The heterogeneity of human populations is a challenge to mathematical descriptions of epidemic outbreaks. Numerical simulations are deployed to account for the many factors influencing the spreading dynamics. Yet, the results from numerical simulations are often as complicated as the reality, leaving us with a sense of confusion about how the different factors account for the simulation results. Here, using a multitype branching together with a graph tensor product approach, I derive a single equation for the effective reproductive number of an infectious disease outbreak. Using this equation I deconvolute the impact of crowd management, targeted testing, contact heterogeneity, stratified vaccination, mask use, and smartphone tracing app use. This equation can be used to gain a basic understanding of infectious disease outbreaks and their simulations.

The fluctuation-dissipation theorem connects equilibrium to mildly (linearly) perturbed situations in a thermodynamic manner: It involves the observable of interest and the entropy production caused by the perturbation. We derive a relation which connects responses of arbitrary order in perturbation strength to correlations of entropy production of lower order, thereby extending the fluctuation-dissipation theorem to cases far from equilibrium in a thermodynamic way. The relation is validated and studied for a four-state model which is coarse-grained to a non-Markovian two-state model.

Neodyme spherical magnets are inexpensive objects that demonstrate how dipolar particles self-assemble into various structures ranging from 1D chains to 3D crystals. The dipole-dipole interactions confer the stability to these particular architectures. In the present paper, we explore ordered structures only, and we evidence that hybrid magnetocrystals, alternating hexagonal planes of antiparallel dipoles, have the lowest magnetic energy. This cohesion is the magnetic counterpart of the Madelung lattice energy found for ionic solids.

We distinguish traditional implementations of autonomous Maxwell demons from related linear devices that were recently proposed, not relying on the notions of measurements and feedback control. In both cases a current seems to flow against its spontaneous direction (imposed, e.g., by a thermal or electric gradient) without external energy intake. However, in the latter case, this current inversion may only be apparent. Even if the currents exchanged between a system and its reservoirs are inverted (by creating additional independent currents between system and demon), this is not enough to conclude that the original current through the system has been inverted. We show that this distinction can be revealed locally by measuring the fluctuations of the system-reservoir currents.

We consider point particles in a table made of two circular cavities connected by two rectangular channels, forming a closed loop under periodic boundary conditions. In the first channel, a bounce-back mechanism acts when the number of particles flowing in one direction exceeds a given threshold $T$. In that case, the particles invert their horizontal velocity, as if colliding with vertical walls. The second channel is divided in two halves parallel to the first but located in the opposite sides of the cavities. In the second channel, motion is free. We show that, suitably tuning the sizes of cavities of the channels and of $T$, nonequilibrium phase transitions take place in the $N\to \infty$ limit. This induces a stationary current in the circuit, thus modeling a kind of battery, although our model is deterministic, conservative, and time reversal invariant.

The phenomenon of degeneracy of an $N$-plet of bound states is studied in the framework of the quasi-Hermitian (a.k.a. $\mathcal{P}T$-symmetric) formulation of quantum theory of closed systems. For a general non-Hermitian Hamiltonian $H=H\left(\lambda \right)$ such a degeneracy may occur at a real Kato's exceptional point ${\lambda }^{\left(\text{EPN}\right)}$ of order $N$ and of the geometric multiplicity alias  clusterization index $K$. The corresponding unitary process of collapse (loss of observability) can be then interpreted as a generic quantum phase transition. The dedicated literature deals, predominantly, with the non-numerical benchmark models of the simplest processes where $K=1$. In our present paper it is shown that in the “anomalous” dynamical scenarios with $1<K\le N/2$ an analogous approach is applicable. A multiparametric anharmonic-oscillator-type exemplification of such systems is constructed as a set of real-matrix $N$ by $N$ Hamiltonians which are exactly solvable, maximally non-Hermitian, and labeled by specific ad hoc  partitionings $\mathcal{R}\left(N\right)$ of $N$.

