations for Brownian Particle Systems with Killing

Particle approximations for certain nonlinear and nonlocal reaction-diffusion equations are studied using a system of Brownian motions with killing. The system is described by a collection of i.i.d. Brownian particles where each particle is killed independently at a rate determined by the empirical sub-probability measure of the states of the particles alive. A large deviation principle (LDP) for such sub-probability measure-valued processes is established. Along the way a convenient variational representation, which is of independent interest, for expectations of nonnegative functionals of Brownian motions together with an i.i.d. sequence of random variables is established. Proof of the LDP relies on this variational representation and weak convergence arguments.

On the role of adhesion in single-file dynamics

For a one-dimensional interacting system of Brownian particles with hard-core interactions (a single-file model), we study the effect of adhesion on both the collective diffusion (diffusion of the entire system with respect to its center of mass) and the tracer diffusion (diffusion of the individual tagged particles). For the case with no adhesion, all properties of these particle systems that are independent of particle labeling (symmetric in all particle coordinates and velocities) are identical to those of non-interacting particles (Lebowitz and Percus, 1967). We clarify this last fact twice. First, we derive our analytical predictions that show that the probability-density functions of single-file (ρsf) and ordinary (ρord) diffusion are identical, ρsf=ρord, predicting a nonanomalous (ordinary) behavior for the collective single-file diffusion, where the average second moment with respect to the center of mass, 〈x(t)2〉, is calculated from ρ for both diffusion processes. Second, for single-file diffusion, we show, both analytically and through large-scale simulations, that 〈x(t)2〉 grows linearly with time, confirming the nonanomalous behavior. This nonanomalous collective behavior comes in contrast to the well-known anomalous sub-diffusion behavior of the individual tagged particles (Harris, 1965). We introduce adhesion to single-file dynamics as a second inter-particle interaction rule and, interestingly, we show that adding adhesion does reduce the magnitudes of both 〈x(t)2〉 and the mean square displacement per particle Δx2; but the diffusion behavior remains intact independent of adhesion in both cases. Moreover, we study the dependence of both the collective diffusion constant D and the tracer diffusion constant DT on the adhesion coefficient α.

Evaluating linear response in active systems with no perturbing field

We present a method for the evaluation of time-dependent linear response functions for systems of active particles propelled by a persistent (colored) noise from unperturbed simulations. The method is inspired by the Malliavin weights sampling method proposed by Warren and Allen (Phys. Rev. Lett., 109 (2012) 250601) for out-of-equilibrium systems of passive Brownian particles. We illustrate our method by evaluating two linear response functions for a single active particle in an external harmonic potential. As an application, we calculate the time-dependent mobility function and an effective temperature, defined through the Einstein relation between the self-diffusion and mobility coefficients, for a system of many active particles interacting via a screened Coulomb potential. We find that this effective temperature decreases with increasing persistence time of the self-propulsion. Initially, for not too large persistence times, it changes rather slowly, but then it decreases markedly when the persistence length of the self-propelled motion becomes comparable with the particle size.

Phase separation in driven granular gases: exploring the elusive character of nonequilibrium steady states

The emergence of patterns and phase separation in many-body systems far from thermal equilibrium is discussed using the example of driven granular gases. It is shown that phase separation follows a similar mechanism as in the systems of active Brownian particles. Depending on the quantities chosen for observation, it may or may not be easy to find functionals analogous to the free energy in equilibrium statistical physics. We argue that although such functionals can always be derived from the dynamics, it is of only limited value for predicting relevant aspects of the nonequilibrium steady state of the system. Consequently, although there is indeed a 'principle' governing the selection of collective nonequilibrium steady states (and the corresponding large deviation functional can be identified), it is not generally useful for predicting the behaviour of the system.

Two-Sided Infinite Systems of Competing Brownian Particles

Two-sided infinite systems of Brownian particles with rank-dependent dynamics, indexed by all integers, exhibit different properties from their one-sided infinite counterparts, indexed by positive integers, and from finite systems. Consider the gap process, which is formed by spacings between adjacent particles. In stark contrast with finite and one-sided infinite systems, two-sided infinite systems can have one- or two-parameter family of stationary gap distributions, or the gap process weakly converging to zero as time goes to infinity.

Propagation of chaos for large Brownian particle system with Coulomb interaction

We investigate a system of N Brownian particles with the Coulomb interaction in any dimension \(d\ge 2\), and we assume that the initial data are independent and identically distributed with a common density \(\rho _0\) satisfying \(\int _{\mathbb {R}^{d}}\rho _0\ln \rho _0\,\hbox {d}x<\infty \) and \(\rho _0\in L^{\frac{2d}{d+2}} (\mathbb {R}^{d}) \cap L^1(\mathbb {R}^{d}, (1+|x|^2)\,\hbox {d}x)\). We prove that there exists a unique global strong solution for this interacting partsicle system and there is no collision among particles almost surely. For \(d=2\), we rigorously prove the propagation of chaos for this particle system globally in time without any cutoff in the following sense. When \(N\rightarrow \infty \), the empirical measure of the particle system converges in law to a probability measure and this measure possesses a density which is the unique weak solution to the mean-field Poisson–Nernst–Planck equation of single component.

