Active matter beyond mean-field: ring-kinetic theory for self-propelled particles

Breakdown of Boltzmann-type models for the alignment of self-propelled rods

Studies in the collective motility of organisms use a range of analytical approaches to formulate continuous kinetic models of collective dynamics from rules or equations describing agent interactions. However, the derivation of these kinetic models often relies on Boltzmann's "molecular chaos" hypothesis, which assumes that correlations between individuals are short-lived. While this assumption is often the simplest way to derive tractable models, it is often not valid in practice due to the high levels of cooperation and self-organization present in biological systems. In this work, we illustrated this point by considering a general Boltzmann-type kinetic model for the alignment of self-propelled rods where rod reorientation occurs upon binary collisions. We examine the accuracy of the kinetic model by comparing numerical solutions of the continuous equations to an agent-based model that implements the underlying rules governing microscopic alignment. Even for the simplest case considered, our comparison demonstrates that the kinetic model fails to replicate the discrete dynamics due to the formation of rod clusters that violate statistical independence. Additionally, we show that introducing noise to limit cluster formation helps improve the agreement between the analytical model and agent simulations but does not restore the agreement completely. These results highlight the need to both develop and disseminate improved moment-closure methods for modeling biological and active matter systems.

A decade of thermostatted kinetic theory models for complex active matter living systems

In the last decade, the thermostatted kinetic theory has been proposed as a general paradigm for the modeling of complex systems of the active matter and, in particular, in biology. Homogeneous and inhomogeneous frameworks of the thermostatted kinetic theory have been employed for modeling phenomena that are the result of interactions among the elements, called active particles, composing the system. Functional subsystems contain heterogeneous active particles that are able to perform the same task, called activity. Active matter living systems usually operate out-of-equilibrium; accordingly, a mathematical thermostat is introduced in order to regulate the fluctuations of the activity of particles. The time evolution of the functional subsystems is obtained by introducing the conservative and the nonconservative interactions which represent activity-transition, natural birth/death, induced proliferation/destruction, and mutation of the active particles. This review paper is divided in two parts: In the first part the review deals with the mathematical frameworks of the thermostatted kinetic theory that can be found in the literature of the last decade and a unified approach is proposed; the second part of the review is devoted to the specific mathematical models derived within the thermostatted kinetic theory presented in the last decade for complex biological systems, such as wound healing diseases, the recognition process and the learning dynamics of the human immune system, the hiding-learning dynamics and the immunoediting process occurring during the cancer-immune system competition. Future research perspectives are discussed from the theoretical and application viewpoints, which suggest the important interplay among the different scholars of the applied sciences and the desire of a multidisciplinary approach or rather a theory for the modeling of every active matter system.

Nematic Torques in Scalar Active Matter: When Fluctuations Favor Polar Order and Persistence

We study the impact of nematic alignment on scalar active matter in the disordered phase. We show that nematic torques control the emergent physics of particles interacting via pairwise forces and can either induce or prevent phase separation. The underlying mechanism is a fluctuation-induced renormalization of the mass of the polar field that generically arises from nematic torques. The correlations between the fluctuations of the polar and nematic fields indeed conspire to increase the particle persistence length, contrary to what phenomenological computations predict. This effect is generic and our theory also quantitatively accounts for how nematic torques enhance particle accumulation along confining boundaries and opposes demixing in mixtures of active and passive particles.

Noisy multistate voter model for flocking in finite dimensions

We study a model for the collective behavior of self-propelled particles subject to pairwise copying interactions and noise. Particles move at a constant speed v on a two-dimensional space and, in a single step of the dynamics, each particle adopts the direction of motion of a randomly chosen neighboring particle within a distance R=1, with the addition of a perturbation of amplitude η (noise). We investigate how the global level of particles' alignment (order) is affected by their motion and the noise amplitude η. In the static case scenario v=0 where particles are fixed at the sites of a square lattice and interact with their first neighbors, we find that for any noise η>0 the system reaches a steady state of complete disorder in the thermodynamic limit, while for η=0 full order is eventually achieved for a system with any number of particles N. Therefore, the model displays a transition at zero noise when particles are static, and thus there are no ordered steady states for a finite noise (η>0). We show that the finite-size transition noise vanishes with N as η_{c}^{1D}∼N^{-1} and η_{c}^{2D}∼(NlnN)^{-1/2} in one- and two-dimensional lattices, respectively, which is linked to known results on the behavior of a type of noisy voter model for catalytic reactions. When particles are allowed to move in the space at a finite speed v>0, an ordered phase emerges, characterized by a fraction of particles moving in a similar direction. The system exhibits an order-disorder phase transition at a noise amplitude η_{c}>0 that is proportional to v, and that scales approximately as η_{c}∼v(-lnv)^{-1/2} for v≪1. These results show that the motion of particles is able to sustain a state of global order in a system with voter-like interactions.

