Brownian Systems Research Articles

Long ranged stress correlations in the hard sphere liquid.

The smooth emergence of shear elasticity is a hallmark of the liquid to glass transition. In a liquid, viscous stresses arise from local structural rearrangements. In the solid, Eshelby has shown that stresses around an inclusion decay as a power law r-D, where D is the dimension of the system. We study glass-forming hard sphere fluids by simulation and observe the emergence of the unscreened power-law Eshelby pattern in the stress correlations of the isotropic liquid state. By a detailed tensorial analysis, we show that the fluctuating force field, viz., the divergence of the stress field, relaxes to zero with time in all states, while the shear stress correlations develop spatial power-law structures inside regions that grow with longitudinal and transverse sound propagation. We observe the predicted exponents r-D and r-D-2. In Brownian systems, shear stresses relax diffusively within these regions, with the diffusion coefficient determined by the shear modulus and the friction coefficient.

Extremal invariant distributions of infinite Brownian particle systems with rank dependent drifts

Extremal invariant distributions of infinite Brownian particle systems with rank dependent drifts

Stretched-exponential relaxation in weakly confined Brownian systems through large deviation theory.

Stretched-exponential relaxation is a widely observed phenomenon found in ordered ferromagnets as well as glassy systems. One modeling approach connects this behavior to a droplet dynamics described by an effective Langevin equationfor the droplet radius with an r^{2/3} potential. Here, we study a Brownian particle under the influence of a general confining, albeit weak, potential field that grows with distance as a sublinear power law. We find that for this memoryless model, observables display stretched-exponential relaxation. The probability density function of the system is studied using a rate-function ansatz. We obtain analytically the stretched-exponential exponent along with an anomalous power-law scaling of length with time. The rate function exhibits a point of nonanalyticity, indicating a dynamical phase transition. In particular, the rate function is double valued both to the left and right of this point, leading to four different rate functions, depending on the choice of initial conditions and symmetry.

Optimal policy for two-stage Brownian inventory systems under the long-run average cost criterion

Optimal policy for two-stage Brownian inventory systems under the long-run average cost criterion

Brownian-molecular dynamic correspondence in two-dimensional hard-disk fluids

It has been both experimentally and theoretically observed that the structural and dynamical properties of glass-forming fluids can be scaled and described within the dynamic universality class of hard spheres. One of the key predictions of this universality is the long-time dynamic equivalence between Brownian and Molecular dynamics systems governed by the same kind of interaction potential. However, the application and extent of this principle has not been validated for two-dimensional systems. To this end, we employ Brownian and Molecular Dynamics simulations to validate this principle in the two-dimensional case, analysing the long-time properties of systems made up of monodisperse hard disks. By introducing a suitable density-dependent scaling factor, we successfully extend the principle of dynamic equivalence to two dimensions. This dynamic correspondence will facilitate the interpretation of long-time molecular dynamic properties in the context of colloidal dynamics. For instance, this correspondence will allow us to establish the extension of the colloidal freezing criterion to the molecular case.

Linear growth of circuit complexity from Brownian dynamics

How rapidly can a many-body quantum system generate randomness? Using path integral methods, we demonstrate that Brownian quantum systems have circuit complexity that grows linearly with time. In particular, we study Brownian clusters of N spins or fermions with time-dependent all-to-all interactions, and calculate the Frame Potential to characterize complexity growth in these models. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using path integral methods. Within this framework it is straightforward to show that the kth Frame Potential comes within ϵ of the Haar value after a time of order t ~ kN + k log k + log ϵ−1. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary k-design after a time of order t ~ kN. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a k-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are analytically tractable and are directly applicable to practical applications calling for unitary k-designs.

Dynamics of semiflexible ring polymer in chiral Brownian particles bath

Molecular dynamics simulations are used to investigate the dynamic behavior of ring polymer in a bath containing a high concentration of chiral Brownian particles. The chiral Brownian particles around the chain drive the ring polymer to deform, and torque acting on ring is produced and drive the ring to rotate in one direction. Furthermore, the relationship between the ring's rotation speed and the angular velocity of the surrounding Brownian particles is not monotonically linear. The rotation of the ring chain increases initially, then decreases as the angular velocity of the Brownian particle increases. These findings offer a theoretical framework for investigating the conformation and motion properties of polymer macromolecules in Brownian particle systems.

