Complex Wadati-type potentials of the form $V\left(x\right)=-w^2\left(x\right)+i{w}_x\left(x\right)$, where $w\left(x\right)$ is a real-valued function, are known to possess a number of intriguing features, unusual for generic non-Hermitian potentials. In the present paper, we introduce a class of nonlinear Schrödinger-type problems which generalize the Wadati potentials by assuming that the base function $w\left(x\right)$ depends not only on the transverse spatial coordinate, but also on the amplitude of the field. Several examples of prospective physical relevance are discussed, including models with the nonlinear dispersion or with the derivative nonlinearity. The numerical study indicates that the generalized model inherits the remarkable features of standard Wadati potentials, such as the existence of continuous soliton families, the possibility of symmetry-breaking bifurcations when the model obeys the parity-time symmetry, the existence of constant-amplitude waves, and the eigenvalue quartets in the linear-instability spectra. Our results deepen the current understanding of the interplay between nonlinearity and non-Hermiticity and expand the class of systems which enjoy the exceptional combination of properties unusual for generic dissipative nonlinear models.

We investigate the time evolution of the density of active links and of the entropy of the distribution of agents among opinions in multistate voter models with all-to-all interaction and on uncorrelated networks. Individual realizations undergo a sequence of eliminations of opinions until consensus is reached. After each elimination the population remains in a metastable state. The density of active links and the entropy in these states varies from realization to realization. Making some simple assumptions we are able to analytically calculate the average density of active links and the average entropy in each of these states. We also show that, averaged over realizations, the density of active links decays exponentially, with a timescale set by the size and geometry of the graph, but independent of the initial number of opinion states. The decay of the average entropy is exponential only at long times when there are at most two opinions left in the population. Finally, we show how metastable states comprising only a subset of opinions can be artificially engineered by introducing precisely one zealot in each of the prevailing opinions.

A typical feature of living tissues is proliferation by division: it is a fundamental aspect of many biological processes, including embryonic development, morphogenesis, and cancer growth. Here, we study the energetics of cell division occurring in epithelia, highlighting the interplay of the key parameters ruling mitosis. We find the existence of a region, in the parameter space, which is independent of the cell elasticity and weakly dependent on the ratio between mother and daughter cells areas. In this region, cell division is energetically favorable. Our results may lead to an exact characterization of cells having anomalous proliferation.

We propose a simple scaling theory describing critical effects at rounded meniscus osculation transitions which occur when the Laplace radius of a condensed macroscopic drop of liquid coincides with the local radius of curvature $R_w$ in a confining parabolic geometry. We argue that the exponent ${\beta }_{\mathrm{osc}}$ characterizing the scale of the interfacial height ${\ell }_0\propto R_w^{{\beta }_{\mathrm{osc}}}$ at osculation, for large $R_w$, falls into two regimes representing fluctuation-dominated and mean-field-like behavior, respectively. These two regimes are separated by an upper critical dimension, which is determined here explicitly and depends on the range of the intermolecular forces. In the fluctuation-dominated regime, representing the universality class of systems with short-range forces, the exponent is related to the value of the interfacial wandering exponent $\zeta$ by ${\beta }_{\mathrm{osc}}=3\zeta /(4-\zeta )$. In contrast, in the mean-field regime, which was not previously identified and which occurs for systems with longer-range forces (and higher dimensions), the exponent ${\beta }_{\mathrm{osc}}$ takes the same value as the exponent ${\beta }_s^{\mathrm{co}}$ for complete wetting, which is determined directly by the intermolecular forces. The prediction ${\beta }_{\mathrm{osc}}=3/7$ in $d=2$ for systems with short-range forces (corresponding to $\zeta =1/2$) is confirmed using an interfacial Hamiltonian model which determines the exact scaling form for the decay of the interfacial height probability distribution function. A numerical study in $d=3$, based on a microscopic model density-functional theory, determines that ${\beta }_{\mathrm{osc}}\approx {\beta }_s^{\mathrm{co}}\approx 0.326$ close to the predicted value of $1/3$ appropriate to the mean-field regime for dispersion forces.

