We report finite-size numerical investigations and mean-field analysis of a Kuramoto model with inertia for fully coupled and diluted systems. In particular, we examine, for a gaussian distribution of the frequencies, the transition from incoherence to coherence for increasingly large system size and inertia. For sufficiently large inertia the transition is hysteretic, and within the hysteretic region clusters of locked oscillators of various sizes and different levels of synchronization coexist. A modification of the mean-field theory developed by Tanaka, Lichtenberg, and Oishi [Physica D 100, 279 (1997)] allows us to derive the synchronization curve associated to each of these clusters. We have also investigated numerically the limits of existence of the coherent and of the incoherent solutions. The minimal coupling required to observe the coherent state is largely independent of the system size, and it saturates to a constant value already for moderately large inertia values. The incoherent state is observable up to a critical coupling whose value saturates for large inertia and for finite system sizes, while in the thermodinamic limit this critical value diverges proportionally to the mass. By increasing the inertia the transition becomes more complex, and the synchronization occurs via the emergence of clusters of whirling oscillators. The presence of these groups of coherently drifting oscillators induces oscillations in the order parameter. We have shown that the transition remains hysteretic even for randomly diluted networks up to a level of connectivity corresponding to a few links per oscillator. Finally, an application to the Italian high-voltage power grid is reported, which reveals the emergence of quasiperiodic oscillations in the order parameter due to the simultaneous presence of many competing whirling clusters.

In the warm dense matter (WDM) regime, material properties like diffusion and viscosity can be obtained from lengthy quantum molecular dynamics simulations, where the quantum behavior of the electrons is represented using either Kohn-Sham or orbital-free density functional theory. To reduce the simulation duration, we fit the time dependence of the autocorrelation functions (ACFs) and then use the fit to find values of the diffusion and viscosity. This fitting procedure avoids noise in the long time behavior of the ACFs. We present a detailed analysis of the functional form used to fit the ACFs, which is always a more efficient means to obtain mass transport properties. We use the fits to estimate the statistical error of the transport properties. We apply this methodology to a dense correlated plasma of copper and a mixture of carbon and hydrogen. Both systems show structure in their ACFs and exhibit multiple time scales. The mixture contains different structural forms of the ACFs for each component in the mixture.

It is commonly believed as a fundamental principle that energy-momentum conservation of a physical system is the result of space-time symmetry. However, for classical particle-field systems, e.g., charged particles interacting through self-consistent electromagnetic or electrostatic fields, such a connection has only been cautiously suggested. It has not been formally established. The difficulty is due to the fact that the dynamics of particles and the electromagnetic fields reside on different manifolds. We show how to overcome this difficulty and establish the connection by generalizing the Euler-Lagrange equation, the central component of a field theory, to a so-called weak form. The weak Euler-Lagrange equation induces a new type of flux, called the weak Euler-Lagrange current, which enters conservation laws. Using field theory together with the weak Euler-Lagrange equation developed here, energy-momentum conservation laws that are difficult to find otherwise can be systematically derived from the underlying space-time symmetry.

We demonstrate controllable parabolic-cylinder optical rogue waves in certain inhomogeneous media. An analytical rogue wave solution of the generalized nonlinear Schrödinger equation with spatially modulated coefficients and an external potential in the form of modulated quadratic potential is obtained by the similarity transformation. Numerical simulations are performed for comparison with the analytical solutions and to confirm the stability of the rogue wave solution obtained. These optical rogue waves are built by the products of parabolic-cylinder functions and the basic rogue wave solution of the standard nonlinear Schrödinger equation. Such rogue waves may appear in different forms, as the hump and paw profiles.