Gutzwiller approximation for indistinguishable interacting Brownian particles on a lattice.

Nonequilibrium Brownian systems can be described using a creation and annihilation operator formalism for classical indistinguishable particles. This formalism has recently been used to derive a many-body master equationfor Brownian particles on a lattice with interactions of arbitrary strength and range. One advantage of this formalism is the possibility of using solution methods for analogous many-body quantum systems. In this paper, we adapt the Gutzwiller approximation for the quantum Bose-Hubbard model to the many-body master equationfor interacting Brownian particles in a lattice in the large-particle limit. Using the adapted Gutzwiller approximation, we numerically explore the complex behavior of nonequilibrium steady-state drift and number fluctuations throughout the full range of interaction strengths and densities for on-site and nearest-neighbor interactions.

Complex optical transport, dynamics, and rheology of intermediately attractive emulsions

Introducing short-range attractions in Brownian systems of monodisperse colloidal spheres can substantially impact their structures and consequently their optical transport and rheological properties. Here, for size-fractionated colloidal emulsions, we show that imposing an intermediate strength of attraction, well above but not much larger than thermal energy (approx 5.6k_{textrm{B}}T), through micellar depletion leads to a striking notch in the measured inverse mean free path of optical transport, 1/ell ^*, as a function of droplet volume fraction, phi. This notch, which appears between the hard-sphere glass transition, phi _{textrm{g}}, and maximal random jamming, phi _{textrm{MRJ}}, implies the existence of a greater population of compact dense clusters of droplets, as compared to tenuous networks of droplets in strongly attractive emulsion gels. We extend a prior decorated core-shell network model for strongly attractive colloidal systems to include dense non-percolating clusters that do not contribute to shear rigidity. By constraining this extended model using the measured 1/ell ^*(phi ), we improve and expand the microrheological interpretation of diffusing wave spectroscopy (DWS) experiments made on attractive colloidal systems. Our measurements and modeling demonstrate richness and complexity in optical transport and shear rheological properties of dense, disordered colloidal systems having short-range intermediate attractions between moderately attractive glasses and strongly attractive gels.

Phase Separation in Systems of Interacting Active Brownian Particles

The aim of this paper is to discuss the mathematical modeling of Brownian active particle systems, a recently popular paradigmatic system for self-propelled particles. We present four microscopic models with different types of repulsive interactions between particles and their associated macroscopic models, which are formally obtained using different coarse-graining methods. The macroscopic limits are integro-differential equations for the density in phase space (positions and orientations) of the particles and may include nonlinearities in both the diffusive and advective components. In contrast to passive particles, systems of active particles can undergo phase separation without any attractive interactions, a mechanism known as motility-induced phase separation (MIPS). We explore the onset of such a transition for each model in the parameter space of occupied volume fraction and P\'eclet number via a linear stability analysis and numerical simulations at both the microscopic and macroscopic levels. We establish that one of the models, namely the mean-field model which assumes long-range repulsive interactions, cannot explain the emergence of MIPS. In contrast, MIPS is observed for the remaining three models that assume short-range interactions that localize the interaction terms in space.

Linear Response Theory of Scale-Dependent Viscoelasticity for Overdamped Brownian Particle Systems

We show the linear response theory of spatial-scale-dependent relaxation moduli for overdamped Brownian particle systems. We employ the Irving-Kirkwood stress tensor field as the microscopic stress tensor field. We show that the scale-dependent relaxation modulus tensor, which characterizes the response of the stress tensor field to the applied velocity gradient field, can be expressed by using the correlation function of the Irving-Kirkwood stress tensor field. The spatial Fourier transform of the relaxation modulus tensor gives the wavenumber-dependent relaxation modulus. For isotropic and homogeneous systems, the relaxation modulus tensor has only two independent components. The transverse and longitudinal deformation modes give the wavenumber-dependent shear relaxation modulus and the wavenumber-dependent bulk relaxation modulus. As simple examples, we derive the explicit expressions for the relaxation moduli for two simple models the non-interacting Brownian particles and the harmonic dumbbell model.

