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A one-dimensional flocking model using active Ising spins is studied, where the system evolves through the reinforcement learning approach \textit{via} defining state, action, and cost function for each spin. The orientation of spin with respect to its neighbouring spins defines its state. The state of spin is updated by altering its spin orientation in accordance with the $\varepsilon$-greedy algorithm (action) and selecting a finite step from a uniform distribution to update position. The $\varepsilon$ parameter is analogous to the thermal noise in the system. The cost function addresses cohesion among the spins. By exploring the system in the plane of the self-propulsion speed and $\varepsilon$ parameter, four distinct phases are found: disorder, flocking, flipping, and oscillatory. In the flipping phase, a condensed flock reverses its direction of motion stochastically. The mean reversal time $\langle T \rangle $ exponentially decays with $\varepsilon$. A new phase, an oscillatory phase, is also found, which is a chaotic phase with a positive Lyapunov exponent. The findings obtained from the reinforcement learning approach for the active Ising model system exhibit similarities with the outcomes of other conventional techniques, even without defining any explicit interaction among the spins.

Phagocytosis is the process by which cells, which are 5 to 10 times larger than the particle size, engulf particles, holding substantial importance in various biological contexts ranging from the nutrient uptake of unicellular organisms to immune system of humans, animals etc. While the previous studies focused primarily on the mechanism of phagocytosis, in this study we have a taken a different route by studying the dynamics of the phagocytes in a system consisting of many bacteria and a small number of phagocytes. We put forward a minimalist framework that models bacteria and phagocytes as active and passive circular disks, respectively. The interactions are governed by directional forces: phagocytes are attracted toward bacteria, while bacteria experience a repulsive force in proximity to phagocytes. Bacteria are capable of reproduction at a fixed rate, and the balance between bacterial reproduction and phagocytic engulfment is governed by the interplay of the two opposing forces. In attraction dominated regimes, bacterial populations decrease rapidly, while in repulsion dominated regimes, bacterial clusters grow and impede phagocytes, often resulting in phagocyte trapping. Conversely, in attraction-dominated scenarios, only a few bacteria remain at later times, rendering the motion of the phagocytes diffusive. Further, the transition between the two regimes occurs through a regime of bi-stability. Our study further describes the dynamics of both species using the tools of statistical analysis, offering insights into the internal dynamics of this system., Comment: 9 pages, 6 figures

Many microswimmers are inherently chiral, and this chirality can introduce fascinating behaviors in a collection of microswimmers. The dynamics become even more intriguing when two types of microswimmers with distinct chirality are mixed. Our study considers a mixture of self-propelled particles with opposite chirality, examining how the system characteristics evolve as the magnitude of chirality is tuned. In weakly chiral systems, the particles exhibit similar behavior, leading to a globally flocking phase where both types of particles are well mixed. However, in an intermediate range of chirality, the particles demix and follow their trajectories, creating a competition between chirality and self-propulsion. This competition results in interesting phases within the system. We explore the characteristics of these different phases in detail, focusing on the roles of self-propulsion speed and chirality., Comment: 13 pages,10 figures

The field of active matter explores the behaviors of self propelled agents out of equilibrium, with active suspensions, such as swimming bacteria in solutions, serving as impactful models. These systems exhibit spatio-temporal patterns akin to active turbulence, driven by internal energy injection. While bacterial turbulence in dense suspensions is well studied, the dynamics in growing bacterial suspensions are less understood. This work presents a phenomenological coarse-grained model for growing bacterial suspensions, incorporating hydrodynamic equations for bacterial density, orientation, and fluid velocity, with birth and death terms for colony growth. Starting with low density and random orientations, the model shows the development of local ordering as bacterial density increases. As density continues to rise, the model captures four distinct phases; dilute, clustered, turbulent, and trapped based on structural patterns and dynamics, with the turbulent phase characterized by spatio-temporal vortex structures, aligning with observations in dense bacterial suspensions., Comment: 11 pages, 9 figures

Active nematics, in their pure form, have demonstrated a plethora of dynamic and steady state behaviors, including large-scale dynamic structures, collective flows, and intricate multi-spatial temporal dynamics. This complexity further increases in the presence of external polar agents. We investigate active nematics interspersed with polar microswimmers, akin to active apolar cells infused with active impurities, microswimmers. Our comprehensive numerical study reveals that varying the microswimmers' motility induces a novel spatiotemporal state in the active nematics backdrop. This state is marked by macroscopic swirl-like structures and a reduction in the overall order of the active nematics. Interestingly, this state emerges at intermediate motility levels, where microswimmers form local clusters and exhibit coherent motion. However, at higher motility levels, the swirls become less coherent, and microswimmer clustering intensifies. We show that the effect of the polar microswimmers on active nematics can be interpreted as a spatiotemporally correlated colored noise on active nematics, which promotes bend instability in active nematics, leading to the observed swirling dynamics. Our findings indicate that the spatiotemporal states are highly sensitive to the microswimmers' motility, offering potential avenues for pathogen identification based on known motility characteristics, Comment: 31 pages, 18 figures