We propose a possibility to simulate the exciton-polariton (EP) system in the lossless limit, which is not currently available in semiconductor microcavities, by means of a simple optical dual-core waveguide, with one core carrying the nonlinearity and operating close to the zero-group-velocity-dispersion point, and the other core being linear and dispersive. Both two-dimensional (2D) and one-dimensional (1D) EP systems may be emulated by means of this optical setting. In the framework of this system, we find that, while the uniform state corresponding to the lower branch of the nonlinear dispersion relation is stable against small perturbations, the upper branch is always subject to the modulational instability. The stability and instability are verified by direct simulations too. We analyze collective excitations on top of the stable lower-branch state, which include a Bogoliubov-like gapless mode and a gapped one. Analytical results are obtained for the corresponding sound velocity and energy gap. The effect of a uniform phase gradient (superflow) on the stability is considered too, with a conclusion that the lower-branch state becomes unstable above a critical wave number of the flux. Finally, we demonstrate that the stable 1D state may carry robust dark solitons.

Relaxation processes driven by a Laplacian matrix can be found in many real-world big-data systems, for example, in search engines on the World Wide Web and the dynamic load-balancing protocols in mesh networks. To numerically implement such processes, a fast-running algorithm for the calculation of the pseudoinverse of the Laplacian matrix is essential. Here we propose an algorithm which computes quickly and efficiently the pseudoinverse of Markov chain generator matrices satisfying the detailed-balance condition, a general class of matrices including the Laplacian. The algorithm utilizes the renormalization of the Gaussian integral. In addition to its applicability to a wide range of problems, the algorithm outperforms other algorithms in its ability to compute within a manageable computing time arbitrary elements of the pseudoinverse of a matrix of size millions by millions. Therefore our algorithm can be used very widely in analyzing the relaxation processes occurring on large-scale networked systems.

We employ a simple model for rotational diffusivity $D_R$ of dumbbells in porous media in order to study spatially heterogeneous and non-Gaussian dynamics at Fickian time scales. We obtain the distribution $P\left(D_R\right)$ of $D_R$'s of single dumbbells for both ergodic and nonergodic systems. When a pore percolating network disappears beyond the pore percolation transition and the rotational dynamics becomes nonergodic, each single dumbbell undergoes Gaussian rotational dynamics but with different $D_R$, which depends solely on the local pore structure. We also construct a map of heterogeneous dynamic regions and illustrate that such seemingly Fickian but non-Gaussian dynamics could be understood as the linear combination of the Gaussian rotational displacement distribution functions of each dumbbell. With a percolating pore network, the rotational dynamics becomes ergodic, and $P(D_R$) is a $\delta$ function at the average value of $D_R$.

We derive the critical nearest-neighbor connectivity $g_{\mathrm{n}}$ as $3/4$, $3(7-9p_c^{\mathrm{tri}})/4(5-4p_c^{\mathrm{tri}})$, and $3(2+7p_c^{\mathrm{tri}})/4(5-p_c^{\mathrm{tri}})$ for bond percolation on the square, honeycomb, and triangular lattice, respectively, where $p_c^{\mathrm{tri}}=2sin(\pi /18)$ is the percolation threshold for the triangular lattice, and confirm these values via Monte Carlo simulations. On the square lattice, we also numerically determine the critical next-nearest-neighbor connectivity as $g_{\mathrm{nn}}=0.6875000\left(2\right)$, which confirms a conjecture by Mitra and Nienhuis [J. Stat. Mech. (2004) P10006], implying the exact value $g_{\mathrm{nn}}=11/16$. We also determine the connectivity on a free surface as $g_{\mathrm{n}}^{\mathrm{surf}}=0.6250001\left(13\right)$ and conjecture that this value is exactly equal to $5/8$. In addition, we find that at criticality, the connectivities depend on the linear finite size $L$ as $\sim L^{y_t-d}$, and the associated specific-heat-like quantities $C_{\mathrm{n}}$ and $C_{\mathrm{nn}}$ scale as $\sim L^{2y_t-d}ln(L/L_0)$, where $d$ is the lattice dimensionality, $y_t=1/\nu$ the thermal renormalization exponent, and $L_0$ a nonuniversal constant. We provide an explanation of this logarithmic factor within the theoretical framework reported recently by Vasseur et al. [J. Stat. Mech. (2012) L07001].