Shape-driven, emergent behavior in active particle mixtures

Active particle systems can vary greatly from one-component systems of spheres to mixtures of particle shapes at different composition ratios. We investigate computationally the combined effect of anisotropy and stoichiometry on the collective behavior of two-dimensional active colloidal mixtures of polygons. We uncover three emergent phenomena not yet reported in active Brownian particle systems. First, we find that mixtures containing hexagons exhibit micro-phase separation with large grains of hexagonal symmetry. We quantify a measurable, implicit ‘steric attraction’ between the active particles as a result of shape anisotropy and activity. This calculation provides further evidence that implicit interactions in active systems, even without explicit attraction, can lead to an effective preferential attraction between particles. Next, we report stable fluid clusters in mixtures containing one triangle or square component. We attribute the fluidization of the dense cluster to the interplay of cluster destabilizing particles, which introduce grain boundaries and slip planes into the system, causing solid-like clusters to break up into fluid clusters. Third, we show that fluid clusters can coexist with solid clusters within a sparse gas of particles in a steady state of three coexisting phases. Our results highlight the potential for a wide variety of behavior to be accessible to active matter systems and establish a route to control active colloidal systems through simple parameter designs.

Minimal dissipation protocols of an instantaneous equilibrium Brownian particle under time-dependent temperature and potential variations

We consider the instantaneous equilibrium (ieq) transition of an underdamped Brownian particle under arbitrary time-dependent temperature and potential variations and derive analytic results for the temperature and potential protocols for minimal dissipative work. Explicit results for the time-dependent minimal dissipation protocols and the associated energetics are obtained for the cases of pure temperature variation, pure potential parameter (stiffness) variation, and isentropic ieq processes. The minimal dissipation condition enforces the forward and backward protocols to be time-reversal related, with the same minimal dissipative work. Remarkably, it is shown that the minimal dissipation path is also the isentropic ieq transition. The energetics in the mdieq transitions are analyzed in detail with emphasis on the conditions for maximizing the mean work and power that can be extracted. Furthermore, exact results for the overdamped limit are derived, indicating that there is great freedom to choose the minimal dissipation protocols, thus allowing the realization of mdieq transitions in Brownian colloidal systems with relative ease.

An infinite set of integral formulae for polar, nematic, and higher order structures at the interface of motility-induced phase separation

Motility-induced phase separation (MIPS) is a purely non-equilibrium phenomenon in which self-propelled particles phase separate without any attractive interactions. One surprising feature of MIPS is the emergence of polar, nematic, and higher order structures at the interfacial region, whose underlying physics remains poorly understood. Starting with a model of MIPS in which all many-body interactions are captured by an effective speed function and an effective pressure function that depend solely on the local particle density, I derive analytically an infinite set of integral formulae for the ordering structures at the interface. I then demonstrate that half of these IF are in fact exact for generic active Brownian particle systems. Finally, I test these integral formulae by applying them to numerical data from direct particle dynamics simulation and find that they remain valid to a great extent.

Dynamic Entropy of Two-Dimensional Active Brownian Systems in Colloidal Plasmas.

We analyze the experimental data on the motion of active Brownian micrograins in RF discharge plasmas. In the experiments, two types of microparticles were used: first—plastic grains fully covered with metal, and second—Janus particles with a thin metal cap. We have tracked the trajectories of the separate grains and plotted the pair correlation functions of the observed structures. To examine the motion of the grains, we studied the dependencies of the MFPT dynamic entropy on the coarsening parameter, the fractal dimension of the system on its mean kinetic temperature, and the mean localization area of the grain on its mean kinetic temperature. Based on the obtained results, we conclude that the character of motion of our active Brownian systems changes as the power of an illuminating laser (and, therefore, the mean kinetic temperature of the grains) increases. Janus particles change their trajectories from more chaotic to spiral-like ones; in the case of fully covered particles, we observe the dynamical phase transition from the more ordered structure to the less ordered one.