How do update rules affect the dynamical and steady state properties of a flock? In this study, we have explored the active Ising spins (s = +-1) in one dimension, where spin updates its orientation according to the Metropolis algorithm (based on the neighbors) via two different update rules. (i) Parallel, and (ii) Random-sequential. We explore the effect of Parallel and Random-sequential updates on the dynamical properties of flocks in one dimension. Due to the inherent asynchronous nature of the Random-sequential update, the directional switching of the flock is increased compared to the Parallel one. The nature of phase transition is affected by the difference in the updating mechanism: discontinuous for Parallel and continuous for Random-sequential updates., Comment: 7 pages, 6 figures. arXiv admin note: text overlap with arXiv:1704.04041

This study explores the effect of quenched disorder on the characteristic of self-propelled particles in one-dimension. Here,particles interact with disorder which serve as directional cues. The study investigates how the density of the disorder influence the emergence of ordering and clustering in the collection of the self propelled particles. We introduce the microscopic model as well as corresponding coarse-grained equations of motion for the local density and the orientation of particle. Disorder affects the macroscopic ordering in the system, the size of the ordered clusters decays algebraically with disorder. Further, the disorder also affects the clustering of particles; in the presence of disorder, a big macroscopic cluster breaks into small clusters, leads to the localization of particles around it and results in high density around the disorder.

We perform a comprehensive study on the role of thermal noise on the ordering kinetics of a collection of active Brownian particles modeled using coarse-grained conserved active model B (AMB). The ordering kinetics of the system is studied for the critical mixture when quenched from high to a low temperature. The structure of the growing domains changes from isolated droplet type for AMB without noise to bi-continuous type for active model B with noise (AMBN). Unlike the passive counterpart of the AMB, the noise is relevant for the growth kinetics of the AMB. We use extensive numerical study, as well as dynamic scaling hypothesis to characterize the kinetics of the system. We find that the asymptotic growth law for AMBN is diffusive Lifshitz-Slyozov (LS) type, whereas it was reported previously that the asymptotic growth law for the AMB without noise is slower, with a growth exponent 4. Moreover, the kinetics of the growing domains show a strong time dependent growth for AMBN. The growth law shows a crossover from early time 1/3 value to intermediate time 1/4 value, and it again traverses from 1/4 to 1/3 asymptotically. The two different scaling functions are found for intermediate time and late time with growth law 1/4 and 1/3 respectively., Comment: 9 Pages, 7 figures

We take a top-down approach to theoretically model the non-equilibrium steady-state behavior of active disks and use this model to derive the effective dynamics of a single particle. By analyzing clustering dynamics in simulations, we introduce a local density variable that captures interaction effects. We numerically calculate the mean and temporal auto-correlation of local density in the steady state, and use these to analytically derive an exact expression for a particle's diffusivity as a function of global packing density. Remarkably, our effective model closely matches numerical diffusivity in high packing density regime where prominent clusters are formed and shows striking agreements with the well-known existence of phase transition beyond a critical packing density. This model could be useful in experimental setups where fluctuations in local density around a tagged particle are measurable.

We present a minimal computational model to mimic the crowd in marathon race. We aim to examine the influence of frontliners on crowd dynamics by comparing the simulated races with and without their presence. The primary outcome of our study revealed that the local velocity and density of the participants exhibit a wave pattern similar to what is observed in actual races. Another important result we obtained is that the travelling wave in the crowd consistently propagates with a constant speed, irrespective of the system size under consideration. The dynamic of participants in the longitudinal direction mainly contributes for the velocity fluctuation and the fluctuation in the transverse direction is suppressed. In the absence of frontliners, the fluctuations in density and velocity weakens without significantly influencing the other statistical and dynamical characteristics of the crowd. It is also observed that the density wave travels faster than the velocity wave. Through this research, we aim to enhance our understanding of crowd motion, which can inform the development of effective crowd management strategies and contribute to the successful control of such events., Comment: 10 pages, 9 figures

Previous studies on the generalized XY model have concentrated on the equilibrium phase diagram and the equilibrium nature of distinct phases under varying parameter conditions. We direct our attention towards examining the systems evolution towards equilibrium states across different parameter values, specifically by varying the relative strengths of ferromagnetic and nematic interactions. We study the kinetics of the system, using the temporal annihilation of defects at varying temperatures and its impact on the coarsening behavior of the system. For both pure polar and pure nematic systems, we observe temperature-dependent decay of the exponent, leading to a decelerated growth of domains within the system. At parameter values where both ferromagnetic and nematic interactions are simultaneously present, we show a phase diagram highlighting three low-temperature phases : polar, nematic, and coexistence, alongside a high-temperature disordered phase. Our study provides valuable insights into the complex interplay of interactions, offering a comprehensive understanding of the systems behavior during its evolution towards equilibrium., Comment: 16 pages, 12 figures

Many microorganisms use chemical `signaling' - a quintessential self-organizing strategy in non-equilibrium - that can induce spontaneous aggregation and coordination in behavior. Using synthetic signaling as a design principle, we construct a minimal model of active Brownian particles (ABPs) having soft repulsive interactions on a chemically quenched patterned substrate. The interplay between chemo-phoretic interactions and activity is numerically investigated for a proposed variant of the Keller-Segel model for chemotaxis. Such competition not only results in a chemo-motility-induced phase-separated state but also a new cohesive clustering phase with synchronized rotations. Our results suggest that rotational order can emerge in systems by virtue of activity and repulsive interactions alone without an explicit alignment interaction. These rotations can also be exploited by designing mechanical devices that can generate reorienting torques using active particles., Comment: 8 pages, 9 figures