We present a theoretical study and numerical simulation of Marangoni convection within ellipsoidal isotropic droplets embedded in free-standing smectic films (FSSFs). The thermocapillary flows are analyzed for both isotropic droplets spontaneously formed in FSSF overheated above the bulk smectic-isotropic transition and oil lenses deposited on the surface of the smectic film. The realistic model for which the upper drop interface is free from the smectic layers, while at the lower drop surface the smectic layering persists is considered in detail. For isotropic droplets and oil lenses this leads effectively to a sticking of fluid motion at the border with a smectic shell. The above mentioned asymmetric configuration is realized experimentally when the temperature of the upper side of the film is higher than at the lower one. The full set of stationary solutions for Stokes stream functions describing the Marangoni convection flows within the ellipsoidal drops are derived analytically. The temperature distribution in the ellipsoidal drop and the surrounding air is determined in the frame of the perturbation theory. As a result, the analytical solutions for the stationary thermocapillary convection are obtained for different droplet ellipticity ratios and the heat conductivity of the liquid crystal and air. In parallel, the numerical hydrodynamic calculations of the thermocapillary motion in drops are made. Both analytical and numerical simulations predict the axially symmetric circulatory convection motion determined by the Marangoni effect at the droplet-free surface. Due to a curvature of the drop interface a temperature gradient along its free surface always exists. Thus, the thermocapillary convection within the ellipsoidal droplets in overheated FSSF is possible for the arbitrarily small Marangoni numbers. Possible experimental observations enabling the checking of our predictions are proposed.

Natural snowflakes exhibit remarkable diversity with a fascinating sixfold symmetry and fractal structure. Efforts to discover the snow crystal growth pattern and mechanism behind it have been made in the past 50 years. Among those, cellular automata (CA) have made certain progress. To investigate the influence of latent heat release in snow crystal growth and have a deeper understanding of crystal growth characteristics, an improved CA model combined with vapor diffusion and heat diffusion is proposed to investigate the snowflake growth process in a two-dimensional and nonisothermal field in this paper. Simulation results show that obtained snow crystal growth behavior is consistent with previous experimental observations and satisfies the crystal growth dynamic theory. Latent heat release impacts the small structure of snow crystals, and it could have a more significant influence on snowflake morphology when vapor diffusion is slow and heat diffusion is dominating.

We present the general theory of Ising transitions in isotropic elastic media with vanishing thermal expansion. By constructing a minimal model with appropriate spin-lattice couplings, we show that in two dimensions near a continuous transition the elasticity is anomalous in unusual ways: the system either significantly stiffens with a hitherto unknown unique, positional order logarithmically stronger than quasi-long-range order, or, as the inversion asymmetry of the order parameter in its coupling with strain increases, it destabilizes when system size $L$ exceeds a finite threshold. In three dimensions, stronger inversion-asymmetric couplings induce instability to the long-range positional order for all $L$. Sufficiently strong order parameter-displacement couplings can also turn the phase transition first order at all dimensions, concomitant with finite jumps in the elastic modulii across the transition. Our theory establishes a one-to-one correspondence between the order of the phase transitions and anomalous elasticity near the transitions.

We present a microscopic derivation of the laws of continuum mechanics of nonideal ordered solids including dissipation, defect diffusion, and heat transport. The starting point is the classical many-body Hamiltonian. The approach relies on the Zwanzig-Mori projection operator formalism to connect microscopic fluctuations to thermodynamic derivatives and transport coefficients. Conservation laws and spontaneous symmetry breaking, implemented via Bogoliubov's inequality, determine the selection of the slow variables. Density fluctuations in reciprocal space encode the displacement field and the defect concentration. Isothermal and adiabatic elastic constants are obtained from equilibrium correlations, while transport coefficients are given as Green-Kubo formulas, providing the basis for their measurement in atomistic simulations or colloidal experiments. The approach to the linearized continuum mechanics and results are compared to others from the literature.

The time-evolution operator obtained from the fractional-time Schrödinger equation (FTSE) is said to be nonunitary since it does not preserve the norm of the vector state in time. As done in the time-dependent non-Hermitian quantum formalism, for a traceless non-Hermitian two-level quantum system, we demonstrate that it is possible to map the nonunitary time-evolution operator in a unitary one. It is done by considering a dynamical Hilbert space with a time-dependent metric operator, constructed from a Hermitian time-dependent Dyson map, in respect to which the system evolves in a unitary way, and the standard quantum mechanics interpretation can be made properly. To elucidate our approach, we consider three examples of Hamiltonian operators and their corresponding unitary dynamics obtained from the solutions of FTSE, and the respective Dyson maps.

We investigate the propagation of waves in one-dimensional systems with Lévy-type disorder. We perform a complete analysis of nonrelativistic and relativistic wave transmission submitted to potential barriers whose width, separation, or both follow Lévy distributions characterized by an exponent $0<\alpha <1$. For the first two cases, where one of the parameters is fixed, nonrelativistic and relativistic waves present anomalous localization, $\langle T\rangle \propto L^{-\alpha }$. However, for the latter case, in which both parameters follow a Lévy distribution, nonrelativistic and relativistic waves present a transition between anomalous and standard localization as the incidence energy increases relative to the barrier height. Moreover, we obtain the localization diagram delimiting anomalous and standard localization regimes, in terms of incidence angle and energy. Finally, we verify that transmission fluctuations, characterized by its standard deviation, are universal, independent of barrier architecture, wave equation type, incidence energy, and angle, further extending earlier studies on electronic localization.

