Ordering kinetics in the active Ising model.

We undertake a numerical study of the ordering kinetics in the two-dimensional (2D) active Ising model (AIM), a discrete flocking model with a conserved density field coupled to a nonconserved magnetization field. We find that for a quench into the liquid-gas coexistence region and in the ordered liquid region, the characteristic length scale of both the density and magnetization domains follows the Lifshitz-Cahn-Allen growth law, R(t)∼t^{1/2}, consistent with the growth law of passive systems with scalar order parameter and nonconserved dynamics. The system morphology is analyzed with the two-point correlation function and its Fourier transform, the structure factor, which conforms to the well-known Porod's law, a manifestation of the coarsening of compact domains with smooth boundaries. We also find the domain growth exponent unaffected by different noise strengths and self-propulsion velocities of the active particles. However, transverse diffusion is found to play the most significant role in the growth kinetics of the AIM. We extract the same growth exponent by solving the hydrodynamic equations of the AIM.

Active XY model on a substrate: Density fluctuations and phase ordering.

We explore the generic long-wavelength properties of an active XY model on a substrate, consisting of a collection of nearly phase-ordered active XY spins in contact with a diffusing, conserved species, as a representative system of active spinners with a conservation law. The spins rotate actively in response to the local density fluctuations and local phase differences, on a solid substrate. We investigate this system by Monte Carlo simulations of an agent-based model, which we set up, complemented by the hydrodynamic theory for the system. We demonstrate that this system can phase-synchronize without any hydrodynamic interactions. Our combined numerical and analytical studies show that this model, when stable, displays hitherto unstudied scaling behavior: As a consequence of the interplay between the mobility, active rotation, and number conservation, such a system can be stable over a wide range of the model parameters characterized by a novel correspondence between the phase and density fluctuations. In different regions of the phase space where the phase-ordered system is stable, it displays generalized quasi-long-range order (QLRO): It shows phase ordering which is generically either logarithmically stronger than the conventional QLRO found in its equilibrium limit, together with "miniscule number fluctuations," or logarithmically weaker than QLRO along with "giant number fluctuations," showing a novel one-to-one correspondence between phase ordering and density fluctuations in the ordered states. Intriguingly, these scaling exponents are found to depend explicitly on the model parameters. We further show that in other parameter regimes there are no stable, ordered phases. Instead, two distinct types of disordered states with short-range phase order are found, characterized by the presence or absence of stable clusters of finite sizes. In a surprising connection, the hydrodynamic theory for this model also describes the fluctuations in a Kardar-Parisi-Zhang (KPZ) surface with a conserved species on it, or an active fluid membrane with a finite tension, without momentum conservation and a conserved species living on it. This implies the existence of stable fluctuating surfaces that are only logarithmically smoother or rougher than the Edward-Wilkinson surface at two dimensions (2D) can exist, in contrast to the 2D pure KPZ-like "rough" surfaces.

Mobility-induced order in active XY spins on a substrate.

We elucidate that the nearly phase-ordered active XY spins in contact with a conserved, diffusing species on a substrate can be stable. For wide-ranging model parameters, it has stable uniform phases robust against noises. These are distinguished by generalized quasi-long-range (QLRO) orientational order logarithmically stronger or weaker than the well-known QLRO in equilibrium, together with miniscule (i.e., hyperuniform) or giant number fluctuations, respectively. This illustrates a direct correspondence between the two. The scaling of both phase and density fluctuations in the stable phase-ordered states is nonuniversal: they depend on the nonlinear dynamical couplings. For other parameters, it has no stable uniformly ordered phase. Our model, a theory for active spinners, provides a minimal framework for wide-ranging systems, e.g., active superfluids on substrates, synchronization of oscillators, active carpets of cilia and bacterial flagella, and active membranes.

Flocking of two unfriendly species: The two-species Vicsek model.

We consider the two-species Vicsek model (TSVM) consisting of two kinds of self-propelled particles, A and B, that tend to align with particles from the same species and to antialign with the other. The model shows a flocking transition that is reminiscent of the original Vicsek model: it has a liquid-gas phase transition and displays micro-phase-separation in the coexistence region where multiple dense liquid bands propagate in a gaseous background. The interesting features of the TSVM are the existence of two kinds of bands, one composed of mainly A particles and one mainly of B particles, the appearance of two dynamical states in the coexistence region: the PF (parallel flocking) state in which all bands of the two species propagate in the same direction, and the APF (antiparallel flocking) state in which the bands of species A and species B move in opposite directions. When PF and APF states exist in the low-density part of the coexistence region they perform stochastic transitions from one to the other. The system size dependence of the transition frequency and dwell times show a pronounced crossover that is determined by the ratio of the band width and the longitudinal system size. Our work paves the way for studying multispecies flocking models with heterogeneous alignment interactions.

Jamming and flocking in the restricted active Potts model.