We study a collection of polar self-propelled particles or polar flock on a two dimensional substrate with birth and death. Most of the previous studies of polar flock with birth and death have assumed the compressible flock, such that the local density of flock is completely ignored. Effect of birth and death of particles on the flock with moderate density is focus of our study. System is modeled using coarse-grained hydrodynamic equations of motion for local density and velocity of the flock and solved using numerical integration of the nonlinear coupled partial differential equations of motion and linearised hydrodynamics about the broken symmetry state. We studied the ordering kinetics as well as the steady state properties of the immortal flock and flock with finite birth and death rate. The ordering kinetics of the velocity field remains unaffected whereas the density field shows a crossover from asymptotic growth exponent $5/6$ for the immortal flock to diffusive limit $1/3$ for large birth and death rates. In the steady state, the presence of birth and death rate leads to the suppression of speed of sound wave and density fluctuations in the system., Comment: 8 pages, 7 figures

We model the active polar fluid as a collection of orientable objects supplied with active stresses and momentum damping coming from the viscosity of bulk fluid medium. The growth kinetics of local orientation field is studied. The effect of active fluid is contractile or extensile depending upon the sign of the active stress. We explore the growth kinetics for different activities. We observe that for both extensile and contractile cases the growth is altered by a prefactor when compared to the equilibrium Model A. We find that the extensile fluid enhances the domain growth whereas the contractile fluid supresses it. The asymptotic growth becomes pure algebraic for large magnitudes of activity. We also find that the domain morphology remains unchanged due to activity and system shows the good dynamic scaling for all activities. Our study provides the understanding of ordering kinetics in active polar gel.

Active Brownian particles (ABPs) are termed out to be a successful way of modeling the moving microorganism on the substrate. In recent studies, it is shown that such organisms can sense the characteristics of the substrate. Motivated by such work, we studied the dynamics and the steady state of ABP moving on a substrate with space-dependent activity. On the substrate, some regions are marked as high in activity, and other regions are such that particles behave as passive Brownian particles. The system is studied in two dimensions with step, sigmoid, Gaussian and cone shape distribution of activity profile on the substrate. The whole interface of the activity profile is symmetrically divided into two regions. This lead to the flow of particles from the active region to the passive region. The final steady state of particle density profile, polarisation and flux very much follows the structure of the inhomogeneous activity and the density in high activity region is lower, maximum at the interface and nearly constant with mean density in the passive region. Further, the steady state density profile for various shapes and designs on two-dimensional substrates. Hence the collection of ABPs on an inhomogeneous substrate can mimic the inhomogeneity of the substrate.

We studied a collection of chiral active particles (CAP) on a two dimensional substrate using extensive numerical study. Particles interact through soft repulsive interaction. The activity and chirality of particles is tuned by varying their self-propulsion speed and angular velocity respectively. Kinetics and steady state properties of particles are studied for different chirality and activity. The phase diagram of system on the plane of activity and chirality shows three distinct phases. For small chirality when activity is dominant, particles show enhanced dynamics and macroscopic phase separation of ordered clusters is observed. For moderate chirality, micro clustered phase is observed in which small clusters with moderate ordering are formed. For large chirality, when chirality dominates, no clustering is found because particle motion is mainly confined to its location. Our study gives a detail insight into the effect of chirality on the properties of collection of CAP, which can be useful to understand the dynamics and steady state of many natural micro swimmers.

We studied a system of polar self-propelled particles (SPPs) on a thin rectangular channel designed into three regions of order-disorder-order. The division of the three regions is made on the basis of the noise SPPs experience in the respective regions. The noise in the two wide region is chosen lower than the critical noise of order-disorder transition and noise in the middle region or interface is higher than the critical noise. This make the geometry of the system analogous to the Josephson Junction (JJ) in solid state physics. Keeping all other parameters fixed, we study the properties of the moving SPPs in the bulk as well as along the interface for different widths of the junction. On increasing interface width, system shows a order-to-disorder transition from coherent moving SPPs in the whole system to the interrupted current for large interface width. Surprisingly, inside the interface we observed the current reversal for intermediate widths of the interface. Such current reversal is due to the strong randomness present inside the interface, that makes the wall of the interface reflecting. Hence Our study give a new interesting collective properties of SPPs at the interface which can be useful to design devices like switch using active agents.

We model a binary mixture of passive and active Brownian particles in two dimensions using the effective interaction between passive particles in the active bath. The activity of active particles and the size ratio of two types of particles are two control parameters in the system. The effective interaction is calculated from the average force on two particles generated by the active particles. The effective interaction can be attractive or repulsive, depending on the system parameters. The passive particles form four distinct structural orders for different system parameters viz; disorder (D), disordered cluster (DC), ordered cluster (OC), and poly-crystalline order (P C). The change in structure is dictated by the change in nature of the effective interaction. We further confirm the four structures using full microscopic simulation of active and passive mixture. Our study is useful to understand the different collective behaviour in non-equilibrium systems., Comment: 8-pages and 6-figures

We study a collection of self-propelled polar particles on a two-dimensional substrate with birth and death. We introduce a minimal lattice model for the system using active Ising spins, where each particle can have two possible orientations. The activity is modeled as a biased movement of the particle along its direction of orientation. The particles also align with their nearest neighbors using Metropolis Monte-Carlo algorithm. System shows a disorder-to-order transition by tuning the temperature of the system. Additionally, the birth and death of the particles is introduced through a birth and death rate $\gamma$. The system is studied near the disorder-to-order transition. The nature of disorder-to-order transition shows a crossover from first order, discontinuous to continuous type as we tune $\gamma$ from zero to finite values. We also write the effective free energy of the local order parameter using renormalised mean field theory and it confirms the dependence of the nature of phase transition on the birth and death rate parameter.