Quantitative kinetic theory of flocking with three-particle closure

We consider aligning self-propelled particles in two dimensions. Their motion is given by Langevin equations and includes nonadditive N-particle interactions. The qualitative behavior is as for the famous Vicsek model. We develop a kinetic theory of flocking beyond mean field. In particular, we self-consistently take into account the full pair correlation function. We find excellent quantitative agreement of the pair correlations with direct agent-based simulations within the disordered regime. Furthermore we use a closure relation to incorporate spatial correlations of three particles. In that way we achieve good quantitative agreement of the onset of flocking with direct simulations. Compared to mean-field theory, the flocking transition is shifted significantly toward lower noise because directional correlations favor disorder. We compare our theory with a recently developed Landau-kinetic theory.

Multiple Particle Correlation Analysis of Many-Particle Systems: Formalism and Application to Active Matter

We introduce a fast spatial point pattern analysis technique that is suitable for systems of many identical particles giving rise to multiparticle correlations up to arbitrary order. The obtained correlation parameters allow us to quantify the quality of mean field assumptions or theories that incorporate correlations of limited order. We study the Vicsek model of self-propelled particles and create a correlation map marking the required correlation order for each point in phase space incorporating up to ten-particle correlations. We find that multiparticle correlations are important even in a large part of the disordered phase. Furthermore, the two-particle correlation parameter serves as an excellent order parameter to locate both phase transitions of the system, whereas two different order parameters were required before.

Transport coefficients of self-propelled particles: Reverse perturbations and transverse current correlations

The reverse perturbation method [Phys. Rev. E 59, 4894 (1999)1063-651X10.1103/PhysRevE.59.4894] for shearing simple liquids and measuring their viscosity is extended to the Vicsek model (VM) of active particles [Phys. Rev. Lett. 75, 1226 (1995)PRLTAO0031-900710.1103/PhysRevLett.75.1226] and its metric-free version. The sheared systems exhibit a phenomenon that is similar to the skin effect of an alternating electric current: Momentum that is fed into the boundaries of a layer decays mostly exponentially toward the center of the layer. It is shown how two transport coefficients, i.e., the shear viscosity ν and the momentum amplification coefficient λ, can be obtained by fitting this decay with an analytical solution of the hydrodynamic equations for the VM. The viscosity of the VM consists of two parts, a kinetic and a collisional contribution. While analytical predictions already exist for the former, a novel expression for the collisional part is derived by an Enskog-like kinetic theory. To verify the predictions for the transport coefficients, Green-Kubo relations were evaluated and transverse current correlations were measured in independent simulations. Not too far to the transition to collective motion, we find excellent agreement between the different measurements of the transport coefficients. However, the measured values of ν and 1-λ are always slightly higher than the mean-field predictions, even at large mean free paths and at state points quite far from the threshold to collective motion, that is, far in the disordered phase. These findings seem to indicate that the mean-field assumption of molecular chaos is much less reliable in systems with velocity-alignment rules such as the VM, compared to models obeying detailed balance such as multiparticle collision dynamics.

Contrarian compulsions produce exotic time-dependent flocking of active particles

Animals having a tendency to align their velocities to an average of those of their neighbors may flock as illustrated by the Vicsek model and its variants. If, in addition, they feel a systematic contrarian trend, the result may be a time periodic adjustment of the flock or period doubling in time. These exotic phases are predicted from kinetic theory and numerically found in a modified two-dimensional Vicsek model of self-propelled particles. Numerical simulations demonstrate striking effects of alignment noise on the polarization order parameter measuring particle flocking: maximum polarization length is achieved at an optimal nonzero noise level. When contrarian compulsions are more likely than conformist ones, nonuniform polarized phases appear as the noise surpasses threshold.