A wake of vortices with sufficiently spaced cores may be represented via the point-vortex model from classical hydrodynamics. We use potential theory representations of vortices to examine the emergence and stability of complex vortex wakes, more particularly the von Kármán vortex street composed of regular polygonal-like clusters of same-signed vortices. We investigate the existence and stability of these streets represented through spatially periodic vortices. We introduce a physically inspired point-vortex model that captures the stability of infinite vortex streets with a finite number of procedurally generated vortices, allowing for numerical analysis of the behavior of vortex streets as they dynamically form.

By means of analytical and numerical methods, we address the modulational instability (MI) in chiral condensates governed by the Gross-Pitaevskii equation including the current nonlinearity. The analysis shows that this nonlinearity partly suppresses the MI driven by the cubic self-focusing, although the current nonlinearity is not represented in the system's energy (although it modifies the momentum), hence it may be considered as zero-energy nonlinearity. Direct simulations demonstrate generation of trains of stochastically interacting chiral solitons by MI. In the ring-shaped setup, the MI creates a single traveling solitary wave. The sign of the current nonlinearity determines the direction of propagation of the emerging solitons.

The Kuramoto model serves as an illustrative paradigm for studying the synchronization transitions and collective behaviors in large ensembles of coupled dynamical units. In this paper, we present a general framework for analytically capturing the stability and bifurcation of the collective dynamics in oscillator populations by extending the global coupling to depend on an arbitrary function of the Kuramoto order parameter. In this generalized Kuramoto model with rotation and reflection symmetry, we show that all steady states characterizing the long-term macroscopic dynamics can be expressed in a universal profile given by the frequency-dependent version of the Ott-Antonsen reduction, and the introduced empirical stability criterion for each steady state degenerates to a remarkably simple expression described by the self-consistent equation [Iatsenko et al., Phys. Rev. Lett. 110, 064101 (2013)]. Here, we provide a detailed description of the spectrum structure in the complex plane by performing a rigorous stability analysis of various steady states in the reduced system. More importantly, we uncover that the empirical stability criterion for each steady state involved in the system is completely equivalent to its linear stability condition that is determined by the nontrivial eigenvalues (discrete spectrum) of the linearization. Our study provides a new and widely applicable approach for exploring the stability properties of collective synchronization, which we believe improves the understanding of the underlying mechanisms of phase transitions and bifurcations in coupled dynamical networks.

The counterintuitive phenomenon of coherence resonance describes a nonmonotonic behavior of the regularity of noise-induced oscillations in the excitable regime, leading to an optimal response in terms of regularity of the excited oscillations for an intermediate noise intensity. We study this phenomenon in populations of FitzHugh-Nagumo (FHN) neurons with different coupling architectures. For networks of FHN systems in an excitable regime, coherence resonance has been previously analyzed numerically. Here we focus on an analytical approach studying the mean-field limits of the globally and locally coupled populations. The mean-field limit refers to an averaged behavior of a complex network as the number of elements goes to infinity. We apply the mean-field approach to the globally coupled FHN network. Further, we derive a mean-field limit approximating the locally coupled FHN network with low noise intensities. We study the effects of the coupling strength and noise intensity on coherence resonance for both the network and the mean-field models. We compare the results of the mean-field and network frameworks and find good agreement in the globally coupled case, where the correspondence between the two approaches is sufficiently good to capture the emergence of coherence resonance, as well as of anticoherence resonance.