With quenched disorder, we introduce two-dimensional active nematics suspended in an incompressible fluid. We write the coarse-grained hydrodynamic equations of motion for slow variables, viz. density, orientation and flow fields. The quenched disorder is introduced such that it interacts with the local orientation at every point with some strength. Disorder strength is tuned from zero to large values. We numerically study the defect dynamics and system's kinetics and find that the finite disorder slows the ordering. The presence of fluid induces large fluctuation in the orientation field, further disturbing the ordering. The large fluctuation in the orientation field due to the fluid is so dominant that it reduces the effect of the quenched disorder. We have also found that the disorder's effect is almost the same for both the contractile and extensile nature of active stresses in the system. This study can help {\color{black} to} understand the impact of quenched {\color{black} disorder} on the ordering kinetics of active gels with nematic interaction among the constituent objects., Comment: 14 figures

We study two-dimensional Ising spins, evolving through reinforcement learning using their state, action, and reward. The state of a spin is defined as whether it is in the majority or minority with its nearest neighbours. The spin updates its state using an {\epsilon}-greedy algorithm. The parameter {\epsilon} plays the role equivalent to the temperature in the Ising model. We find a phase transition from long-ranged ordered to a disordered state as we tune {\epsilon} from small to large values. In analogy with the phase transition in the Ising model, we calculate the critical {\epsilon} and the three critical exponents {\beta}, {\gamma}, {\nu} of magnetization, susceptibility, and correlation length, respectively. A hyper-scaling relation d{\nu} = 2{\beta} + {\gamma} is obtained between the three exponents. The system is studied for different learning rates. The exponents approach the exact values for two-dimensional Ising model for lower learning rates.

We study a binary mixture of disk-shaped {\it active run and tumble } particles (${ARNPs}$) and passive particles on a two-dimensional substrate. Both types of particles are athermal. The particles interact through the soft repulsive potential. The activity of $ARNPs$ is controlled by tuning their tumbling rate. The system is studied for various sizes of passive particles keeping size of $ARNPs$ fixed. Hence the variables are, size ratio (S) of passive particles and $ARNPs$, and activity of $ARNPs$ is $v$. The characterstics dynamics of both $ARNPs$ and passive particles show a crossover from early time ballistic to later time diffusive. Furthermore, we observed that passive particles dynamics changes from diffusive to subdiffusive with respect to their size. Moreover, late time effective diffusivity $D_{eff}$ of passive particles decreases with increasing their size as in the corresponding equilibrium Stokes systems. We calculated the effective temperatures, using $D_{eff}$, $T_{a,eff}(\Delta)$ and also using speed distribution $T_{a,eff}(v)$ and compared them. The both $T_{a,eff}(\Delta)$ and $T_{a,eff}(v)$ increases linearly with activity and are in agreement with each other. Hence we can say that an effective equilibrium can be establish in such binary mixture. Our study can be useful to study the various biological systems like; dynamics of passive organels in cytoplasm, colloids etc.

The steady state and phase ordering kinetics in a pure active Borwnian particle system are studied in recent years. In binary mixture of active and passive Brownian particles passive particles are used as probe to understand the properties of active medium. In our present study we study the mixture of passive and active Brownian particles. Here we aim to understand the steady state and kinetics of small passive particles in the mixture. In our system,the passive particles are small in size and large in number, whereas ABPs are large in size and small in number. The system is studied on a two-dimensional substrate using overdamp Langevin dynamic simulation. The steady state and kinetics of passive particles are studied for various size and activity of active particles. Passive particles are purely athermal in nature and have dynamics only due to bigger ABPs. For small size ratio and activity the passive particles remain homogeneous in the system, whereas on increasing size ratio and activity they form periodic hexagonal close pack (HCP) spanning clusters in the system. We have also studied the kinetics of growing passive particle clusters. The mass of the largest cluster shows a much slower growth kinetics in contrast to conserved growth kinetics in ABP system. Our study provides an understanding of steady state and kinetics of passive particles in the presence of bigger active particles. The mixture can be thought of as effect of big microorganism moving in passive medium.

We study the ordering kinetics of an assembly of {\it active Brownian particles} (ABPs) on a two-dimensional substrate. We use a coarse-grained equation for the composition order parameter $\psi ({\bf r},t)$, where ${\bf r}$ and $t$ denote space and time, respectively. The model is similar to the {\it Cahn-Hilliard equation} or {\it Model B} (MB) for a conserved order parameter with an additional activity term of strength $\lambda$. This model has been introduced by Wittkowski et al., Nature Comm. {\bf 5}, 4351 (2014), and is termed {\it Active Model B} (AMB). We study domain growth kinetics and dynamical scaling of the correlation function for the AMB with critical and off-critical compositions. The quantity $P = \mbox{sign}(\lambda \times \psi_0)$ governs the asymptotic growth kinetics for the off-critical AMB, where $\psi_0$ denotes the average order parameter. For negative $P$, the domain growth law is the usual Lifshitz-Slyozov growth law with $L(t,\lambda) \sim t^{1/3}$. For positive $P$, the growth law shows a crossover to a novel growth law $L(t,\lambda) \sim t^{1/4}$. Further, the correlation function shows good dynamical scaling for the off-critical AMB but the scaling function has a dependency on $\psi_0$ and $\lambda$. We also study the effects of both additive and multiplicative noise on the AMB.