Coherence-resonance chimera was discovered in [Phys. Rev. Lett. 117, 014102 (2016)], which combines the effect of coherence-resonance and classical chimeras in the presence of noise in a network of type-II excitable systems. However, the same in a network of type-I excitable units has not been observed yet. In this paper we report the occurrence of coherence-resonance chimera in coupled type-I excitable systems. We consider a paradigmatic model of type-I excitability, namely, the saddle-node infinite period model, and show that the coherence-resonance chimera appears over an optimum range of noise intensity. Moreover, we discover a unique chimera pattern that is a mixture of classical chimera and the coherence-resonance chimera. We support our results using quantitative measures and map them in parameter space. This study reveals that the coherence-resonance chimera is a general chimera pattern and thus it deepens our understanding of role of noise in coupled excitable systems.

Ensemble models of graphs are one of the most important theoretical tools to study complex networks. Among them, exponential random graphs (ERGs) have proven to be very useful in the analysis of social networks. In this paper we develop a technique, borrowed from the statistical mechanics of lattice gases, to solve Strauss's model of transitive networks. This model was introduced long ago as an ERG ensemble for networks with high clustering and exhibits a first-order phase transition above a critical value of the triangle interaction parameter where two different kinds of networks with different densities of links (or, alternatively, different clustering) coexist. Compared to previous mean-field approaches, our method describes accurately even small networks and can be extended beyond Strauss's classical model—e.g., to networks with different types of nodes. This allows us to tackle, for instance, models with node homophily. We provide results for the latter and show that they accurately reproduce the outcome of Monte Carlo simulations.

With the use of simulations of a stochastic microscopic traffic model in the framework of the three-phase traffic theory, we have revealed the statistical physics of a traffic flow instability with respect to a transition from synchronized flow (S) to free flow (F) (Kerner's $\mathrm{S}\to \mathrm{F}$ instability) at a moving bottleneck (MB) occurring through a slow-moving vehicle in vehicular traffic. We have found that the $\mathrm{S}\to \mathrm{F}$ instability can occur at the MB more frequently than at an on-ramp bottleneck. From a comparison of the occurrence of the $\mathrm{S}\to \mathrm{F}$ instability at the MB and on-ramp bottleneck at the same probability of traffic breakdown and the same flow rate it has been found that, whereas the frequency of the $\mathrm{S}\to \mathrm{F}$ instability at the on-ramp bottleneck barely changes, the larger the velocity of the MB, the more frequently the $\mathrm{S}\to \mathrm{F}$ instability occurs at the MB. Contrarily, when the MB velocity decreases considerably, then rather than the $\mathrm{S}\to \mathrm{F}$ instability, in synchronized flow at the MB the classical traffic flow instability leading to the emergence of wide-moving jams ($\mathrm{S}\to \mathrm{J}$ instability) occurs. It has been found that the physics of the intensification of the $\mathrm{S}\to \mathrm{F}$ instability at the MB with the increase in the MB velocity is associated with the increase in the mean space gap (mean time headway) between vehicles in synchronized flow. For this reason, when the MB velocity increases, there is an MB velocity at which the $\mathrm{S}\to \mathrm{F}$ instability dominates the $\mathrm{S}\to \mathrm{J}$ instability: The MB velocity influences considerably on the competition between the $\mathrm{S}\to \mathrm{F}$ and classical traffic flow instabilities in synchronized flow.

The thermal conductivity of glasses exhibits an unusual temperature dependence compared to their crystalline counterparts. Sound attenuation due to disorder in glasses was proposed to be important in rationalizing this special behavior. Simulation studies suggest that in the harmonic approximation, the sound attenuation follows Rayleigh scattering scaling at small wave vector in both two-dimensional (2D) and 3D glasses. The influence of the anharmonicity on sound attenuation has very recently been investigated numerically, but only in 3D glasses. Hence, it remains unknown in simulations how sound attenuation changes with the wave vector in 2D glasses when the anharmonicity comes into play. Here, we address this issue by performing computer simulations in low-temperature 2D glasses over a large range of glass stabilities. We find that the way the anharmonicity affects sound attenuation in 2D glasses is the same as that in 3D, thus revealing that numerically the influence of the anharmonicity on sound attenuation does not rely on the spatial dimension.

We propose a simulation method for Brownian dynamics of hard rods in one dimension for arbitrary continuous external force fields. It is an event-driven procedure based on the fragmentation and mergers of clusters formed by particles in contact. It allows one to treat particle interactions in addition to the hard-sphere exclusion as long as the corresponding interaction forces are continuous functions of the particle coordinates. We furthermore develop a treatment of sticky hard spheres as described by Baxter's contact interaction potential.