In this work we have used lattice Monte Carlo to determine the orientational order of a system of biaxial particles confined between two walls inducing perfect order and subjected to an electric field perpendicular to the walls. The particles are set to interact with their nearest neighbors through a biaxial version of the Lebwohl-Lasher potential. A particular set of values for the molecular reduced polarizabilities defining the potential used was considered; the Metropolis sampling algorithm was used in the Monte Carlo simulations. The relevant order parameters were determined in the middle plane of the sample and for some cases across the whole thickness of the sample. We have determined the temperature–electric field phase diagram for this system and found, as expected, five different system configurations corresponding to three different mesophases. At low temperatures and low fields the system finds itself in an undistorted biaxial phase. On increasing the field at low temperatures, a Freedericksz transition takes place and the secondary directors reorient towards the field while the primary director stays undistorted and parallel to the walls. On increasing the field further, a second Freedericksz transition occurs and the primary director orients also towards the field direction. The orientational order measured at the field strengths tested is not affected by the field. On increasing the temperature, a transition to a uniaxial phase occurs and within the range of this phase a field increase leads also to a Freedericksz transition where the main director reorients towards the field. At higher temperature a transition to a disordered phase is found. We have performed finite size scaling analysis for the Freedericksz critical fields and found that they scale with the distance $L$ between the walls as $L^{–1}$ as expected from continuum theory. From these fields we have also determined the temperature dependence of two elastic constant ratios. Critical exponents and critical temperatures for the order parameter and the correlation length for the biaxial-uniaxial phase transition and the uniaxial to disordered phase transition were also determined by finite size scaling and are discussed.

The rarefied, oscillatory, pressure-driven binary gas mixture flow between parallel plates is computationally investigated in terms of the mixture molar fraction and molecular mass ratio of the species, in a wide range of gas rarefaction and oscillation frequency. Modeling is based on the McCormack kinetic model. The output quantities are in dimensionless form and include the flow rate, wall shear stress and pumping power of the mixture, as well as the velocity and shear stress distributions and flow rates of the species. The presented results are for He-Xe and Ne-Ar. The heavier species are affected more drastically than the lighter ones from the inertial forces, resulting to large differences between the flow rate amplitudes of the species, which are increased as the flow becomes less rarefied, provided that the oscillation frequency is adequately high. At very high frequencies the ratio of the flow rate amplitudes of the light over the heavy species tends to the inverse of their molecular mass ratio in the whole range of gas rarefaction. The velocity overshooting effect becomes more pronounced as the molecular mass is increased. The mixture flow rate amplitude is larger, while its phase angle is smaller, than the corresponding ones of single gas, and they both vary nonmonotonically with the molar fraction. The effect of the mixture composition on the wall shear stress and pumping power is small. The present work may be useful in the design of gas separation devices, operating at moderate and high frequencies in rarefied and dense atmospheres.

Acousotophoretic particle separations in counterpropagating surface acoustic wave (SAW) fields, e.g., standing SAWs (SSAWs), phase modulated SSAWs, tilted angle SSAWs, and partial standing SAWs, have proven successful. But there still lacks analytical tools for predicting the particle trajectory and optimizing the device designs. Here, we study the acoustophoresis of spherical Rayleigh particles in counterpropagating SAW fields and find that particle motions can be characterized into two distinct modes, the drift mode and the locked mode. Through theoretical studies, we provide analytical expressions of particle trajectories in different fields and different moving patterns. Based on these, we obtain theory-based protocols for designing such SAW acoustofluidic particle separation chips, which are demonstrated through finite-element simulations. The results here provide theoretical guidelines for designing high throughput and high efficiency particle separation devices.

In a geographically distributed population, assortative clustering plays an important role in evolution by modifying local environments. To examine its effects in a linear habitat, we consider a one-dimensional grid of cells, where each cell is either empty or occupied by an organism whose replication strategy is genetically inherited to offspring. The strategy determines whether to have offspring in surrounding cells, as a function of the neighborhood configuration. If more than one offspring compete for a cell, then they can be all exterminated due to the cost of conflict depending on environmental conditions. We find that the system is more densely populated in an unfavorable environment than in a favorable one because only the latter has to pay the cost of conflict. This observation agrees reasonably well with a mean-field analysis which takes assortative clustering of strategies into consideration. Our finding suggests a possibility of intrinsic nonlinearity between environmental conditions and population density when an evolutionary process is involved.

