An oscillatory system called a plastic bottle oscillator is studied, in which the downflow of water and upflow of air alternate periodically in an upside-down plastic bottle containing water. It is demonstrated that a coupled two-bottle system exhibits in- and antiphase synchronization according to the nature of coupling. A simple ordinary differential equation is deduced to interpret the characteristics of a single oscillator. This model is also extended to coupled oscillators, and the model reproduces the essential features of the experimental observations.

The small-scale dynamo is a process by which turbulent kinetic energy is converted into magnetic energy, and thus it is expected to depend crucially on the nature of the turbulence. In this paper, we present a model for the small-scale dynamo that takes into account the slope of the turbulent velocity spectrum $v\left(\ell \right)\propto {\ell }^{\vartheta }$, where $\ell$ and $v\left(\ell \right)$ are the size of a turbulent fluctuation and the typical velocity on that scale. The time evolution of the fluctuation component of the magnetic field, i.e., the small-scale field, is described by the Kazantsev equation. We solve this linear differential equation for its eigenvalues with the quantum-mechanical WKB approximation. The validity of this method is estimated as a function of the magnetic Prandtl number Pm. We calculate the minimal magnetic Reynolds number for dynamo action, ${\mathrm{Rm}}_{\mathrm{crit}}$, using our model of the turbulent velocity correlation function. For Kolmogorov turbulence ($\vartheta =1/3$), we find that the critical magnetic Reynolds number is ${\mathrm{Rm}}_{\mathrm{crit}}^{\mathrm{K}}\approx 110$ and for Burgers turbulence ($\vartheta =1/2$) ${\mathrm{Rm}}_{\mathrm{crit}}^{\mathrm{B}}\approx 2700$. Furthermore, we derive that the growth rate of the small-scale magnetic field for a general type of turbulence is $\Gamma \propto {\mathrm{Re}}^{(1-\vartheta )/(1+\vartheta )}$ in the limit of infinite magnetic Prandtl number. For decreasing magnetic Prandtl number (down to $\mathrm{Pm}\gtrsim 10$), the growth rate of the small-scale dynamo decreases. The details of this drop depend on the WKB approximation, which becomes invalid for a magnetic Prandtl number of about unity.

We derive a number of solutions for one-dimensional dynamics of relativistic magnetized plasma that can be used as benchmark estimates in relativistic hydrodynamic and magnetohydrodynamic numerical codes. First, we analyze the properties of simple waves of fast modes propagating orthogonally to the magnetic field in relativistically hot plasma. The magnetic and kinetic pressures obey different equations of state, so that the system behaves as a mixture of gases with different polytropic indices. We find the self-similar solutions for the expansion of hot strongly magnetized plasma into vacuum. Second, we derive linear hodograph and Darboux equations for the relativistic Khalatnikov potential, which describe arbitrary one-dimensional isentropic relativistic motion of cold magnetized plasma and find their general and particular solutions. The obtained hodograph and Darboux equations are very powerful: A system of highly nonlinear, relativistic, time-dependent equations describing arbitrary (not necessarily self-similar) dynamics of highly magnetized plasma reduces to a single linear differential equation.

We study theoretically the effect of the variation of the elastic constants on the stability of cholesteric blue phases (BPs). We demonstrate that the stability of BPs is greatly enhanced when the bend elastic constant $K_{33}$ is smaller, in agreement with recent experimental findings. Larger splay ($K_{11}$) and twist ($K_{22}$) elastic constants also stabilize BPs. Our argument sheds light on the subtle effect of the variation of the elastic constants on the stability of BPs, and could guide the synthesis of liquid crystals with a wider temperature range of stable BPs.

Shock wave propagation in multiphase media is typically dominated by the relative compressibility of the two components of the mixture. The difference in the compressibility of the components results in a shock-induced variation in the effective volume fraction of the suspension tending toward the random-close-packing limit for the system, and a disordered solid can take form within the suspension. The present study uses a Hugoniot-based model to demonstrate this variation in the volume fraction of the solid phase as well as a simple hard-sphere model to investigate the formation of disordered structures within uniaxially compressed model suspensions. Both models are discussed in terms of available experimental plate impact data in dense suspensions. Through coordination number statistics of the mesoscopic hard-sphere model, comparisons are made with the trends of the experimental pressure-volume fraction relationship to illustrate the role of these disordered structures in the bulk properties of the suspensions. A criterion for the dynamic stiffening of suspensions under high-rate dynamic loading is suggested as an analog to quasi-static jamming based on the results of the simulations.

