Transport coefficients in dense active Brownian particle systems: mode-coupling theory and simulation results

Utility of Brownian dynamics simulations in chemistry and biology: A comprehensive review

Brownian dynamics (BD) simulations, a powerful computer simulation tool that has gained significant attraction in investigating the intricate dynamics of chemical and biological systems. By meticulously modeling the diffusive motion of molecules and their intricate interactions, BD simulations offer invaluable insights into a diverse array of phenomena, including reaction kinetics, molecular transport, and biomolecular association. This comprehensive review delves into the utility of BD simulations within the realms of chemistry and biology. We meticulously explore the theoretical underpinnings of the technique, critically analyze its strengths and limitations, and showcase its diverse applications across various scientific domains. By providing a comprehensive analysis of the existing literature, we aim to illuminate the potential of BD simulations to significantly advance our understanding of complex chemical and biological systems, ultimately contributing to the development of innovative therapeutic solutions serving a broad range of biomedical applications.

Emergent mesoscale correlations in active solids with noisy chiral dynamics

We present the linear response theory for an elastic solid composed of active Brownian particles with intrinsic individual chirality, deriving both a normal mode formulation and a continuum elastic formulation. Using this theory, we compute analytically the velocity correlations and energy spectra under different conditions, showing an excellent agreement with simulations. We generate the corresponding phase diagram, identifying chiral and achiral disordered regimes (for high chirality or noise levels), as well as chiral and achiral states with mesoscopic-range order (for low chirality and noise). The chiral ordered states display mesoscopic spatial correlations and oscillating time correlations, but no wave propagation. In the high chirality regime, we find a peak in the elastic energy spectrum that leads to a non-monotonic behavior with increasing noise strength that is consistent with the emergence of the 'hammering state' recently identified in chiral glasses. Finally, we show numerically that our theory, despite its linear response nature, can be applied beyond the idealized homogeneous solid assumed in our derivations. Indeed, by increasing the level of activity, we show that it remains a good approximation of the system dynamics until just below the melting transition. In addition, we show that there is still an excellent agreement between our analytical results and simulations when we extend our results to heterogeneous solids composed of mixtures of active particles with different intrinsic chirality and noise levels. The derived linear response theory is therefore robust and applicable to a broad range of real-world active systems. Our work provides a thorough analytical and numerical description of the emergent states in a densely packed system of chiral self-propelled Brownian disks, thus allowing a detailed understanding of the phases and dynamics identified in a minimal chiral active system.

Anomalous diffusion of active Brownian particles in responsive elastic gels: Nonergodicity, non-Gaussianity, and distributions of trapping times

Understanding actual transport mechanisms of self-propelled particles (SPPs) in complex elastic gels-such as in the cell cytoplasm, in in vitro networks of chromatin or of F-actin fibers, or in mucus gels-has far-reaching consequences. Implications beyond biology/biophysics are in engineering and medicine, with a particular focus on microrheology and on targeted drug delivery. Here, we examine via extensive computer simulations the dynamics of SPPs in deformable gellike structures responsive to thermal fluctuations. We treat tracer particles comparable to and larger than the mesh size of the gel. We observe distinct trapping events of active tracers at relatively short times, leading to subdiffusion; it is followed by an escape from meshwork-induced traps due to the flexibility of the network, resulting in superdiffusion. We thus find crossovers between different transport regimes. We also find pronounced nonergodicity in the dynamics of SPPs and non-Gaussianity at intermediate times. The distributions of trapping times of the tracers escaping from "cages" in our quasiperiodic gel often reveal the existence of two distinct timescales in the dynamics. At high activity of the tracers these timescales become comparable. Furthermore, we find that the mean waiting time exhibits a power-law dependence on the activity of SPPs (in terms of their Péclet number). Our results additionally showcase both exponential and nonexponential trapping events at high activities. Extensions of this setup are possible, with the factors such as anisotropy of the particles, different topologies of the gel network, and various interactions between the particles (also of a nonlocal nature) to be considered.

Extremely persistent dense active fluids

We study the dynamics of dense three-dimensional systems of active particles for large persistence times τp at constant average self-propulsion force f. These systems are fluid counterparts of previously investigated extremely persistent systems, which in the large persistence time limit relax only on the time scale of τp. We find that many dynamic properties of the systems we study, such as the mean-squared velocity, the self-intermediate scattering function, and the shear-stress correlation function, become τp-independent in the large persistence time limit. In addition, the large τp limits of many dynamic properties, such as the mean-square velocity and the relaxation times of the scattering function, and the shear-stress correlation function, depend on f as power laws with non-trivial exponents. We conjecture that these systems constitute a new class of extremely persistent active systems.