Three-body correlations and conditional forces in suspensions of active hard disks

Self-propelled Brownian particles show rich out-of-equilibrium physics, for instance, the motility-induced phase separation (MIPS). While decades of studying the structure of liquids have established a deep understanding of passive systems, not much is known about correlations in active suspensions. In this work we derive an approximate analytic theory for three-body correlations and forces in systems of active Brownian disks starting from the many-body Smoluchowski equation. We use our theory to predict the conditional forces that act on a tagged particle and their dependence on the propulsion speed of self-propelled disks. We identify preferred directions of these forces in relation to the direction of propulsion and the positions of the surrounding particles. We further relate our theory to the effective swimming speed of the active disks, which is relevant for the physics of MIPS. To test and validate our theory, we additionally run particle-resolved computer simulations, for which we explicitly calculate the three-body forces. In this context, we discuss the modeling of active Brownian swimmers with nearly hard interaction potentials. We find very good agreement between our simulations and numerical solutions of our theory, especially for the nonequilibrium pair-distribution function. For our analytical results, we carefully discuss their range of validity in the context of the different levels of approximation we applied. This discussion allows us to study the individual contribution of particles to three-body forces and to the emerging structure. Thus, our work sheds light on the collective behavior, provides the basis for further studies of correlations in active suspensions, and makes a step towards an emerging liquid state theory.

On effective temperature in network models of collective behavior

Collective behavior of self-propelled units is studied analytically within the Vectorial Network Model (VNM), a mean-field approximation of the well-known Vicsek model. We propose a dynamical systems framework to study the stochastic dynamics of the VNM in the presence of general additive noise. We establish that a single parameter, which is a linear function of the circular mean of the noise, controls the macroscopic phase of the system-ordered or disordered. By establishing a fluctuation-dissipation relation, we posit that this parameter can be regarded as an effective temperature of collective behavior. The exact critical temperature is obtained analytically for systems with small connectivity, equivalent to low-density ensembles of self-propelled units. Numerical simulations are conducted to demonstrate the applicability of this new notion of effective temperature to the Vicsek model. The identification of an effective temperature of collective behavior is an important step toward understanding order-disorder phase transitions, informing consistent coarse-graining techniques and explaining the physics underlying the emergence of collective phenomena.

Nontoxic colloidal particles impede antibiotic resistance of swarming bacteria by disrupting collective motion and speed

A monolayer of swarming B. subtilis on semisolid agar is shown to display enhanced resistance against antibacterial drugs due to their collective behavior and motility. The dynamics of swarming motion, visualized in real time using time-lapse microscopy, prevents the bacteria from prolonged exposure to lethal drug concentrations. The elevated drug resistance is significantly reduced when the collective motion of bacteria is judiciously disrupted using nontoxic polystyrene colloidal particles immobilized on the agar surface. The colloidal particles block and hinder the motion of the cells, and force large swarming rafts to break up into smaller packs in order to maneuver across narrow spaces between densely packed particles. In this manner, cohesive rafts rapidly lose their collectivity, speed, and group dynamics, and the cells become vulnerable to the drugs. The antibiotic resistance capability of swarming B. subtilis is experimentally observed to be negatively correlated with the number density of colloidal particles on the engineered surface. This relationship is further tested using an improved self-propelled particle model that takes into account interparticle alignment and hard-core repulsion. This work has pertinent implications on the design of optimal methods to treat drug resistant bacteria commonly found in swarming colonies.

Polar Pattern Formation in Driven Filament Systems Require Non-Binary Particle Collisions

Living matter has the extraordinary ability to behave in a concerted manner, which is exemplified throughout nature ranging from the self-organisation of the cytoskeleton to flocks of animals [1-4]. The microscopic dynamics of constituents have been linked to the system's meso- or macroscopic behaviour in silico via the Boltzmann equation for propelled particles [5-10]. Thereby, simplified binary collision rules between the constituents had to be assumed due to the lack of experimental data. We report here experimentally determined binary collision statistics by studying the recently introduced molecular system, the high density actomyosin motility assay [11-13]. We demonstrate that the alignment effect of the binary collision statistics is too weak to account for the observed ordering transition. The transition density for polar pattern formation decreases quadratically with filament length, which indicates that multi-filament collisions drive the observed ordering phenomenon and that a gas-like picture cannot explain the transition of the system to polar order. The presented findings demonstrate that the unique properties of biological active matter systems require a description that goes well beyond a gas-like picture developed in the framework of kinetic theories.