The nucleation and growth of point islands during submonolayer deposition on a one-dimensional substrate is simulated for critical island size $i=0,1,2,3$. The small- and large-size asymptotics for the gap-size and capture-zone distributions (CZDs) are studied. Comparisons to theoretical predictions from fragmentation equation analyses are made, along with those from the recently proposed generalized Wigner surmise (GWS). We find that the simulation data can be fully understood in the framework provided by the fragmentation equations, while highlighting the theoretical areas that require further development. The GWS works well for the small-size CZD behavior, but completely fails to describe the large-size CZD asymptotics of the one-dimensional system.

The fluid dynamics of animal locomotion, such as that of an undulating fish, are of great interest to both biologists and engineers. However, experimentally studying these fluid dynamics is difficult and time consuming. Model studies can be of great help because of their simpler and more detailed analysis. Their insights may guide empirical work. Particularly the recently introduced multiparticle collision dynamics method may be suitable for the study of moving organisms because it is computationally fast, simple to implement, and has a continuous representation of space. As regards the study of hydrodynamics of moving organisms, the method has only been applied at low Reynolds numbers (below 120) for soft, permeable bodies, and static fishlike shapes. In the present paper we use it to study the hydrodynamics of an undulating fish at Reynolds numbers 1100–1500, after confirming its performance for a moving insect wing at Reynolds number 75. We measure (1) drag, thrust, and lift forces, (2) swimming efficiency and spatial structure of the wake, and (3) distribution of forces along the fish body. We confirm the resemblance between the simulated undulating fish and empirical data. In contrast to theoretical predictions, our model shows that for steadily undulating fish, thrust is produced by the rear 2/3 of the body and that the slip ratio $U/V$ (with $U$ the forward swimming speed and $V$ the rearward speed of the body wave) correlates negatively (instead of positively) with the actual Froude efficiency of swimming. Besides, we show that the common practice of modeling individuals while constraining their sideways acceleration causes them to resemble unconstrained fish with a higher tailbeat frequency.

We study the statistical properties of the generation of random graphs according to the configuration model, in which one assigns randomly degrees to nodes. This model is often used, for example, for the scale-free degree distribution $\sim d^{-\gamma }$. For the efficient variant, where nonfeasible edges are rejected and the construction of a graph continues, there exists a bias, which we calculate explicitly for a small sample ensemble. We find that this bias does not disappear with growing system size. This becomes visible, for example, also for scale-free graphs when measuring quantities such as the graph diameter. Hence the efficient generation of general scale-free graphs with a very broad distribution ($\gamma <2$) remains an open problem.

Improving the accuracy of forecast models for physical systems such as the atmosphere is a crucial ongoing effort. The primary focus of recent research on these highly nonlinear systems has been errors in state estimation, but as that error has been successfully diminished, the role of model error in forecast uncertainty has duly increased. The present study is an investigation of an empirical model correction procedure involving the comparison of short forecasts with a reference “truth” system during a training period, in order to calculate systematic (1) state-independent model bias and (2) state-dependent error patterns. An estimate of the likelihood of the latter error component is computed from the current state at every time step of model integration. The effectiveness of this technique is explored in a realistic scenario, in which the model is structurally different (in dynamics, dimension, and parametrization) from the target system. Results suggest that the correction procedure is more effective for reducing error and prolonging forecast usefulness than parameter tuning. However, the cost of this increase in average forecast accuracy is the creation of substantial qualitative differences between the dynamics of the corrected model and the true system. A method to mitigate dynamical ramifications and further increase forecast accuracy is presented.

We thoroughly analyze the level statistics and eigenfunctions in concentric as well as nonconcentric square torus billiards. We confirm the characteristics of quantum and classical correspondence and the existence of scarred and superscarred modes in concentric square torus billiards. Furthermore, we not only verify that the transition from regular to chaotic behaviors can be manifested in nonconcentric square torus billiards, but also develop an analytical distribution to excellently fit the numerical level statistics. Finally, we intriguingly observe that numerous eigenstates commonly exhibit the wave patterns to be an ensemble of classical diamond trajectories, as the effective wavelengths are considerably shorter than the size of internal hole.