Rescaled mode-coupling scheme for the quantitative description of experimentally observed colloid dynamics

We describe experimentally observed collective dynamics in colloidal suspensions of model hard-sphere particles using a modified mode coupling theory (MCT). This rescaled MCT is capable of describing quantitatively the wave-vector and time-dependent diffusion in these systems. Intermediate scattering functions of liquidlike structured dispersions are determined by means of static and dynamic light-scattering experiments. The structure and short-time dynamics of the systems can be described quantitatively employing a multicomponent Percus-Yevick ansatz for the partial structure factors and an effective, one-component description of hydrodynamic interactions based on the semianalytical δγ expansion. Combined with a recently proposed empirical modification of MCT in which memory functions are calculated using effective structure factors at rescaled number densities, the scheme is able to model the collective dynamics over the entire accessible time and wave-vector range and predicts the volume-fraction-dependence of long-time self-diffusion coefficients and the zero-shear viscosity quantitatively. This highlights the potential of MCT as a practical tool for the quantitative analysis and prediction of experimental observations.

Microscopic theory for nonequilibrium correlation functions in dense active fluids

One of the key hallmarks of dense active matter in the liquid, supercooled, and solid phases is the so-called equal-time velocity correlations. Crucially, these correlations can emerge spontaneously, i.e., they require no explicit alignment interactions, and therefore represent a generic feature of dense active matter. This indicates that for a meaningful comparison or possible mapping between active and passive liquids one not only needs to understand their structural properties, but also the impact of these velocity correlations. This has already prompted several simulation and theoretical studies, though they are mostly focused on athermal systems and thus overlook the effect of translational diffusion. Here, we present a fully microscopic method to calculate nonequilibrium correlations in two-dimensional systems of thermal active Brownian particles (ABPs). We use the integration through transients formalism together with (active) mode-coupling theory and analytically calculate qualitatively consistent static structure factors and active velocity correlations. We complement our theoretical results with simulations of both thermal and athermal ABPs which exemplify the disruptive role that thermal noise has on velocity correlations.

Fluid-Glass-Jamming Rheology of Soft Active Brownian Particles

We numerically study the shear rheology of a binary mixture of soft active Brownian particles, from the fluid to the disordered solid regime. At low shear rates, we find a Newtonian regime, where a Green-Kubo relation with an effective temperature provides the linear viscosity. It is followed by a shear-thinning regime at high shear rates. At high densities, solidification is signaled by the emergence of a finite yield stress. We construct a "fluid-glass-jamming" phase diagram with activity replacing temperature. While both parameters gauge fluctuations, activity also changes the exponent characterizing the decay of the diffusivity close to the glass transition and the shape of the yield stress surface. The dense disordered active solid appears to be mostly dominated by athermal jamming rather than glass rheology.

Computer simulations and mode-coupling theory of glass-forming confined hard-sphere fluids

We present mode-coupling theory (MCT) results for densely packed hard-sphere fluids confined between two parallel walls and compare them quantitatively to computer simulations. The numerical solution of MCT is calculated using the full system of matrix-valued integro-differential equations. We investigate several dynamical properties of supercooled liquids including scattering functions, frequency-dependent susceptibilities, and mean-square displacements. Close to the glass transition, we find quantitative agreement between the coherent scattering function predicted from theory and that evaluated from simulations, which enables us to make quantitative statements on caging and relaxation dynamics of the confined hard-sphere fluid.

Glassy Dynamics in Chiral Fluids

Chiral active matter is enjoying a rapid increase of interest, spurred by the rich variety of asymmetries that can be attained in, e.g., the shape or self-propulsion mechanism of active particles. Though this has already led to the observance of so-called chiral crystals, active chiral glasses remain largely unexplored. A possible reason for this could be the naive expectation that interactions dominate the glassy dynamics and the details of the active motion become increasingly less relevant. Here, we show that quite the opposite is true by studying the glassy dynamics of interacting chiral active Brownian particles. We demonstrate that when our chiral fluid is pushed to glassy conditions, it exhibits highly nontrivial dynamics, especially compared to a standard linear active fluid such as common active Brownian particles. Despite the added complexity, we are still able to present a full rationalization for all identified dynamical regimes. Most notably, we introduce a new "hammering" mechanism, unique to rapidly spinning particles in high-density conditions, that can fluidize a chiral active solid.