Glassy dynamics of Brownian particles with velocity-dependent friction.

We consider a two-dimensional model system of Brownian particles in which slow particles are accelerated while fast particles are damped. The motion of the individual particles is described by a Langevin equation with Rayleigh-Helmholtz velocity-dependent friction. In the case of noninteracting particles, the time evolution equations lead to a non-Gaussian velocity distribution. The velocity-dependent friction allows negative values of the friction or energy intakes by slow particles, which we consider active motion, and also causes breaking of the fluctuation dissipation relation. Defining the effective temperature proportional to the second moment of velocity, it is shown that for a constant effective temperature the higher the noise strength, the lower the number of active particles in the system. Using the Mori-Zwanzig formalism and the mode-coupling approximation, the equations of motion for the density autocorrelation function are derived. The equations are solved using the equilibrium structure factors. The integration-through-transients approach is used to derive a relation between the structure factor in the stationary state considering the interacting forces, and the conventional equilibrium static structure factor.

Calculation of displacement correlation tensor indicating vortical cooperative motion in two-dimensional colloidal liquids.

As an indicator of cooperative motion in a system of Brownian particles that models two-dimensional colloidal liquids, a displacement correlation tensor is calculated analytically and compared with numerical results. The key idea for the analytical calculation is to relate the displacement correlation tensor, which is a kind of four-point space-time correlation, to the Lagrangian two-time correlation of the deformation gradient tensor. Tensorial treatment of the statistical quantities, including the displacement correlation itself, allows capturing the vortical structure of the cooperative motion. The calculated displacement correlation also implies a negative long-time tail in the velocity autocorrelation, which is a manifestation of the cage effect. Both the longitudinal and transverse components of the displacement correlation are found to be expressible in terms of a similarity variable, suggesting that the cages are nested to form a self-similar structure in the space-time.

Systems of Brownian particles with asymmetric collisions

Nous etudions des systemes de particules browniennes sur l’axe reel qui interagissent de facon asymetrique en fonction de leurs temps locaux de collision. Nous prouvons l’existence et l’unicite au sens fort de tels processus et les identifions avec un systeme de particules constitue d’une famille de processus browniens ordonnes ou les coefficients de derive et de diffusion, ainsi que les temps locaux de collision entre les particules, dependent de leurs rangs. Ces systemes browniens peuvent etre compris comme des generalisations de processus stochastiques issus des marches boursiers et de la gestion de portefeuilles. Nous pouvons pallier certaines lacunes de ces modeles tout en conservant la possibilite d’effectuer des calculs explicites. Nous montrons aussi, qu’en plus de leur interet intrinseque, de tels systemes de particules browniennes apparaissent comme des limites universelles de processus de sauts sur un reseau avec des interactions locales. Une etape clef dans la preuve est l’analyse d’une generalisation des applications de Skorokhod qui inclut les temps locaux a l’intersection des faces de l’orthant positif. Le resultat generalise la convergence du processus d’exclusion simple totalement asymetrique (TASEP) vers sa version continue. Finalement, nous identifions parmi les systemes de particules browniennes ceux qui possedent une structure probabiliste determinantale.

Configurational entropy and effective temperature in systems of active Brownian particles.

We propose a method to determine the effective density of states and configurational entropy in systems of active Brownian particles by measuring the probability distribution function of potential energy at varying temperatures. Assuming that the entropy is a continuous and monotonically increasing function of energy, we provide support that two-dimensional systems of purely repulsive active Brownian spheres can be mapped onto an equilibrium system with a Boltzmann-like distribution and an effective temperature. We find that the effective temperature depends even for a large number of particles on system size, suggesting that active systems are non-extensive. In addition, the effective Helmholtz free energy can be derived from the configurational entropy. We verify our results regarding the configurational entropy by using thermodynamic integration of the effective Helmholtz free energy with respect to temperature.

Displacement correlation as an indicator of collective motion in one-dimensional and quasi-one-dimensional systems of repulsive Brownian particles

While the slow dynamics in glassy liquids are known to be accompanied by collective motions undetectable with static structure factor and requiring four-point space-time correlations for their detection, it is usually difficult to calculate such correlations analytically. In the present study, a system of Brownian particles in a (quasi-)one-dimensional passageway is taken as an example to demonstrate the usefulness of displacement correlation. In the purely one-dimensional case (known as the single-file diffusion) with overtaking forbidden, the diffusion slows down and collective motion is captured by displacement correlation both calculated here numerically and analytically. On the other hand, displacement correlation vanishes if overtaking is allowed, which leads to normal diffusion.