We study the binary phase separation in active model B, on a two-dimensional substrate with inhomogeneous activity. The activity was introduced with a maximum value at the center of the box and spread as a Bivariate-Gaussian distribution as we move away from the center. The system was studied for three different intensities of the distribution. Towards the boundary of the box, activity is zero or the model is similar to the passive model B. We start from the random homogeneous distribution of density of particles, and the system evolves towards a structured distribution of density. With time, density starts to phase separately with maximum density at the center of the box and decreases as we move away from the center of the box. The width of the density profile at the center increases as a power lawexponent{\alpha}(t) remains close between 2/3 to 3/4 up to some moderate time and then decays to zero in the steady state. Hence, our result shows the response of density in an active binary system with respect to the patterned substrate. It can be used to design devices useful for the trapping and segregation of active particles., Comment: 15 pages, 6 figures

Collection of self-propelled particles (SPPs) exhibit coherent motion and show true long-range order in two-dimensions. Inhomogeneity, in general destroys the usual long-range order of the polar SPPs. We model a system of polar self-propelled particles with inhomogeneous interaction strength or bond disorder. The system is studied near the order-to-disorder transition for different strengths of the disorder. The nature of phase transition changes from discontinuous to continuous type by tuning the strength of the disorder. The bond disorder also enhances the ordering near the transition due to the formation of a homogeneous flock state for the large disorder. It leads to faster information transfer in the system and enhances the system information entropy. Our study gives a new understanding of the effect of intrinsic inhomogeneity in the self-propelled particle system., Comment: 7 pages, 7 figures. arXiv admin note: text overlap with arXiv:2005.03862

We undertake a detailed numerical study of the {\it Active Model B} proposed by Wittkowski et al. [Nature Comm. {\bf 5}, 4351 (2014)]. We find that the introduction of activity has a drastic effect on the ordering kinetics. First, the domain growth law shows a crossover from the usual Lifshitz-Slyozov growth law for phase separation ($L \sim t^{1/3}$, where $t$ is the time) to a novel growth law ($L \sim t^{1/4}$) at late times. Second, the correlation function exhibits dynamical scaling for a given activity strength $\lambda$, but the scaling function depends on $\lambda$.

We numerically study the dynamics and the phases of self-propelled disk-shaped particles of different sizes with soft repulsive potential in two dimensions. Size diversity is introduced by the polydispersity index (PDI) $\epsilon$, which is the width of the uniform distribution of the particle's radius. The self-propulsion speed of the particles controls the activity $v$. We observe enhanced dynamics for large size diversity among the particles. We calculate the effective diffusion coefficient $D_{eff}$ in the steady-state. The system exhibits four distinct phases, jammed phase with small $D_{eff}$ for small activity and liquid phase with enhanced $D_{eff}$ for large activity. The number fluctuation is larger and smaller than the equilibrium limit in the liquid and jammed phase, respectively. Further, the jammed phase is of two types: solid-jammed and liquid jammed for small and large PDI. Whereas the liquid phase is called motility induced phase separation (MIPS)-liquid for small PDI and for large PDI, we find enhanced diffusivity and call it the {\em pure liquid} phase. The system is studied for three packing densities $\phi$, and the response of the system for polydispersity is the same for all $\phi$'s. Our study can help understand the behavior of cells of various sizes in a tissue, artificial self-driven granular particles, or living organisms of different sizes in a dense environment., Comment: 8 pages, 10 figures

Swimming bacteria in passive nematics in the form of lyotropic liquid crystals are defined as a new class of active matter known as living liquid crystals in recent studies. It has also been shown that liquid crystal solutions are promising candidates for trapping and detecting bacteria. We ask the question, can a similar class of matter be designed for background nematics which are also active? Hence, we developed a minimal model for the mixture of polar particles in active nematics. It is found that the active nematics in such a mixture are highly sensitive to the presence of polar particles, and show the formation of large scale higher order structures for a relatively low polar particle density. Upon increasing the density of polar particles, different phases of active nematics are found and it is observed that the system shows two phase transitions. The first phase transition is a first order transition from quasi-long ranged ordered active nematics to disordered active nematics with larger scale structures. On further increasing density of polar particles, the system transitions to a third phase, where polar particles form large, mutually aligned clusters. These clusters sweep the whole system and enforce local order in the nematics. The current study can be helpful for detecting the presence of very low densities of polar swimmers in active nematics and can be used to design and control different structures in active nematics.