Despite the fact that conserved currents have dimensions that are determined solely by dimensional analysis (and hence no anomalous dimensions), Nature abounds in examples of anomalous diffusion in which $L\propto t^{\gamma }$, with $\gamma \ne 1/2$, and heat transport in which the thermal conductivity diverges as $L^{\alpha }$. Aside from breaking of Lorentz invariance, the true common link in such problems is an anomalous dimension for the underlying conserved current, thereby violating the basic tenet of field theory. We show here that the phenomenological nonlocal equations of motion that are used to describe such anomalies all follow from Lorentz-violating gauge transformations arising from Noether's second theorem. The generalizations lead to a family of diffusion and heat transport equations that systematize how nonlocal gauge transformations generate more general forms of Fick's and Fourier's laws for diffusive and heat transport, respectively. In particular, the associated Goldstone modes of the form $\omega \propto k^{\alpha }, \alpha \in \mathbb{R}$ are direct consequences of fractional equations of motion.

We derive and asymptotically analyze mass-action models for disease spread that include transient pair formation and dissociation. Populations of unpaired susceptible individuals and infected individuals are distinguished from the population of three types of pairs of individuals: both susceptible, one susceptible and one infected, and both infected. Disease transmission can occur only within a pair consisting of one susceptible individual and one infected individual. We use perturbation expansion to formally derive uniformly valid approximations for the dynamics of the total infected and susceptible populations under different conditions including combinations of fast association, fast transmission, and fast dissociation limits. The effective equations are derived from the fundamental mass-action system without implicitly imposing transmission mechanisms, such as those used in frequency-dependent models. Our results represent submodels that show how effective nonlinear transmission can arise from pairing dynamics and are juxtaposed with density-based mass-action and frequency-based models.

Cellulose nanocrystals (CNCs) are interesting for the construction of biomaterials for energy delivery and packaging purposes. The corresponding processing of CNCs can be optimized through the variation of intercellulose interactions by employing different types of solvents, and thereby varying the degree of cellulose hydrogen bonding. The aim of this work is (i) to show how different types of solvents affect the self-assembled morphology of CNCs, (ii) to study the microscopic dynamics and averaged orientations on the CNCs in aqueous suspensions, including the effect of externally imposed electric fields, and (iii) to explore the nonlinear optical response of CNCs. The homogeneity of self-assembled chiral-nematic phase depends on both the polarity of the solvent and the CNC concentration. The variation of the chiral-nematic pitch length with concentration, as determined from real-space and Fourier images, is found to be strongly solvent dependent. The anisotropic microdynamics of CNCs suspension exhibits two modes, related to diffusion parallel and perpendicular to the (chiral-) nematic director. We have found also the coupling between translational and orientational motion, due to existing correlation length of twisted nematic elasticity. Preliminary second-harmonic generation experiments are performed, which reveal that relatively high field strengths are required to reorient chiral-nematic domains of CNCs.

We consider the mutator model with unidirected transitions from the wild type to the mutator type, with different fitness functions for the wild types and mutator types. We calculate both the fraction of mutator types in the population and the surpluses, i.e., the mean number of mutations in the regular part of genomes for the wild type and mutator type, which have never been derived exactly. We identify the phase structure. Beside the mixed (ordinary evolution phase with finite fraction of wild types at large genome length) and the mutator phase (the absolute majority is mutators), we find another new phase as well—it has the mean fitness of the mixed phase but an exponentially small (in genome length) fraction of wild types. We identify the phase transition point and discuss its implications.

We study a system of Kuramoto oscillators arranged on a two-dimensional periodic lattice where the oscillators interact with their nearest neighbors, and all oscillators have the same natural frequency. The initial phases of the oscillators are chosen to be distributed uniformly between $(-\pi ,\pi ]$. During the relaxation process to the final stationary phase, we observe different features in the phase field of the oscillators: initially, the state is randomly oriented, then clusters form. As time evolves, the size of the clusters increases and vortices that constitute topological defects in the phase field form in the system. These defects, being topological, annihilate in pairs; i.e., a given defect annihilates if it encounters another defect with opposite polarity. Finally, the system ends up either in a completely phase synchronized state in case of complete annihilation or a metastable phase locked state characterized by presence of vortices and antivortices. The basin volumes of the two scenarios are estimated. Finally, we carry out a duality transformation similar to that carried out for the $\mathit{XY}$ model of planar spins on the Hamiltonian version of the Kuramoto model to expose the underlying vortex structure.