In the context of slow quenching dynamics of a $p$-wave superconducting chain, it has been shown that a Majorana edge state cannot be adiabatically transported from one topological phase to the other separated by a quantum critical line. On the other hand, the inclusion of a phase factor in the hopping term, which breaks the effective time-reversal invariance, results in an extended gapless region between two topological phases. We show that for a finite chain with an open boundary condition, there exists a nonzero probability that a Majorana edge state can be adiabatically transported from one topological phase to the other across this gapless region following a slow quench of the superconducting term; this happens for an optimum transit time, which is proportional to the system size and diverges for a thermodynamically large chain. We attribute this phenomenon to the mixing of the Majorana only with low-lying inverted bulk states. Remarkably, the Majorana state always persists with the same probability even after the quenching is stopped. For a periodic chain, on the other hand, we find a Kibble-Zurek scaling of the defect density with a renormalized rate of quenching.

Shear modulus of frictionless granular materials near the jamming transition under oscillatory shear is numerically investigated. It is found that the shear modulus $G$ satisfies a scaling law to interpolate between $G\sim {(\varphi -{\varphi }_J)}^{1/2}$ and $G\sim {\gamma }_0^{-1/2}(\varphi -{\varphi }_J)$ for a linear spring model of the elastic interaction between contacting grains, where $\varphi$, ${\varphi }_J$, and ${\gamma }_0$ are, respectively, the volume fraction of grains, the fraction at the jamming point, and the amplitude of the oscillatory shear. The linear relation between the shear modulus and $\varphi -{\varphi }_J$ can be understood by slip avalanches.

The full triaxial stress state within individual particles in a monodisperse spherical granular assembly has been measured. This was made possible by neutron imaging and computed tomography combined with neutron diffraction strain measurement techniques and associated stress reconstruction. The assembly in question consists of 549 precision steel ball bearings under an applied axial load of 85 MPa in a cylindrical die. Clear evidence of force chains was observed in terms of both the shape of the probability distribution function for normal stresses and the network formed by highly loaded particles. An extensive analysis of the source and magnitude of uncertainty in these measurements is also presented.

There is a longstanding belief that in social networks with simultaneous friendly and hostile interactions (signed networks) there is a general tendency to a global balance. Balance represents a state of the network with a lack of contentious situations. Here we introduce a method to quantify the degree of balance of any signed (social) network. It accounts for the contribution of all signed cycles in the network and gives, in agreement with empirical evidence, more weight to the shorter cycles than to the longer ones. We found that, contrary to what is generally believed, many signed social networks, in particular very large directed online social networks, are in general very poorly balanced. We also show that unbalanced states can be changed by tuning the weights of the social interactions among the agents in the network.

We present a family of exact solutions describing discrete solitary waves in a nonintegrable Fermi-Pasta-Ulam chain. The family is sufficiently rich to cover the whole spectrum of known behaviors from delocalized quasicontinuum waves moving with near-sonic velocities to highly localized anticontinuum excitations with only one particle moving at a time.

Molecular dynamics simulation barostat schemes are derived for achieving a given normal pressure for a thin liquid or solid layer confined between two parallel walls. This work builds on the boundary-controlled barostat scheme of Lupkowski and van Swol [J. Chem. Phys. 93, 737 (1990)]. Two classes of barostat are explored, one in which the external load is applied to a virtual regular lattice to which the wall atoms are bound using a tethering potential. The other type of barostat applies the external force directly to the wall atoms, which are not tethered. The extent to which the wall separation distribution is Gaussian is shown to be an effective measure of the quality of the barostat. The first class of barostat can suffer from anomalous dynamical signatures, even resonances, which are sensitive to the effective mass of the virtual lattice, whose value lacks any rigorous definition. The second type of barostat performs much better under equilibrium and wall-sliding nonequilibrium conditions and in not being so prone to resonance instabilities in the wall separation and does not require so many largely arbitrary parameters. The results of exploratory simulations which characterize the dynamical response of the model systems for both dry and wet or lubricated systems using the different barostats are presented. The barostats which have an inherent damping mechanism, such as the ones analogous to a damped harmonic oscillator, reduce the occurrence of large fluctuations and resonances in the separation between the two walls, and they also achieve a new target pressure more quickly. Near a nonequilibrium phase boundary the attributes of the barostat can have a marked influence on the observed behavior.