We study the dynamics of circular active particles (AP) on a two dimensional periodic undulated surface. Each particle has an internal energy mechanism which is modeled by an active friction force and it is controlled by an activity parameter $v_0$. It acts as negative friction if the speed of the particle is smaller than $v_0$ and normal friction otherwise. Surface undulation is modeled by the periodic undulation of fixed amplitude and wavelength and is measured in terms of a dimensionless ratio of amplitude and wavelength, $\bar{h}$. The dynamics of the particle is studied for different activities, $v_0$ and surface undulations (SU), $\bar{h}$. Three types of particle dynamics are observed on varying activity and SU. For small $v_0 \lesssim 0.1$ and $\bar{h}$ $\gtrsim 0.8$, particles remain confined in a surface minimum, for moderate $v_0 \lesssim \bar{h}$, dynamics of particle shows an intermediate subdiffusion to late time diffusion and for large $v_0 \gtrsim \bar{h}$, it shows initial superdiffusion to late time diffusion. For all $v_0$'s and $\bar{h} \lesssim 0.2$, the dynamics of particle, satisfies the Green-Kubo relation between the effective diffusivity and velocity auto-correlation function. Systematic deviation is found on increasing $\bar{h}$. Hence, an effective equilibrium can be established for a range of system parameters in this nonequilibirum system., Comment: 6 pages, 6 figures

In this study, we introduce a minimal model for a collection of polar self-propelled particles (SPPs) on a two-dimensional substrate where each particle has a different ability to interact with its neighbours. The SPPs interact through a short-range alignment interaction and interaction strength of each particle is obtained from a uniform distribution. Moreover, the volume exclusion among the SPPs is taken care of by introducing a repulsive interaction among them. We characterise the ordered steady state and kinetics of the system for different strengths of the disorder. We find that the presence of the disorder does not destroy the usual long-range ordering in the system. To our surprise, we note that the density clustering is enhanced in the presence of the disorder. Moreover, the disorder leads to the formation of a random network of different interaction strengths, which makes the alignment weaker and it results in the slower dynamics. Hence, the disorder leads to more cohesion among the particles. Furthermore, we note that the kinetics of the ordered state remains unaffected in the presence of the disorder. Size of orientationally ordered domains and density clusters grow with time with dynamic growth exponents $z_{o} \sim 2$ and $z_{\rho} \sim 4$, respectively.

We introduce a two-dimensional active nematic with quenched disorder. We write the coarse-grained hydrodynamic equations of motion for slow variables, viz. density, and orientation. Disorder strength is tuned from zero to large values. Results from the numerical solution of equations of motion as well as the calculation of two-point orientation correlation function using linear approximation show that the ordered steady-state follows a disorder-dependent crossover from quasi long-range order (QLRO) to short-range order (SRO). Such crossover is due to the pinning of +1/2 and -1/2 topological defects in the presence of finite disorder, which breaks the system in uncorrelated domains. Finite disorder slows the dynamics of +1/2 defect, and it leads to slower growth dynamics. The two-point correlation functions for the density and orientation fields show good dynamic scaling but no static scaling for the different disorder strengths. Our findings can motivate experimentalists to verify the results and find applications in living and artificial apolar systems in the presence of a quenched disorder., Comment: 10 figures

We study the dynamics of a particle moving in a square two-dimensional Lorentz lattice-gas. The underlying lattice-gas is occupied by two kinds of rotators, "right-rotator (R)" and "left-rotator (L)" and some of the sites are empty {\it{viz.}} vacancy "V".The density of $R$ and $L$ are the same and density of $V$ is one of the key parameters of our model. The rotators deterministically rotate the direction of a particle's velocity to the right or left and vacancies leave it unchanged. We characterise the dynamics of particle motion for different densities of vacancies. Since the system is deterministic, the particle forms a closed trajectory asymptotically. The probability of the particle being in a closed or open trajectory at time $t$ is a function of the density of vacancies. \textcolor{black}{The motion of the particle is {\it{uniform}} throughout in a fully occupied lattice. However, it is divided in two distinct phases in partially vacant lattices}: The first phase of the motion, which is the focus of this study, is characterised by anomalous diffusion and a power-law decay of the probability of being in an open trajectory. The second phase of the motion is characterised by subdiffusive motion and an exponential decay of the probability of being in an open trajectory. For lattices with a non-zero density of vacancies, the first phase of motion lasts for a longer period of time as the density of vacancies increases.

A collection of self-propelled particles (SPPs) shows coherent motion and exhibits a true long range ordered (LRO) state in two dimensions. Various studies show that the presence of spatial inhomogeneities can destroy the usual long-range ordering in the system. However, the effects of inhomogeneity due to the intrinsic properties of the particles are barely addressed. In this paper we consider a collection of polar SPPs moving with inhomogeneous speed (IS) on a two dimensional substrate, which can arise due to varying physical strength of the individual particle. To our surprise, the IS not only preserves the usual long-range ordering present in the homogeneous speed models but also induces faster ordering in the system. Furthermore, The response of the flock to an external perturbation is also faster, compared to Vicsek like model systems, due to the frequent update of neighbors of each SPP in the presence of the IS. Therefore, our study shows that the IS can help in faster information transfer in the moving flock.

We introduce a minimal model for a two-dimensional polar flock with nonquenched rotators, and show that the rotators make the usual macroscopic long-range order of the flock more robust than the clean system. The rotators memorise the flock-information which helps in establishing the robustness. Moreover, the memory of the rotators assists in probing the moving flock. We also formulate a hydrodynamic framework for the microscopic model that makes our study comprehensive. Using linearised hydrodynamics, it is shown that the presence of such nonquenched heterogeneities increases the sound speeds of the flock. The enhanced sound speeds lead to faster convection of information and consequently the robust ordering in the system. We argue that similar nonquenched heterogeneities may be useful in monitoring and controlling large crowds.