The evaporation characteristics of self-rewetting fluids have attracted much attention in recent years. However, the evaporation dynamics as well as the underlying evaporation mechanism of self-rewetting fluid droplets has not been well understood. In this paper, we numerically investigate the evaporation performance and the dynamic behavior of self-rewetting fluid droplets on chemically patterned surfaces using a thermal multiphase lattice Boltzmann model with liquid-vapor phase change. First, it is shown that a self-rewetting fluid droplet can spontaneously separate into two droplets during its evaporation on a hydrophilic surface with a hydrophobic stripe, while no separation occurs during the evaporation of a conventional fluid droplet. The positive surface tension gradient of the self-rewetting fluid is found to play an important role in the spontaneous separation of the self-rewetting fluid droplet during the evaporation. Meanwhile, the separation behavior of the self-rewetting fluid droplet can effectively increase the length of the triple-phase contact line, which leads to a significant increase in the evaporation rate as compared with that of a conventional fluid droplet. Moreover, by investigating the evaporation performance of self-rewetting fluid droplets on chemically stripe-patterned surfaces with different values of the widths of the hydrophilic and hydrophobic stripes, it is found that the stripe width and the initial location of the droplet significantly affect the dynamic behavior and the evaporation efficiency of the self-rewetting fluid droplet. For different relative positions between the droplet and the stripes, the droplet may spontaneously separate into two or three droplets and achieve much better evaporation efficiency when the stripe width is within an optimal range.

Magnetic reconnection in laser-produced magnetized plasma is investigated by using optical diagnostics. The magnetic field is generated via the Biermann battery effect, and the inversely directed magnetic field lines interact with each other. It is shown by self-emission measurement that two colliding plasmas stagnate on a midplane, forming two planar dense regions, and that they interact later in time. Laser Thomson scattering spectra are distorted in the direction of the self-generated magnetic field, indicating asymmetric ion velocity distribution and plasma acceleration. In addition, the spectra perpendicular to the magnetic field show different peak intensity, suggesting an electron current formation. These results are interpreted as magnetic field dissipation, reconnection, and outflow acceleration. Two-directional laser Thomson scattering is, as discussed here, a powerful tool for the investigation of microphysics in the reconnection region.

We present a detailed description of the essentially entropic lattice Boltzmann model. The entropic lattice Boltzmann model guarantees unconditional numerical stability by iteratively solving the nonlinear entropy evolution equation. In this paper we explain the construction of closed-form analytic solutions to this equation. We demonstrate that near equilibrium this analytic solution reduces to the standard lattice Boltzmann model. We consider a few test cases to show that the analytic solution does not exhibit any significant deviation from the iterative solution. We also extend the analytical solution for the Ellipsoidal Statistical (ES)–Bhatnagar-Gross-Krook model to remove the limitation on the Prandtl number for heat transfer problems. The simplicity of the analytic solution removes the computational overhead and algorithmic complexity associated with the entropic lattice Boltzmann models.

The lattice Boltzmann method is adopted to solve the liquid-vapor phase change problems in this article. By modifying the collision term for the temperature evolution equation, a thermal lattice Boltzmann model is constructed. As compared with previous studies, the most striking feature of the present approach is that it could avoid the calculations of both the Laplacian term of temperature [$\mathbf{\nabla }·\left(\kappa \mathbf{\nabla }T\right)$] and the gradient term of heat capacitance [$\mathbf{\nabla }\left(\rho {\mathrm{c}}_v\right)$]. In addition, since the present approach adopts a simple linear equilibrium distribution function, it is possible to use the D2Q5 lattice for the two-dimensional cases considered here. Thus, the present model is more efficient than previous models in which the lattice is usually limited to the D2Q9. The proposed model is first validated by the problems of droplet evaporation in open space and droplet evaporation on a heated surface, and the numerical results show good agreement with the analytical results and the finite difference method. Then it is used to model the nucleate boiling problem, and the relationship between detachment bubble diameter and gravitational acceleration obtained with the present approach fits well with previous works.

Neural networks have been used successfully in a variety of fields, which has led to a great deal of interest in developing a theoretical understanding of how they store the information needed to perform a particular task. We study the weight matrices of trained deep neural networks using methods from random matrix theory (RMT) and show that the statistics of most of the singular values follow universal RMT predictions. This suggests that they are random and do not contain system specific information, which we investigate further by comparing the statistics of eigenvector entries to the universal Porter-Thomas distribution. We find that for most eigenvectors the hypothesis of randomness cannot be rejected, and that only eigenvectors belonging to the largest singular values deviate from the RMT prediction, indicating that they may encode learned information. In addition, a comparison with RMT predictions also allows to distinguish networks trained in different learning regimes—from lazy to rich learning. We analyze the spectral distribution of the large singular values using the Hill estimator and find that the distribution cannot in general be characterized by a tail index, i.e., is not of power-law type.