We study the active Potts model with either site occupancy restriction or on-site repulsion to explore jamming and kinetic arrest in a flocking model. The incorporation of such volume exclusion features leads to a surprisingly rich variety of self-organized spatial patterns. While bands and lanes of moving particles commonly occur without or under weak volume exclusion, strong volume exclusion along with low temperature, high activity, and large particle density facilitates jams due to motility-induced phase separation. Through several phase diagrams, we identify the phase boundaries separating the jammed and free-flowing phases and study the transition between these phases which provide us with both qualitative and quantitative predictions of how jamming might be delayed or dissolved. We further formulate and analyze a hydrodynamic theory for the restricted APM which predicts various features of the microscopic model.

Studying the progress of COVID-19 outbreak in India using SIRD model.

We explore a standard epidemiological model, known as the SIRD model, to study the COVID-19 infection in India, and a few other countries around the world. We use (a) the stable cumulative infection of various countries and (b) the number of infection versus the tests carried out to evaluate the model. The time-dependent infection rate is set in the model to obtain the best fit with the available data. The model is simulated aiming to project the probable features of the infection in India, various Indian states, and other countries. India imposed an early lockdown to contain the infection that can be treated by its healthcare system. We find that with the current infection rate and containment measures, the total active infection in India would be maximum at the end of June or beginning of July 2020. With proper containment measures in the infected zones and social distancing, the infection is expected to fall considerably from August. If the containment measures are relaxed before the arrival of the peak infection, more people from the susceptible population will fall sick as the infection is expected to see a threefold rise at the peak. If the relaxation is given a month after the peak infection, a second peak with a moderate infection will follow. However, a gradual relaxation of the lockdown started well ahead of the peak infection, leads to a nearly twofold increase of the peak infection with no second peak. The model is further extended to incorporate the infection arising from the population showing no symptoms. The preliminary finding suggests that random testing needs to be carried out within the asymptomatic population to contain the spread of the disease. Our model provides a semi-quantitative overview of the progression of COVID-19 in India, with model projections reasonably replicating the current progress. The projection of the model is highly sensitive to the choice of the parameters and the available data.

SEIRD model to study the asymptomatic growth during COVID-19 pandemic in India.

According to the current perception, symptomatic, presymptomatic and asymptomatic infectious persons can infect the healthy population susceptible to the SARS-CoV-2. More importantly, various reports indicate that the number of asymptomatic cases can be several-fold higher than the reported symptomatic cases. In this article, we take the reported cases in India and various states within the country till September 1, as the specimen to understand the progression of the COVID-19. Employing a modified SEIRD model, we predict the spread of COVID-19 by the symptomatic as well as asymptomatic infectious population. Considering reported infection primarily due to symptomatic, we compare the model predicted results with the available data to estimate the dynamics of the asymptomatically infected population. Our data indicate that in the absence of the asymptomatic infectious population, the number of symptomatic cases would have been much less. Therefore, the current progress of the symptomatic infection can be reduced by quarantining the asymptomatically infectious population via extensive or random testing. This study is motivated strictly toward academic pursuit; this theoretical investigation is not meant for influencing policy decisions or public health practices.

Flocking with a q-fold discrete symmetry: Band-to-lane transition in the active Potts model.

We study the q-state active Potts model (APM) on a two-dimensional lattice in which self-propelled particles have q internal states corresponding to the q directions of motion. A local alignment rule inspired by the ferromagnetic q-state Potts model and self-propulsion via biased diffusion according to the internal particle states elicits collective motion at high densities and low noise. We formulate a coarse-grained hydrodynamic theory with which we compute the phase diagrams of the APM for q=4 and q=6 and analyze the flocking dynamics in the coexistence region, where the high-density (polar liquid) phase forms a fluctuating stripe of coherently moving particles on the background of the low-density (gas) phase. A reorientation transition of the phase-separated profiles from transversal band motion to longitudinal lane formation is found, which is absent in the Vicsek model and the active Ising model. The origin of this reorientation transition is revealed by a stability analysis: for large velocities the transverse diffusivity approaches zero and stabilizes lanes. Computer simulations corroborate the analytical predictions of the flocking and reorientation transitions and validate the phase diagrams of the APM.

Ordering kinetics in a q-state random-bond clock model: Role of vortices and interfaces.

In this article, we present a Monte Carlo study of phase transition and coarsening dynamics in the nonconserved two-dimensional random-bond q-state clock model (RBCM) deriving from a pure clock model [Chatterjee et al., Phys. Rev. E 98, 032109 (2018)10.1103/PhysRevE.98.032109]. Akin to the pure clock model, RBCM also passes through two different phases when quenched from a disordered initial configuration representing at infinite temperature. Our investigation of the equilibrium phase transition affirms that both upper (T_{c}^{1}) and lower (T_{c}^{2}) phase transition temperatures decrease with bond randomness strength ε. Effect of ε on the nonequilibrium coarsening dynamics is investigated following independent rapid quenches in the quasi-long-range ordered (QLRO, T_{c}^{2}