We consider the rates of relaxation of a particle in a harmonic well, subject to Lévy noise characterized by its Lévy index $\mu$. Using the propagator for this Lévy–Ornstein-Uhlenbeck process (LOUP), we show that the eigenvalue spectrum of the associated Fokker-Planck operator has the form $(n+m\mu )\nu$ where $\nu$ is the force constant characterizing the well, and $n,m\in \mathbb{N}$. If $\mu$ is irrational, the eigenvalues are all nondegenerate, but rational $\mu$ can lead to degeneracy. The maximum degeneracy is shown to be 2. The left eigenfunctions of the fractional Fokker-Planck operator are very simple while the right eigenfunctions may be obtained from the lowest eigenfunction by a combination of two different step-up operators. Further, we find that the acceptable eigenfunctions should have the asymptotic behavior ${\left|x\right|}^{-n_1-n_2\mu }$ as $\left|x\right|\to \infty$, with $n_1$ and $n_2$ being positive integers, though this condition alone is not enough to identify them uniquely. We also assert that the rates of relaxation of LOUP are determined by the eigenvalues of the associated fractional Fokker-Planck operator and do not depend on the initial state if the moments of the initial distribution are all finite. If the initial distribution has fat tails, for which the higher moments diverge, one can have nonspectral relaxation, as pointed out by $et$ $al$. [Phys. Rev. Lett. 110, 150602 (2013)].

We study the imaginary-time relaxation critical dynamics of a quantum system with a vanishing initial correlation length and an arbitrary initial order parameter $M_0$. We find that in quantum critical dynamics, the behavior of $M_0$ under scale transformations deviates from a simple power law, which was proposed for very small $M_0$ previously. A universal characteristic function is then suggested to describe the rescaled initial magnetization, similar to classical critical dynamics. This characteristic function is shown to be able to describe the quantum critical dynamics in both short- and long-time stages of the evolution. The one-dimensional transverse-field Ising model is employed to numerically determine the specific form of the characteristic function. We demonstrate that it is applicable as long as the system is in the vicinity of the quantum critical point. The universality of the characteristic function is confirmed by numerical simulations of models belonging to the same universality class.

Polypeptide fibrillar transitions are studied using a simplified lattice model, modified from the three-state Potts model, where uniform residues as spins, placed on a cubic lattice, can interact with neighbors to form coil, helical, sheet, or fibrillar structure. Using the transfer matrix method and numerical calculations, we analyzed the partition function and construct phase diagrams. The model manifests phase transitions among coil, helix, sheet, and fibril through parameterizing bond coupling energy ${\varepsilon }_h,{\varepsilon }_s,{\varepsilon }_f$, structural entropies $s_h,s_s,s_f$ of helical, sheet, and fibrillar states, and number density $\rho$. The phase diagrams show the transition sequence is basically governed by ${\varepsilon }_h$, ${\varepsilon }_s$, and ${\varepsilon }_f$, while the transition temperature is determined by the competition among ${\varepsilon }_h$, ${\varepsilon }_s$, and ${\varepsilon }_f$, as well as $s_h$, $s_s$, $s_f$, and $\rho$. Furthermore, the fibrillation is accompanied with an abrupt phase transition from coil, helix, or sheet to fibril even for short polypeptide length, resembling the feature of nucleation-growth process. The finite-size effect in specific heat at transitions for the nonfibrillation case can be described by the scaling form of lattice model. With rich phase-transition properties, our model provides a useful reference for protein aggregation experiments and modeling.