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.

Steady states of active Brownian particles interacting with boundaries

An active Brownian particle is a minimal model for a self-propelled colloid in a dissipative environment. Experiments and simulations show that, in the presence of boundaries and obstacles, active Brownian particle systems approach nontrivial nonequilibrium steady states with intriguing phenomenology, such as accumulation at boundaries, ratchet effects, and long-range depletion interactions. Nevertheless, theoretical analysis of these phenomena has proven difficult. Here, we address this theoretical challenge in the context of non-interacting particles in two dimensions, basing our analysis on the steady-state Smoluchowski equation for the one-particle distribution function. Our primary result is an approximation strategy that connects asymptotic solutions of the Smoluchowski equation to boundary conditions. We test this approximation against the exact analytic solution in a 2D planar geometry, as well as numerical solutions in circular and elliptic geometries. We find good agreement so long as the boundary conditions do not vary too rapidly with respect to the persistence length of particle trajectories. Our results are relevant for characterizing long-range flows and depletion interactions in such systems. In particular, our framework shows how such behaviors are connected to the breaking of detailed balance at the boundaries.

Rate of Entropy Production in Stochastic Mechanical Systems.

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker–Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.

Phase behavior of rotationally asymmetric Brownian kites containing 90° internal angles* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11874277 and 21621004, and 11704276) and the Natural Science Foundation of Tianjin, China (Grant No. 19JCQNJC14900).

Previous Monte Carlo simulations have shown that ordered tetratic phases can emerge in a dense two-dimensional Brownian system of rotationally asymmetric hard kites having 90° internal angles. However, there have been no experimental investigations yet to compare with these simulation results. Here, we have fabricated two types of micron-sized kites having internal angles of 72°–90°–108°–90° and 72°–99°–90°–99°, respectively, and we have experimentally studied their phase behavior in two-dimensional systems. Interestingly and in contrast to the Monte Carlo simulations, the experimental results show a phase sequence of isotropic fluid-hexagonal rotator crystal-square crystal as the area fraction ϕ A increases for both types of kites. The observed square crystal displays not only a quasi-long-range translational order but also (quasi-)long-range 4-fold bond- and molecular-orientational order; these characteristics confirm that tetratic order can emerge even in dense Brownian systems of rotationally asymmetric particles. A model based on local polymorphic configurations (LPCs) is proposed to understand the origin of the square lattice order in these dense kite systems. The results in this study provide a new route to realize custom-designed self-assembly of colloids by controlling LPCs.

Instantaneous equilibrium transition for Brownian systems under time-dependent temperature and potential variations: Reversibility, heat and work relations, and fast isentropic process

The theory of constructing instantaneous equilibrium (ieq) transition under arbitrary time-dependent temperature and potential variation for a Brownian particle is developed. It is shown that it is essential to consider the underdamped dynamics for temperature-changing transitions. The ieq is maintained by a time-dependent auxiliary position and momentum potential, which can be calculated for given time-dependent transition protocols. Explicit analytic results are derived for the work and heat statistics, energy, and entropy changes for harmonic and non-harmonic trapping potential with arbitrary time-dependent potential parameters and temperature protocols. Numerical solutions of the corresponding Langevin dynamics are computed to confirm the theoretical results. Although ieq transition of the reverse process is not the time-reversal of the ieq transition of the forward process due to the odd-parity of controlling parameters, their phase-space distribution functions restore the time-reversal symmetry, and hence the energy and entropy changes of the ieq of the reverse process are simply the negative of that of the forward process. Furthermore, it is shown that it is possible to construct an ieq transition that has zero entropy change at a finite transition rate, i.e., a fast ieq isentropic process, and is further demonstrated by explicit Langevin dynamics simulations. Our theory provides fundamental building blocks for designing controlled microscopic heat engine cycles. Implications for constructing an efficient Brownian heat engine are also discussed.