The length spectra of flat three-dimensional dielectric resonators of circular shape were determined from a microwave experiment. They were compared to a semiclassical trace formula obtained within a two-dimensional model based on the effective index of refraction approximation and a good agreement was found. It was necessary to take into account the dispersion of the effective index of refraction for the two-dimensional approximation. Furthermore, small deviations between the experimental length spectrum and the trace formula prediction were attributed to the systematic error of the effective index of refraction approximation. In summary, the methods developed in this article enable the application of the trace formula for two-dimensional dielectric resonators also to realistic, flat three-dimensional dielectric microcavities and -lasers, allowing for the interpretation of their spectra in terms of classical periodic orbits.

Starting from a classical picture of shear viscosity we construct a stationary velocity gradient in a microscopic parton cascade. Employing the Navier-Stokes ansatz we extract the shear viscosity coefficient $\eta$. For elastic isotropic scatterings we find an excellent agreement with the analytic values. This confirms the applicability of this method. Furthermore, for both elastic and inelastic scatterings with pQCD based cross sections we extract the shear viscosity coefficient $\eta$ for a pure gluonic system and find a good agreement with already published calculations.

We determined the probability distribution of the combined output power from 25 coupled fiber lasers and show that it agrees well with the Tracy-Widom and Majumdar-Vergassola distributions of the largest eigenvalue of Wishart random matrices with no fitting parameters. This was achieved with 500 000 measurements of the combined output power from the fiber lasers, that continuously changes with variations of the fiber lasers lengths. We show experimentally that for small deviations of the combined output power over its mean value the Tracy-Widom distribution is correct, while for large deviations the Majumdar-Vergassola distribution is correct.

For one-dimensional nonlinear lattices with momentum conserving interparticle interactions, intensive studies have suggested that the heat conductivity $\kappa$ diverges with the system size $L$ as $\kappa \sim L^{\alpha }$ and the value of $\alpha$ is universal. But in the Fermi-Pasta-Ulam-$\beta$ lattices with nearest-neighbor (NN) and next-nearest-neighbor (NNN) coupling, we find that $\alpha$ strongly depends on $\gamma$, the ratio of the NNN coupling to the NN coupling. The correlation between the $\gamma$-dependent heat conduction behavior and the in-band discrete breathers is also analyzed.

We investigate finite-size scaling in percolating widthless stick systems with variable aspect ratios in an extensive Monte Carlo simulation study. A generalized scaling function is introduced to describe the scaling behavior of the percolation distribution moments and probability at the percolation threshold. We show that the prefactors in the generalized scaling function depend on the system aspect ratio and exhibit features that are generic to the whole class of the percolating systems. In particular, we demonstrate the existence of a characteristic aspect ratio for which percolation probability at the threshold is scale invariant and definite parity of the prefactors in the generalized scaling function for the first two percolation probability moments.

Starting from the pioneering work of Agarwal [G. S. Agarwal, Zeitschrift für Physik 252, 25 (1972)], we present a unified derivation of a number of modified fluctuation-dissipation relations (MFDR) that relate response to small perturbations around nonequilibrium steady states to steady-state correlations. Using this formalism we show the equivalence of velocity forms of MFDR derived using continuum Langevin and discrete master equation dynamics. The resulting additive correction to the Einstein relation is exemplified using a flashing ratchet model of molecular motors.