We study the dynamics of a particle moving in one-dimensional Lorentz lattice-gas where particle performs mainly three different kinds of motion {\it viz} ballistic motion, diffusion and confinement. There are two different types of scatterers, {\it viz} reflector and transmitters, randomly placed in the lattice. Reflectors are such that they reverse the particle's velocity direction and transmitters let it pass through. Scatterers also change their character with flipping probability $1-\alpha$, once the particle interacts with a scatterer. Hence the system is defined by two sets of parameters, $r$, which is the initial density of reflector/transmitter and $\alpha$. For $\alpha=0$ and $\alpha=1$ dynamics of the particle is purely deterministic else it is probabilistic. In the pure deterministic case dynamics of the particle is either propagation in one direction or confined between two near-by reflectors present. For the probabilistic case $\alpha \ne 1$ and $\ne 0$, although the dynamics of particle shows anomalous diffusion where dynamics is faster, slower and comparable to normal diffusion on the variation of system parameters $(\alpha, r)$, but the asymptotic behaviour of the particle is normal diffusion. We plot the phase diagram for the asymptotic behaviour, in the plane of $\alpha$ and $r$., Comment: 13 pages, 8 figures. J Stat Phys (2019)

We study the collective behavior of binary mixture of self-propelled particles. Particles moves along their heading direction with {\it variable speed} and interact through short range alignment interaction. A variable speed parameter $\gamma >0$ is introduced such that for $\gamma=0.0$ model reduces to {\it constant speed} Vicsek's model. We mix the particles with two different $\gamma$'s and study the steady state behavior of the mixture for different choice of $\gamma$'s and noise strength. One of the $\gamma$ is kept fixed to $1.0$ and another one is varied from small $0.0$ to larger values $8.0$. Properties of system is characterise by two types of order parameters (i) orientation order parameter, which is a measure of ordering in the system and (ii) density order parameter, which measures the phase separation is the system. For all set of $\gamma$'s, system shows a transition from disorder-to-ordered state on the variation of noise strength. The nature of transition and critical noise is independent of value of $\gamma$, which is also supported from coarse-grained hydrodynamic study. On the variation of system parameters, ($\gamma$'s, $\eta$), we find four distinct phases, (i) ordered phase separated, (ii) ordered mixed, (iii) disordered mixed and (iv) disordered phase segragated. Our study shade light on different phases of mixture of different types of active particles., Comment: 18 pages,7 figurs

We study the motion of an active Brownian particle (ABP) using overdamped Langevin dynamics on a two-dimensional substrate with periodic array of obstacles and in a quasi-one-dimensional corrugated channel comprised of periodically arrayed obstacles. The periodic arrangement of the obstacles enhances the persistent motion of the ABP in comparison to its motion in the free space. Persistent motion increases with the activity of the ABP. We note that the periodic arrangement induces directionality in ABP motion at late time, and it increases with the size of the obstacles. We also note that the ABP exhibits a super-diffusive dynamics in the corrugated channel. The transport property is independent of the shape of the channel; rather it depends on the packing fraction of the obstacles in the system. However, the ABP shows the usual diffusive dynamics in the quasi-one-dimensional channel with flat boundary.

We study the disorder-to-order transition in a collection of polar self-propelled particles interacting through a distance dependent alignment interaction. Strength of the interaction, $a^{d}$ ($00$, the system shows a transition from a disordered to an ordered state as a function of noise strength. We calculate the critical noise strength, $\eta_c(a)$ for different $a$ and compare it with the mean-field result. Nature of the disorder-to-order transition continuously changes from discontinuous to continuous with decreasing $a$. We numerically estimate tri-critical point $a_{TCP}$ at which the nature of transition changes from discontinuous to continuous. The density phase separation is large for ${\it a}$ close to unity, and it decays with decreasing $a$. We also write the coarse-grained hydrodynamic equations of motion for general ${\it a}$, and find that the homogeneous ordered state is unstable to small perturbation as ${\it a}$ approaches to $1$. The instability in the homogeneous ordered state is consistent with the large density phase separation for ${\it a}$ close to unity., Comment: 17 pages and 7 figures

We study a collection of polar self-propelled particles (SPPs) on a two-dimensional substrate in the presence of random quenched rotators. These rotators act like obstacles which rotate the orientation of the SPPs by an angle determined by their intrinsic orientations. In the zero self-propulsion limit, our model reduces to the equilibrium $XY$ model with quenched disorder, while for the clean system, it is similar to the Vicsek model for polar flock. We note that a small amount of the quenched rotators destroys the long-range order usually noted in the clean SPPs. The system shows a quasi-long range order state upto some moderate density of the rotators. On further increment in the density of rotators, the system shows a continuous transition from the quasi-long-range order to disorder state at some critical density of rotators. Our linearized hydrodynamic calculation predicts anisotropic higher order fluctuation in two-point structure factors for density and velocity fields of the SPPs. We argue that nonlinear terms probably suppress this fluctuation such that no long-range order but only a quasi-long-range order prevails in the system.