Mode-coupling theory for the dynamics of dense underdamped active Brownian particle system

We present a theory to study the inertial effect on glassy dynamics of the underdamped active Brownian particle (UABP) system. Using the assumption of the nonequilibrium steady-state, we obtain an effective Fokker-Planck equation for the probability distribution function (PDF) as a function of positions and momentums. With this equation, we achieve the evolution equation of the intermediate scattering function through the Zwanzig-Mori projection operator method and the mode-coupling theory (MCT). Theoretical analysis shows that the inertia of the particle affects the memory function and corresponding glass transition by influencing the structure factor and a velocity correlation function. The theory provides theoretical support and guidance for subsequent simulation work.

Active glassy dynamics is unaffected by the microscopic details of self-propulsion

Recent years have seen a rapid increase of interest in dense active materials, which, in the disordered state, share striking similarities with the conventional passive glass-forming matter. For such passive glassy materials, it is well established (at least in three dimensions) that the details of the microscopic dynamics, e.g., Newtonian or Brownian, do not influence the long-time glassy behavior. Here, we investigate whether this still holds true in the non-equilibrium active case by considering two simple and widely used active particle models, i.e., active Ornstein-Uhlenbeck particles (AOUPs) and active Brownian particles (ABPs). In particular, we seek to gain more insight into the role of the self-propulsion mechanism on the glassy dynamics by deriving a mode-coupling theory (MCT) for thermal AOUPs, which can be directly compared to a recently developed MCT for ABPs. Both theories explicitly take into account the active degrees of freedom. We solve the AOUP- and ABP-MCT equations in two dimensions and demonstrate that both models give almost identical results for the intermediate scattering function over a large variety of control parameters (packing fractions, active speeds, and persistence times). We also confirm this theoretical equivalence between the different self-propulsion mechanisms numerically via simulations of a polydisperse mixture of active quasi-hard spheres, thereby establishing that, at least for these model systems, the microscopic details of self-propulsion do not alter the active glassy behavior.

Disordered Collective Motion in Dense Assemblies of Persistent Particles

We explore the emergence of nonequilibrium collective motion in disordered nonthermal active matter when persistent motion and crowding effects compete, using simulations of a two-dimensional model of size polydisperse self-propelled particles. In stark contrast with monodisperse systems, we find that polydispersity stabilizes a homogeneous active liquid at arbitrary large persistence times, characterized by remarkable velocity correlations and irregular turbulent flows. For all persistence values, the active fluid undergoes a nonequilibrium glass transition at large density. This is accompanied by collective motion, whose nature evolves from near-equilibrium spatially heterogeneous dynamics at small persistence, to a qualitatively different intermittent dynamics when persistence is large. This latter regime involves a complex time evolution of the correlated displacement field.

Anomalous Transport of Tracers in Active Baths

We derive the long-time dynamics of a tracer immersed in a one-dimensional active bath. In contrast to previous studies, we find that the damping and noise correlations possess long-time tails with exponents that depend on the tracer symmetry. For generic tracers, shape asymmetry induces ratchet effects that alter fluctuations and lead to superdiffusion and friction that grows with time when the tracer is dragged at a constant speed. In the singular limit of a completely symmetric tracer, we recover normal diffusion and finite friction. Furthermore, for small symmetric tracers, the active contribution to the friction becomes negative: active particles enhance motion rather than oppose it. These results show that, in low-dimensional systems, the motion of a passive tracer in an active bath cannot be modeled as a persistent random walker with a finite correlation time.

Cage Length Controls the Nonmonotonic Dynamics of Active Glassy Matter

Dense active matter is gaining widespread interest due to its remarkable similarity with conventional glass-forming materials. However, active matter is inherently out of equilibrium and even simple models such as active Brownian particles (ABPs) and active Ornstein-Uhlenbeck particles (AOUPs) behave markedly differently from their passive counterparts. Controversially, this difference has been shown to manifest itself via either a speedup, slowdown, or nonmonotonic change of the glassy relaxation dynamics. Here we rationalize these seemingly contrasting views on the departure from equilibrium by identifying the ratio of the short-time length scale to the cage length, i.e., the length scale of local particle caging, as a vital and unifying control parameter for active glassy matter. In particular, we explore the glassy dynamics of both thermal and athermal ABPs and AOUPs upon increasing the persistence time. We find that for all studied systems there is an optimum of the dynamics; this optimum occurs when the cage length coincides with the corresponding short-time length scale of the system, which is either the persistence length for athermal systems or a combination of the persistence length and a diffusive length scale for thermal systems. This new insight, for which we also provide a simple physical argument, allows us to reconcile and explain the manifestly disparate departures from equilibrium reported in many previous studies of dense active materials.