We present a spectral linear analysis in terms of advected Fourier modes to describe the behavior of a fluid submitted to four constraints: shear (with rate $S$), rotation (with angular velocity $\Omega$), stratification, and magnetic field within the linear spectral theory or the shearing box model in astrophysics. As a consequence of the fact that the base flow must be a solution of the Euler-Boussinesq equations, only radial and/or vertical density gradients can be taken into account. Ertel's theorem no longer is valid to show the conservation of potential vorticity, in the presence of the Lorentz force, but a similar theorem can be applied to a potential magnetic induction: The scalar product of the density gradient by the magnetic field is a Lagrangian invariant for an inviscid and nondiffusive fluid. The linear system with a minimal number of solenoidal components, two for both velocity and magnetic disturbance fields, is eventually expressed as a four-component inhomogeneous linear differential system in which the buoyancy scalar is a combination of solenoidal components (variables) and the (constant) potential magnetic induction. We study the stability of such a system for both an infinite streamwise wavelength ($k_1=0$, axisymmetric disturbances) and a finite one ($k_1\ne 0$, nonaxisymmetric disturbances). In the former case ($k_1=0)$, we recover and extend previous results characterizing the magnetorotational instability (MRI) for combined effects of radial and vertical magnetic fields and combined effects of radial and vertical density gradients. We derive an expression for the MRI growth rate in terms of the stratification strength, which indicates that purely radial stratification can inhibit the MRI instability, while purely vertical stratification cannot completely suppress the MRI instability. In the case of nonaxisymmetric disturbances $(k_1\ne 0)$, we only consider the effect of vertical stratification, and we use Levinson's theorem to demonstrate the stability of the solution at infinite vertical wavelength ($k_3=0)$: There is an oscillatory behavior for $\tau >1+|K_2/k_1|$, where $\tau =St$ is a dimensionless time and $K_2$ is the radial component of the wave vector at $\tau =0$. The model is suitable to describe instabilities leading to turbulence by the bypass mechanism that can be relevant for the analysis of magnetized stratified Keplerian disks with a purely azimuthal field. For initial isotropic conditions, the time evolution of the spectral density of total energy (kinetic + magnetic + potential) is considered. At $k_3=0$, the vertical motion is purely oscillatory, and the sum of the vertical (kinetic + magnetic) energy plus the potential energy does not evolve with time and remains equal to its initial value. The horizontal motion can induce a rapid transient growth provided $K_2/k_1\gg 1$. This rapid growth is due to the aperiodic velocity vortex mode that behaves like $K_h/k_h$ where $k_h\left(\tau \right)={[k_1^2+{(K_2-k_1\tau )}^2]}^{1/2}$ and $K_h=k_h\left(0\right)$. After the leading phase ($\tau >K_2/k_1\gg 1)$, the horizontal magnetic energy and the horizontal kinetic energy exhibit a similar (oscillatory) behavior yielding a high level of total energy. The contribution to energies coming from the modes $k_1=0$ and $k_3=0$ is addressed by investigating the one-dimensional spectra for an initial Gaussian dense spectrum. For a magnetized Keplerian disk with a purely vertical field, it is found that an important contribution to magnetic and kinetic energies comes from the region near $k_1=0$. The limit at $k_1=0$ of the streamwise one-dimensional spectra of energies, or equivalently, the streamwise two-dimensional (2D) energy, is then computed. The comparison of the ratios of these 2D quantities with their three-dimensional counterparts provided by previous direct numerical simulations shows a quantitative agreement.

We study the evolution of cooperation in evolutionary spatial games when the payoff correlates with the increasing age of players (the level of correlation is set through a single parameter, $\alpha$). The demographic heterogeneous age distribution, directly affecting the outcome of the game, is thus shown to be responsible for enhancing the cooperative behavior in the population. In particular, moderate values of $\alpha$ allow cooperators not only to survive but to outcompete defectors, even when the temptation to defect is large and the ageless, standard $\alpha =0$ model does not sustain cooperation. The interplay between age structure and noise is also considered, and we obtain the conditions for optimal levels of cooperation.

Noise-induced volatility refers to a phenomenon of increased level of fluctuations in the collective dynamics of bistable units in the presence of a rapidly varying external signal, and intermediate noise levels. The archetypical signature of this phenomenon is that—beyond the increase in the level of fluctuations—the response of the system becomes uncorrelated with the external driving force, making it different from stochastic resonance. Numerical simulations and an analytical theory of a stochastic dynamical version of the Ising model on regular and random networks demonstrate the ubiquity and robustness of this phenomenon, which is argued to be a possible cause of excess volatility in financial markets, of enhanced effective temperatures in a variety of out-of-equilibrium systems, and of strong selective responses of immune systems of complex biological organisms. Extensive numerical simulations are compared with a mean-field theory for different network topologies.