Ordering dynamics of self-propelled particles in an inhomogeneous medium in two-dimensions is studied. We write coarse-grained hydrodynamic equations of motion for coarse-grained density and velocity fields in the presence of an external random disorder field, which is quenched in time. The strength of inhomogeneity is tuned from zero disorder (clean system) to large disorder. In the clean system, the velocity field grows algebraically as $L_{\rm V} \sim t^{0.5}$. The density field does not show clean power-law growth; however, it follows $L_{\rm \rho} \sim t^{0.8}$ approximately. In the inhomogeneous system, we find a disorder dependent growth. For both the density and the velocity, growth slow down with increasing strength of disorder. The velocity shows a disorder dependent power-law growth $L_{\rm V}(t,\Delta) \sim t^{1/\bar z_{\rm V}(\Delta)}$ for intermediate times. At late times, there is a crossover to logarithmic growth $L_{\rm V}(t,\Delta) \sim (\ln t)^{1/\varphi}$, where $\varphi$ is a disorder independent exponent. Two-point correlation functions for the velocity shows dynamical scaling, but the density does not.

We introduce a lattice model for active nematic composed of self-propelled apolar particles,study its different ordering states in the density-temperature parameter space, and compare with the corresponding equilibrium model. The active particles interact with their neighbours within the framework of the Lebwohl-Lasher model, and move anisotropically along their orientation to an unoccupied nearest neighbour lattice site. An interplay of the activity, thermal fluctuations and density gives rise distinct states in the system. For a fixed temperature, the active nematic shows a disordered isotropic state, a locally ordered inhomogeneous mixed state, and bistability between the inhomogeneous mixed and a homogeneous globally ordered state in different density regime. In the low temperature regime, the isotropic to the inhomogeneous mixed state transition occurs with a jump in the order parameter at a density less than the corresponding equilibrium disorder-order transition density. Our analytical calculations justify the shift in the transition density and the jump in the order parameter. We construct the phase diagram of the active nematic in the density-temperature plane.

We study binary mixtures of small active and big passive athermal particles interacting via soft repulsive forces on a frictional substrate. Athermal self propelled particles are known to phase separate into a dense aggregate and a dilute gas-like phase at fairly low packing fractions. Known as {\emph {motility induced phase separation}}, this phenomenon governs the behaviour of binary mixtures for small to intermediate size ratios of the particle species. An effective attraction between passive particles, due to the surrounding active medium, leads to true phase separation for large size ratios and volume fractions of active particles. The effective interaction between active and passive particles can be attractive or repulsive at short range depending on the size ratio and volume fractions of the particles. This affects the clustering of passive particles. We find three distinct phases based on the spatial distribution of passive particles. The cluster size distribution of passive particles decays exponentially in the {\emph{homogeneous phase}}. It decays as a power law with an exponential cutoff in the {\emph{clustered phase}} and tends to a power law as the system approaches the transition to the {\emph{phase separated state}}. We present a phase diagram in the plane defined by the size ratio and volume fraction of passive particles., Comment: 9 pages, 11 figures

In this study the effect of parallel and random-sequential updates on the dynamical properties of flocks in one dimension is considered. It is found that the frequency of directional switching is increased for random-sequential updates as compared to a parallel update. The nature of disorder to order transition is also affected by the difference of updating mechanism: discontinuous for parallel and continuous for random-sequential updates., Comment: 4 pages, 4 figures

We study a collection of polar self-propelled particles confined to a long two-dimensional channel. We write the coupled hydrodynamic equations of motion for density and polarisation order parameter. At two confined boundaries, density is fixed to the mean and orientation is anti-parallel with fixed magnitude of polarisation. Such boundary conditions make our system similar to a sheared suspension of self-propelled particles, which has many practical applications. Antiparallel alignment at the two confined boundaries and alignment inside the channel create rolls of orientation along the long axis of the channel. For zero self-propulsion speed, density and orientation fields are decoupled and density remains homogeneous inside the channel. For finite self-propelled speed, density inhomogeneities develop and these rolls move along the long axis of the channel. Density inhomogeneity increases sharply with increasing the self propulsion speed and then reaches a maximum and again decreases for very large speeds. Formation of rolls is very similar to the classic problem of Rayleigh-Benard convection in fluid dynamics., Comment: 7 pages, 7 figures, accepted in Physica A

We study the order-disorder transition in a collection of polar self-propelled particles, interacting through a distance dependent short range alignment interaction. A distance dependent interaction parameter $a_0$ is introduced such that on decreasing $a_0$ interaction decay faster with distance $d$ and for $a_0=1.0$ model reduces to Vicsek's type. For all $a_0>0.0$, system shows a transition from disorder to long ranged ordered state. We find another phase transition from phase separated to nonphase separated state with decreasing $a_0$: at the same time order-disorder transition changes from discontinuous to continuous type. Hence density phase separation plays an important role in predicting the nature of order-disorder transition. We also calculate the two-point density structure factor using coarse-grained hydrodynamic equations of motion with an introduction of a density dependent alignment term in the equation introduced by Toner and Tu \cite{tonertu}. Density structure factor shows a divergence at a critical wave-vector $q_c$, which decreases with decreasing density dependent alignment term. Alignment term in the coarse-grained equation plays the same role as the distance dependent parameter $a_0$ in the microscopic simulation. Our results can be tested in many biological systems: where particle have tendency to interact strongly with their closest neighbours., Comment: 10 pages, 7 figures