Tracer dynamics in crowded active-particle suspensions

We discuss the dynamics of active Brownian particles (ABPs) in crowded environments through the mean-squared displacement (MSD) of active and passive tracer particles in both active and passive host systems. Exact equations for the MSD are derived using a projection operator technique, extending to dense systems the known solution for a single ABP. The interaction of the tracer particle with the host particles gives rise to strong memory effects. Evaluating these approximately in the framework of a recently developed mode-coupling theory for active Brownian particles (ABP-MCT), we discuss the various dynamical regimes that emerge: While self-propelled motion gives rise to super-diffusive MSD, at high densities, this competes with an interaction-induced sub-diffusive regime. The predictions of the theory are shown to be in good agreement with results obtained from an event-driven Brownian dynamics (ED-BD) simulation scheme for the dynamics of two-dimensional active Brownian hard disks.

Generalized mode-coupling theory for mixtures of Brownian particles

Generalized mode-coupling theory (GMCT) has recently emerged as a promising first-principles theory to study the poorly understood dynamics of glass-forming materials. Formulated as a hierarchical extension of standard mode-coupling theory (MCT), it is able to systematically improve its predictions by including the exact dynamics of higher-order correlation functions into its hierarchy. However, in contrast to Newtonian dynamics, a fully generalized version of the theory based on Brownian dynamics is still lacking. To close this gap, we provide a detailed derivation of GMCT for colloidal mixtures obeying a many-body Smoluchowski equation. We demonstrate that a hierarchy of coupled equations can again be established and show that these, consistent with standard MCT, are identical to the ones obtained from Newtonian GMCT when taking the overdamped limit. Consequently, the nontrivial similarity between Brownian and Newtonian MCT is maintained for our multicomponent GMCT. As a proof of principle, we also solve the generalized mode-coupling equations for the binary Kob-Andersen Lennard-Jones mixture undergoing Brownian dynamics and confirm the improved predictive power of the theory upon using more levels of the GMCT hierarchy of equations.

Mode-coupling theory for tagged-particle motion of active Brownian particles

We derive a mode-coupling theory (MCT) to describe the dynamics of a tracer particle that is embedded in a dense system of active Brownian particles (ABPs) in two spatial dimensions. The ABP undergo translational and rotational Brownian motion and are equipped with a fixed self-propulsion speed along their orientational vector that describes their active motility. The resulting equations of motion for the tagged-particle density-correlation functions describe the various cases of tracer dynamics close to the glass transition: that of a single active particle in a glass-forming passive host suspensions, that of a passive colloidal particle in a suspension of ABP, and that of active tracers in a bath of active particles. Numerical results are presented for these cases assuming hard-sphere interactions among the particles. The qualitative and quantitative accuracy of the theory is tested against event-driven Brownian dynamics (ED-BD) simulations of active and passive hard disks. Simulation and theory are found in quantitative agreement, provided one adjusts the overall density (as known from the passive description of glassy dynamics), and allows for a rescaling of self-propulsion velocities in the active host system. These adjustments account for the fact that ABP-MCT generally overestimates the tendency for kinetic arrest. We confirm in the simulations a peculiar feature of the transient and stationary dynamical density-correlation functions regarding their lack of symmetry under time reversal, demonstrating the nonequilibrium nature of the system and how it manifests itself in the theory.

Collective excitations in active fluids: Microflows and breakdown in spectral equipartition of kinetic energy

The effect of particle activity on collective excitations in active fluids of microflyers is studied. With an in silico study, we observed an oscillating breakdown of equipartition (uniform spectral distribution) of kinetic energy in reciprocal space. The phenomenon is related to short-range velocity-velocity correlations that were realized without forming of long-lived mesoscale vortices in the system. This stands in contrast to well-known mesoscale turbulence operating in active nematic systems (bacterial or artificial) and reveals the features of collective dynamics in active fluids, which should be important for structural transitions and glassy dynamics in active matter.